(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 255454, 4501] NotebookOptionsPosition[ 238598, 4200] NotebookOutlinePosition[ 255538, 4503] CellTagsIndexPosition[ 255495, 4500] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 0, $CellContext`add$$ = True, $CellContext`b$$ = 2 Pi, $CellContext`ColorCode$$ = False, $CellContext`DomainA$$ = 1, $CellContext`DomainB$$ = 1, $CellContext`fcn$$ = $CellContext`f1, $CellContext`font$$ = 0, $CellContext`GRH$$ = {$CellContext`CURVE}, $CellContext`opacity$$ = 0.5, $CellContext`OriginZoom$$ = False, $CellContext`PlotPoints3D$$ = 80, $CellContext`range$$ = 1, $CellContext`Recursion$$ = Automatic, $CellContext`rescale$$ = 1, $CellContext`scaleDomA$$ = 1, $CellContext`scaleDomB$$ = 1, $CellContext`scaleRg$$ = 1, $CellContext`scaleVct$$ = 1, $CellContext`text$$ = False, $CellContext`TEXT$$ = 1, $CellContext`twoD$$ = False, $CellContext`u$$ = 1, $CellContext`UnitSphere$$ = False, $CellContext`VEC$$ = {$CellContext`RVect, $CellContext`VVect, \ $CellContext`AVect, $CellContext`AtVect, $CellContext`AnVect, \ $CellContext`ALPVVect, $CellContext`ALPAVect}, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[ " \!\(\*\nStyleBox[\"Range\",\nFontSize->14,\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0, 0]]\)"], Manipulate`Dump`ThisIsNotAControl}, { Hold[" Scale"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`scaleRg$$], 1, ""}, { Rational[1, 90], Rational[1, 80], Rational[1, 70], Rational[1, 60], Rational[1, 50], Rational[1, 40], Rational[1, 30], Rational[1, 20], Rational[1, 10], 1 -> "\!\(\*\nStyleBox[\"1\",\nFontColor->RGBColor[0, 0, 1]]\)", 10, 20, 30, 40, 50, 60, 70, 80, 90}}, {{ Hold[$CellContext`range$$], 1, ""}, 1, 10}, { Hold[ " \!\(\*\nStyleBox[\"Domain\",\nFontSize->14,\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0, 0]]\)"], Manipulate`Dump`ThisIsNotAControl}, { Hold[" Scale"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`scaleDomB$$], 1, ""}, { Rational[1, 90], Rational[1, 80], Rational[1, 70], Rational[1, 60], Rational[1, 50], Rational[1, 40], Rational[1, 30], Rational[1, 20], Rational[1, 10], 1 -> "\!\(\*\nStyleBox[\"1\",\nFontColor->RGBColor[0, 0, 1]]\)", 10, 20, 30, 40, 50, 60, 70, 80, 90}}, { Hold[" \!\(\*\nStyleBox[\"b\",\nFontColor->RGBColor[0, 0, 1]]\)"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`DomainB$$], 1, ""}, 1, 10}, { Hold[" Scale"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`scaleDomA$$], 1, ""}, { Rational[1, 90], Rational[1, 80], Rational[1, 70], Rational[1, 60], Rational[1, 50], Rational[1, 40], Rational[1, 30], Rational[1, 20], Rational[1, 10], 1 -> "\!\(\*\nStyleBox[\"1\",\nFontColor->RGBColor[0, 0, 1]]\)", 10, 20, 30, 40, 50, 60, 70, 80, 90}}, { Hold[" \!\(\*\nStyleBox[\"a\",\nFontColor->RGBColor[0, 0, 1]]\)"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`DomainA$$], 1, ""}, 10, 1}, { Hold[ " \!\(\*\nStyleBox[\"Vectors\",\nFontVariations->{\"Underline\"->True},\ \nFontColor->RGBColor[1, 0, 0]]\)"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`rescale$$], 1, ""}, 0.5, 2}, { Hold[" Scale"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`scaleVct$$], 1, ""}, { Rational[1, 90], Rational[1, 80], Rational[1, 70], Rational[1, 60], Rational[1, 50], Rational[1, 40], Rational[1, 30], Rational[1, 20], Rational[1, 10], 1 -> "\!\(\*\nStyleBox[\"1\",\nFontColor->RGBColor[0, 0, 1]]\)", 10, 20, 30, 40, 50, 60, 70, 80, 90}}, { Hold[" Reset"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`rescale$$], 1, ""}, {1}}, {{ Hold[$CellContext`fcn$$], $CellContext`f1, Style[ Row[{ Spacer[50], "r(\!\(\*\nStyleBox[\"t\",\nFontColor->RGBColor[0, 0, 1]]\)) = "}], 20, Bold]}, {$CellContext`f1 -> Row[{ Style[ TraditionalForm[{2 Sin[$CellContext`t], Cos[$CellContext`t], 0}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=2\[Pi], Ellipse", 15]}], $CellContext`f2 -> Row[{ Style[ TraditionalForm[{ Cos[2^Rational[-1, 2] $CellContext`t], Sin[2^Rational[-1, 2] $CellContext`t], 2^Rational[-1, 2] $CellContext`t}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=4\!\(\*SqrtBox[\(2\)]\)\[Pi], \ Circilar Helix \!\(\*\nStyleBox[\"\[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]\",\nFontSize->14]\) ", 15]}], $CellContext`f3 -> Row[{ Style[ TraditionalForm[{$CellContext`t, $CellContext`t^2, 0}], 15], Style[ ", a=-2 \[LessEqual] t \[LessEqual] b=2, Parabola", 15]}], $CellContext`f4 -> Row[{ Style[ TraditionalForm[{ Sin[ Sin[$CellContext`t]], Cos[ Sin[$CellContext`t]], Sin[$CellContext`t]}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=2\[Pi], Trigonometric Curve \ \!\(\*\nStyleBox[\"\[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP]\",\nFontSize->14]\) ", 15]}], $CellContext`f5 -> Row[{ Style[ TraditionalForm[{(Rational[1, 3] 3^Rational[-1, 2]) (3 - 2 $CellContext`t)^Rational[3, 2], ( Rational[2, 3] Rational[2, 3]^Rational[1, 2]) $CellContext`t^ Rational[3, 2], 0}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=\!\(\*FractionBox[\(3\), \ \(2\)]\), One Link of Astroid \!\(\*\nStyleBox[\"\[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]\",\nFontSize->14]\) ", 15]}], $CellContext`f6 -> Row[{ Style[ TraditionalForm[{Cos[$CellContext`t]^3, Sin[$CellContext`t]^3, 0}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=2\[Pi], Astroid ", 15]}], $CellContext`f7 -> Row[{ Style[ TraditionalForm[{ Sin[$CellContext`t^Rational[1, 2]], Cos[$CellContext`t^ Rational[1, 2]], ( Rational[1, 2] (4 - $CellContext`t^(-1))^ Rational[1, 2]) $CellContext`t + Rational[-1, 8] Log[-1 + (8 + 4 (4 - $CellContext`t^(-1))^ Rational[1, 2]) $CellContext`t]}], 15], Style[ ", a=1 \[LessEqual] t \[LessEqual] b=50, Root Helix \!\(\*\n\ StyleBox[\"\[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP]\",\nFontSize->14]\) ", 15]}], $CellContext`f8 -> Row[{ Style[ TraditionalForm[{$CellContext`t, Rational[1, 2]/$CellContext`t, 0}], 15], Style[ ", a=-3 \[LessEqual] t \[LessEqual] b=3, Hyperbola ", 15]}], $CellContext`f9 -> Row[{ Style[ TraditionalForm[{ Sin[$CellContext`t], Cos[2 $CellContext`t], Cos[$CellContext`t]}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=2\[Pi], Trigonometric Saddle", 15]}], $CellContext`f10 -> Row[{ Style[ TraditionalForm[{ 2 ArcCos[1 + Rational[-1, 4] $CellContext`t] - Sin[ 2 ArcCos[1 + Rational[-1, 4] $CellContext`t]], ( Rational[-1, 8] (-8 + $CellContext`t)) $CellContext`t, 0}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=8, One Link of Cycloid \!\(\*\ \nStyleBox[\"\[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP]\",\nFontSize->14]\) ", 15]}], $CellContext`f11 -> Row[{ Style[ TraditionalForm[{$CellContext`t - Sin[$CellContext`t], 1 - Cos[$CellContext`t], 0}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=8\[Pi], Cycloid ", 15]}], $CellContext`f12 -> Row[{ Style[ TraditionalForm[{ Sin[$CellContext`t^2], Cos[$CellContext`t^2], $CellContext`t^2}], 15], Style[ ", a=-2 \[LessEqual] t \[LessEqual] b=2, Double Traced Helix", 15]}]}}, {{ Hold[$CellContext`VEC$$], {$CellContext`ShowVect}, Row[{ Spacer[60], Style[ "\!\(\*\nStyleBox[\"VECTORS\",\n\ FontVariations->{\"Underline\"->True}]\)", 12, Bold], Spacer[19]}]}, {$CellContext`RVect -> Style["\!\(\*\nStyleBox[OverscriptBox[\"r\", \"\[Rule]\"],\n\ FontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`VVect -> Style["\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\"v\", \" \"}], \"\ \[Rule]\"],\nFontSize->14,\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`AVect -> Style["\!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[1, 0, 0]]\) ", { GrayLevel[0], 10, Bold}], $CellContext`AtVect -> Style["\!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"t\"], \"\ \[Rule]\"],\nFontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\n\ StyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`AnVect -> Style["\!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"n\"], \"\ \[Rule]\"],\nFontSize->14,\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\n\ StyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ALPVVect -> Style["ALP \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ALPAVect -> Style["ALP \!\(\*\nStyleBox[OverscriptBox[\"a\", \n RowBox[{\" \", \"\ \[Rule]\"}]],\nFontSize->14,\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ShowVect -> Style["\!\(\*\nStyleBox[\"All\",\nFontColor->GrayLevel[0]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)", { GrayLevel[0], 10, Bold}], $CellContext`RemoveVect -> Style["\!\(\*\nStyleBox[\"None\",\nFontColor->GrayLevel[0]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)", { GrayLevel[0], 10, Bold}], $CellContext`TRANSLATEVect -> Style["Translate Vectors", { GrayLevel[0], 10, Bold}]}}, {{ Hold[$CellContext`GRH$$], {$CellContext`CURVE}, Row[{ Spacer[60], Style[ "\!\(\*\nStyleBox[\"GRAPHS\",\n\ FontVariations->{\"Underline\"->True}]\)", 12, Bold], Spacer[19]}]}, {$CellContext`CURVE -> Style["\!\(\*\nStyleBox[OverscriptBox[\"r\", \"\[Rule]\"],\n\ FontSize->14]\)\!\(\*\nStyleBox[\"(\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\"t\",\nFontSize->14]\)\!\(\*\nStyleBox[\")\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`VT -> Style["\!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\"(\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\"t\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\")\",\nFontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`AT -> Style["\!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\"(\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\"t\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\")\",\nFontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`AtT -> Style["\!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"t\"], \"\ \[Rule]\"],\nFontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\n\ StyleBox[\"(\",\nFontSize->14]\)\!\(\*\nStyleBox[\"t\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\")\",\nFontSize->14]\)\!\(\*\nStyleBox[\" \ \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`AnT -> Style["\!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"n\"], \"\ \[Rule]\"],\nFontSize->14,\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\n\ StyleBox[\"(\",\nFontSize->14]\)\!\(\*\nStyleBox[\"t\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\")\",\nFontSize->14]\)\!\(\*\nStyleBox[\" \ \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ALPV -> Style["ALP \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\"(\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\"t\",\nFontSize->14]\)\!\(\*\nStyleBox[\")\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ALPA -> Style["ALP \!\(\*\nStyleBox[OverscriptBox[\"a\", \n RowBox[{\" \", \"\ \[Rule]\"}]],\nFontSize->14,\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\"(\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\"t\",\nFontSize->14]\)\!\(\*\nStyleBox[\")\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ShowGr -> Style["\!\(\*\nStyleBox[\"All\",\nFontColor->GrayLevel[0]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)", { GrayLevel[0], 10, Bold}], $CellContext`RemoveGRphs -> Style["\!\(\*\nStyleBox[\"None\",\nFontColor->GrayLevel[0]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)", { GrayLevel[0], 10, Bold}], $CellContext`TRANSLATEGraphs -> Style["Translate Graphs \!\(\*\nStyleBox[\"MOTION\",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"ALONG\",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"THE\",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"CURVE\",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\) ", { GrayLevel[0], 10, Bold}]}}, {{ Hold[$CellContext`TEXT$$], 1, Row[{ Spacer[60], Style[ "\!\(\*\nStyleBox[\"CONTENT\",\n\ FontVariations->{\"Underline\"->True}]\)", 12, Bold], Spacer[19]}]}, { 1 -> Invisible["1234567"], 2 -> Invisible["123456"], 3 -> Invisible["1234567"], 4 -> Invisible["1234567"], 5 -> Invisible["12345678"], 6 -> Invisible["123456789101"], 7 -> Invisible["123456789101"], 0 -> Row[{"\!\(\*\nStyleBox[\"HELP\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0.5, 0]]\)", Invisible["123456789101213141516171819202122232425262728"]}]}}, { Hold[ " \!\(\*\nStyleBox[\"IMAGE\",\nFontSize->16,\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0, 0]]\)"], Manipulate`Dump`ThisIsNotAControl}, { Hold[" Quality"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`PlotPoints3D$$], 80, ""}, { 1, 2, 3, 4, 5, 10, 20, 30, 40, 50, 60, 70, 80 -> "\!\(\*\nStyleBox[\"80\",\nFontColor->RGBColor[0, 0, 1]]\)", 90, 100, 110, 120, 130, 140, 150}}, { Hold[" Speed"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`Recursion$$], Automatic, ""}, { 2 -> "Fast", Automatic -> "\!\(\*\nStyleBox[\"Auto\",\nFontColor->RGBColor[0, 0, 1]]\)", 15 -> "Slow"}}, { Hold[""], Manipulate`Dump`ThisIsNotAControl}, { Hold[" Color"], Manipulate`Dump`ThisIsNotAControl}, { Hold[" Coded"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`ColorCode$$], False, ""}, {True, False}}, { Hold[""], Manipulate`Dump`ThisIsNotAControl}, { Hold[" 2D View"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`twoD$$], False, ""}, {True, False}}, { Hold[""], Manipulate`Dump`ThisIsNotAControl}, { Hold[ " \!\(\*\nStyleBox[\"Origin\",\nFontSize->14,\n\ FontColor->GrayLevel[0]]\)"], Manipulate`Dump`ThisIsNotAControl}, { Hold[ " \!\(\*\nStyleBox[\"View\",\nFontSize->14,\n\ FontColor->GrayLevel[0]]\)"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`OriginZoom$$], False, ""}, {True, False}}, { Hold[""], Manipulate`Dump`ThisIsNotAControl}, { Hold[" \!\(\*\nStyleBox[\"Unit\",\nFontColor->GrayLevel[0]]\)"], Manipulate`Dump`ThisIsNotAControl}, { Hold[ "\!\(\*\nStyleBox[\" \",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\"Sphere\",\n\ FontColor->GrayLevel[0]]\)"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`UnitSphere$$], False, ""}, {True, False}}, { Hold[" Opacity"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`opacity$$], 0.5, ""}, 0, 1}, { Hold[""], Manipulate`Dump`ThisIsNotAControl}, { Hold[ Style[ "\!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\) = \ \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\ \[Rule]\)]\)n", 13, RGBColor[1, 0, 0], Bold, Background -> RGBColor[0.87, 0.94, 1]]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`add$$], True, ""}, {True, False}}, { Hold[""], Manipulate`Dump`ThisIsNotAControl}, { Hold[""], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`text$$], False, ""}, {True, False}}, { Hold[ " \!\(\*\nStyleBox[\"TEXT\",\nFontSize->16,\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0, 0]]\)"], Manipulate`Dump`ThisIsNotAControl}, { Hold[ Dynamic[ Row[{ Spacer[11], Framed[ Style[$CellContext`font$$, Bold, 15]]}]]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`font$$], 0, ""}, -10, 10, 1}, {{ Hold[$CellContext`u$$], 1, Row[{ "\!\(\*\nStyleBox[\"\[Copyright]\",\nFontSize->14]\)\!\(\*\nStyleBox[\ \" \",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"N\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\".\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Bykov\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\",\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"SJ\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Delta\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"College\",\n\ FontColor->RGBColor[0, 0, 1]]\)", Spacer[300], "\!\(\*\nStyleBox[\"t\",\nFontSize->24,\nFontColor->RGBColor[0, 0, \ 1]]\)"}]}, Dynamic[$CellContext`a$$], Dynamic[$CellContext`b$$], 0.001}, { Hold[$CellContext`a$$], 0, 1}, { Hold[$CellContext`b$$], 0, 1}}, Typeset`size$$ = {980., {293., 298.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`scaleRg$4519$$ = False, $CellContext`range$4520$$ = 0, $CellContext`scaleDomB$4521$$ = False, $CellContext`DomainB$4522$$ = 0, $CellContext`scaleDomA$4523$$ = False, $CellContext`DomainA$4524$$ = 0, $CellContext`rescale$4525$$ = 0, $CellContext`scaleVct$4526$$ = False, $CellContext`fcn$4527$$ = False, $CellContext`VEC$4528$$ = False, $CellContext`opacity$4529$$ = 0, $CellContext`font$4530$$ = 0, $CellContext`u$4531$$ = 0, $CellContext`a$4532$$ = 0, $CellContext`b$4533$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 0, $CellContext`add$$ = True, $CellContext`b$$ = 0, $CellContext`ColorCode$$ = False, $CellContext`DomainA$$ = 1, $CellContext`DomainB$$ = 1, $CellContext`fcn$$ = $CellContext`f1, $CellContext`font$$ = 0, $CellContext`GRH$$ = {$CellContext`CURVE}, $CellContext`opacity$$ = 0.5, $CellContext`OriginZoom$$ = False, $CellContext`PlotPoints3D$$ = 80, $CellContext`range$$ = 1, $CellContext`Recursion$$ = Automatic, $CellContext`rescale$$ = 1, $CellContext`scaleDomA$$ = 1, $CellContext`scaleDomB$$ = 1, $CellContext`scaleRg$$ = 1, $CellContext`scaleVct$$ = 1, $CellContext`text$$ = False, $CellContext`TEXT$$ = 1, $CellContext`twoD$$ = False, $CellContext`u$$ = 1, $CellContext`UnitSphere$$ = False, $CellContext`VEC$$ = {$CellContext`ShowVect}}, "ControllerVariables" :> { Hold[$CellContext`scaleRg$$, $CellContext`scaleRg$4519$$, False], Hold[$CellContext`range$$, $CellContext`range$4520$$, 0], Hold[$CellContext`scaleDomB$$, $CellContext`scaleDomB$4521$$, False], Hold[$CellContext`DomainB$$, $CellContext`DomainB$4522$$, 0], Hold[$CellContext`scaleDomA$$, $CellContext`scaleDomA$4523$$, False], Hold[$CellContext`DomainA$$, $CellContext`DomainA$4524$$, 0], Hold[$CellContext`rescale$$, $CellContext`rescale$4525$$, 0], Hold[$CellContext`scaleVct$$, $CellContext`scaleVct$4526$$, False], Hold[$CellContext`fcn$$, $CellContext`fcn$4527$$, False], Hold[$CellContext`VEC$$, $CellContext`VEC$4528$$, False], Hold[$CellContext`opacity$$, $CellContext`opacity$4529$$, 0], Hold[$CellContext`font$$, $CellContext`font$4530$$, 0], Hold[$CellContext`u$$, $CellContext`u$4531$$, 0], Hold[$CellContext`a$$, $CellContext`a$4532$$, 0], Hold[$CellContext`b$$, $CellContext`b$4533$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> (Off[ MessageName[Power, "infy"]]; Off[ MessageName[Infinity, "indet"]]; Off[ MessageName[ParametricPlot3D, "invmaxrec"]]; Off[ MessageName[Simplify, "fas"]]; Off[ MessageName[Refine, "fas"]]; Framed[ Pane[Off[ MessageName[Det, "inf"]]; Off[ MessageName[Transpose, "nmtx"]]; Off[ MessageName[General, "stop"]]; Switch[$CellContext`fcn$$, $CellContext`f1, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := {2 Sin[$CellContext`t], Cos[$CellContext`t], 0}; $CellContext`a0 = 0; $CellContext`b0 = 2 Pi; $CellContext`Fnumber = 1; $CellContext`ptsize = 0.015; $CellContext`jumps = None; $CellContext`asymptote = Graphics[]; $CellContext`LABEL = Text["ELLIPSE\n\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)and \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graphs coincide with the curve; \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) is the opposite of \ \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\); \!\(\*\nStyleBox[OverscriptBox[\n \ RowBox[{\" \", SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 0]]\)and\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\ \" \", SubscriptBox[\"a\", \"n\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, \ 1, 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 1]]\)graphs turn \ into each other under translation;\n\!\(\*\nStyleBox[\"Things\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"to\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\): Compare \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\),\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\) and \!\(\ \*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 0]]\),\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graphs; Increase the \!\(\*\nStyleBox[\"Domain\",\nFontSlant->\"Italic\"]\)\!\ \(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)to maximum, due to rendering \ errors the \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\) graph will fill in the \ ellipse; \!\(\*\nStyleBox[\"2\",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\ \"Italic\"]\)\!\(\*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\);\n\!\(\*\n\ StyleBox[\"VISTA\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) Get \ \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \ \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\),\!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"n\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 1]]\)and \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)graphs \ and vectors on the screen, deploy \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\) = \ \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\ \[Rule]\)]\)n diagram, go to full screen view, \!\(\*\n\ StyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) graphs and vectors, watch \ in slow motion;"], $CellContext`f2, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := { Cos[$CellContext`t/2^Rational[1, 2]], Sin[$CellContext`t/2^Rational[1, 2]], $CellContext`t/2^ Rational[1, 2]}; $CellContext`a0 = 0; $CellContext`b0 = (4 2^Rational[1, 2]) Pi; $CellContext`Fnumber = 2; $CellContext`ptsize = 0.015; $CellContext`jumps = None; $CellContext`asymptote = Graphics[]; $CellContext`LABEL = Text["CIRCULAR HELIX\n\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]\!\(\*\nStyleBox[\";\",\n\ FontSize->10]\)\!\(\*\nStyleBox[\" \",\nFontSize->10]\)Constant \!\(\*\n\ StyleBox[\"\[Kappa]\",\nFontColor->RGBColor[0.6, 0.4, 0.2]]\), \!\(\*\n\ StyleBox[\"S\",\nFontColor->RGBColor[0, 1, 0]]\) and \!\(\*\nStyleBox[\"a\",\n\ FontColor->RGBColor[1, 0, 0]]\); \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 0]]\) graph is a circle in \ z=\!\(\*FractionBox[\(1\), SqrtBox[\(2\)]]\) plane; \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graph is a circle in z=0 plane; \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \ \", SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)\[Congruent]\!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\"0\",\n\ FontSlant->\"Plain\"], \"\[Rule]\"],\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\" \",\nFontSlant->\"Italic\"]\)and\!\(\*\nStyleBox[OverscriptBox[\n \ RowBox[{\" \", SubscriptBox[\"a\", \"n\"]}], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 1]]\)\[Congruent]\!\(\*\nStyleBox[OverscriptBox[\n \ StyleBox[\"a\",\nFontColor->RGBColor[1, 0, 0]], \"\[Rule]\"],\n\ FontColor->RGBColor[1, 0, 0]]\) (direct consequences of \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP]);\n\!\(\ \*\nStyleBox[\"Things\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"to\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \ \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\): Leave only \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]]\) \ and \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[1, 0, 0]]\) vectors and graphs on the screen, use \!\(\*\n\ StyleBox[\"Origin\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\) \ and \!\(\*\nStyleBox[\"Unit\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \ \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"Sphere\",\n\ FontSlant->\"Italic\"]\);\n\!\(\*\nStyleBox[\"VISTA\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\" \",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\"POINT\",\nFontVariations->{\ \"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\ \nFontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) \ Translated\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", \"v\"}], \n \ RowBox[{\" \", \"\[Rule]\"}]],\nFontColor->RGBColor[0, 1, 0]]\), \!\(\*\n\ StyleBox[OverscriptBox[SubscriptBox[\"a\", \"t\"], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\), \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)form 3 \ helixes. \!\(\*\nStyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) graphs and \ vectors, watch in slow motion;"], $CellContext`f3, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := {$CellContext`t, $CellContext`t^2, 0}; $CellContext`a0 = -2; $CellContext`b0 = 2; $CellContext`Fnumber = 3; $CellContext`ptsize = 0.03; $CellContext`jumps = None; $CellContext`asymptote = { Graphics[{Dashed, Line[{{-3, -2}, {3, -2}}]}], Graphics[{Dashed, Line[{{-3, 2}, {3, 2}}]}]}; $CellContext`LABEL = Text["PARABOLA\n\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): \!\(\*\nStyleBox[\"\[Kappa]\",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\) approaches 0 at \[PlusMinus]\!\(\*\n\ StyleBox[\"\[Infinity]\",\nFontSize->16]\), i.e. the curve flats out; tip of \ \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, \ 1, 0]]\) is on the x=1 line; Constant \!\(\*\nStyleBox[OverscriptBox[\"a\", \ \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\);\n\!\(\*\nStyleBox[\"Things\",\ \nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"to\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\): Increase the \!\(\*\n\ StyleBox[\"Domain\",\nFontSlant->\"Italic\"]\) to watch how \!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"t\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)vector approaches {0,2,0} at \ \[PlusMinus]\!\(\*\nStyleBox[\"\[Infinity]\",\nFontSize->16]\)\!\(\*\n\ StyleBox[\" \",\nFontSize->16]\)and graph fills out a circle; \!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"n\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 1]]\)vector approaches \!\(\*OverscriptBox[\(0\), \ \(\[Rule]\)]\) at \[PlusMinus]\!\(\*\nStyleBox[\"\[Infinity]\",\n\ FontSize->16]\)\!\(\*\nStyleBox[\" \",\nFontSize->16]\)and its graph also \ fills out the same circle; What is the center of this circle? Radius?; Repeat \ for \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \ \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)and\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[OverscriptBox[\"a\", \ \"\[Rule]\"],\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\) graphs; \!\(\*\nStyleBox[\"2\",\nFontSlant->\"Italic\ \"]\)\!\(\*\nStyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\ \nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"View\",\n\ FontSlant->\"Italic\"]\);\n\!\(\*\nStyleBox[\"VISTA\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\" \",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\"POINT\",\nFontVariations->{\ \"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\ \nFontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) \n\ 1. Get \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \ \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\),\!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"n\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 1]]\)and \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)graphs \ and vectors on the screen, deploy\!\(\*OverscriptBox[\(\(\\ \)\(A\)\), \(\ \[Rule]\)]\) = \!\(\*OverscriptBox[\(A\), \ \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)n diagram, go to \ full screen view, \!\(\*\nStyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) \ graphs and vectors, watch in slow motion;\n2. \!\(\*\nStyleBox[\"Translate\",\ \nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\), \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)graphs;"], $CellContext`f4, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := { Sin[ Sin[$CellContext`t]], Cos[ Sin[$CellContext`t]], Cos[$CellContext`t]}; $CellContext`a0 = 0; $CellContext`b0 = 2 Pi; $CellContext`Fnumber = 4; $CellContext`ptsize = 0.015; $CellContext`jumps = None; $CellContext`asymptote = Graphics[]; $CellContext`LABEL = Text["TRIGONOMETRIC CURVE\n\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]\!\(\*\nStyleBox[\";\",\n\ FontSize->10]\)\!\(\*\nStyleBox[\" \",\nFontSize->10]\) \n\!\(\*\n\ StyleBox[\"Things\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"to\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \ \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\): Use \!\(\*\nStyleBox[\"Origin\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\) and \!\(\*\n\ StyleBox[\"Unit\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"Sphere\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)to \ examine \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 0]]\)and\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[OverscriptBox[\"a\", \ \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) graphs;\n\!\(\*\n\ StyleBox[\"VISTA\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) \ \!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)Get \!\(\*\nStyleBox[OverscriptBox[\"a\", \ \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0, 0]]\)graph and vector on the screen, \!\(\*\n\ StyleBox[\"Translate\",\nFontSlant->\"Italic\"]\), watch in slow motion how \ \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, \ 0, 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)reaches an \ endpoint and continuously turns back ;"], $CellContext`f5, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := {(3 - 2 $CellContext`t)^(3/2)/(3 3^Rational[1, 2]), ((2/3) (2/3)^ Rational[1, 2]) $CellContext`t^(3/2), 0}; $CellContext`a0 = 0; $CellContext`b0 = 3/2; $CellContext`Fnumber = 5; $CellContext`ptsize = 0.05; $CellContext`jumps = {0, 3/2}; $CellContext`asymptote = { Graphics[{Dashed, Line[{{0, 0}, {0, 2}}]}], Graphics[{Dashed, Line[{{1.5, 0}, {1.5, 2}}]}]}; $CellContext`LABEL = Text["ASTROID \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP], ONE LINK\n\!\(\*\nStyleBox[\"Curve\",\n\ FontVariations->{\"Underline\"->True}]\): A point on the circumference of a \ circle of radius \!\(\*FractionBox[\(3\), \(4\)]\) rolling around the inside \ of a circle of radius 1 forms the given astroid.\n\!\(\*\n\ StyleBox[\"Notice\",\nFontVariations->{\"Underline\"->True}]\): At endpoints \ \!\(\*\nStyleBox[\"\[Kappa]\",\nFontColor->RGBColor[0.6, 0.4, 0.2]]\), \!\(\*\ \nStyleBox[\"a\",\nFontColor->RGBColor[1, 0, 0]]\), and \!\(\*\n\ StyleBox[SubscriptBox[\"a\", \"n\"],\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)are infinite;\n\!\(\*\n\ StyleBox[\"Things\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"to\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \ \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\): Use \!\(\*\nStyleBox[\"Origin\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\) and \!\(\*\n\ StyleBox[\"Unit\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"Sphere\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)to \ examine \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 0]]\)and\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[OverscriptBox[\"a\", \ \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) graphs; \!\(\*\n\ StyleBox[\"2\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"D\",\nFontSlant->\ \"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\"View\",\nFontSlant->\"Italic\"]\);\n\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\ \*\nStyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"not\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"try\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0.5, 0]]\): \ Domain increase, since [0,\!\(\*FractionBox[\(3\), \(2\)]\)] is the natural \ domain of this parameterization of the astroid (revisit the formulas to \ confirm). The next graph will show astroid parameterization with unrestricted \ domain, but we will lose the \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] property, can not have them both;\n\!\(\*\n\ StyleBox[\"VISTA\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) Get \ \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, \ 1, 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 1]]\)and \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)graphs and vectors on the screen, go to full screen \ view, \!\(\*\nStyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) graphs and \ vectors, watch in slow motion;"], $CellContext`f6, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := { Cos[$CellContext`t]^3, Sin[$CellContext`t]^3, 0}; $CellContext`a0 = 0; $CellContext`b0 = 2 Pi; $CellContext`Fnumber = 6; $CellContext`ptsize = 0.03; $CellContext`jumps = { 0, Pi/2, Pi, 3 (Pi/2), 2 Pi}; $CellContext`asymptote = { Graphics[{Dashed, Line[{{Pi/2, -3}, {Pi/2, 5}}]}], Graphics[{Dashed, Line[{{Pi, -3}, {Pi, 5}}]}], Graphics[{Dashed, Line[{{3 (Pi/2), -3}, {3 (Pi/2), 5}}]}], Graphics[{Dashed, Line[{{2 Pi, -3}, {2 Pi, 5}}]}]}; $CellContext`LABEL = Text["ASTROID\n\!\(\*\nStyleBox[\"Curve\",\n\ FontVariations->{\"Underline\"->True}]\): A point on the circumference of a \ circle of radius 3 rolling around the inside of a circle of radius 4 forms \ the given astroid.\n\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): This is the complete astroid, we \ saw one \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \ link of it in the previous example; At four cusps \!\(\*\nStyleBox[\"\[Kappa]\ \",\nFontColor->RGBColor[0.6, 0.4, 0.2]]\) is infinite, \!\(\*\n\ StyleBox[\"S\",\nFontColor->RGBColor[0, 1, 0]]\) is not smooth (also has \ cusps) and \!\(\*\nStyleBox[SubscriptBox[\"a\", \"t\"],\n\ FontColor->RGBColor[0, 0, 1]]\) is not even continuous (has jump \ discontinuities); \!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\"a\",\n\ FontColor->RGBColor[1, 0, 0]], \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ is rather surprisingly smooth at all cusps; points including cusps and \!\(\*\ \nStyleBox[SubscriptBox[\"a\", \"n\"],\nFontColor->RGBColor[0, 1, 1]]\) is \ constant;\n\!\(\*\nStyleBox[\"Things\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"to\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\): Match the petals of \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]]\),\ \!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[SubscriptBox[\"a\", \"t\"], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\),\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \ \", SubscriptBox[\"a\", \"n\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 1]]\)and\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graphs with links of the astroid, \!\(\*\nStyleBox[\"Translate\",\n\ FontSlant->\"Italic\"]\); \!\(\*\nStyleBox[\"2\",\nFontSlant->\"Italic\"]\)\!\ \(\*\nStyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\);\ \n\!\(\*\nStyleBox[\"JUMPS\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0.5, 0]]\) \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)graph is discontinuous at \ cusps, velocity vector makes a sudden U-turn, when crossing a cusp and jumps \ across the graph, remember, it is always a unit vector and all changes occur \ in direction only. These jumps are indicated with straight segments across \ the circle; Get \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\" \",\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\)graph and \ vector on the screen, click \!\(\*\nStyleBox[\"Play\",\n\ FontSlant->\"Italic\"]\) on \!\(\*\nStyleBox[\"t\",\nFontColor->RGBColor[0, \ 0, 1]]\) slider and set the animation to very slow, you will see \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\)jump; Repeat for \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\);\n\!\(\*\ \nStyleBox[\"VISTA\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) \ \!\(\*\nStyleBox[\"PROPELLER\",\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\). Get \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\" \",\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\)and \ \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)graphs and vectors on the screen, go to full screen \ view, \!\(\*\nStyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) graphs and \ vectors, watch in slow motion;"], $CellContext`f7, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := { Sin[$CellContext`t^Rational[1, 2]], Cos[$CellContext`t^ Rational[ 1, 2]], ((1/2) (4 - 1/$CellContext`t)^ Rational[1, 2]) $CellContext`t - (1/8) Log[-1 + (8 + 4 (4 - 1/$CellContext`t)^ Rational[1, 2]) $CellContext`t]}; $CellContext`a0 = 1; $CellContext`b0 = 50; $CellContext`Fnumber = 7; $CellContext`ptsize = 0.01; $CellContext`jumps = None; $CellContext`asymptote = Graphics[]; $CellContext`LABEL = Text["ROOT HELIX \n\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]; It is a very slowly unwinding \ helix; The idea of the example is for you to compare \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)and\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)graphs of this \ helix and the standard one in one of the previous examples;\n\!\(\*\n\ StyleBox[\"Things\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"to\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \ \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\): Leave only \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)and\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ vectors and graphs on the screen, move \!\(\*\nStyleBox[\"t\",\n\ FontColor->RGBColor[0, 0, 1]]\) slider all the way to the right, use \!\(\*\n\ StyleBox[\"Origin\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\) \ and \!\(\*\nStyleBox[\"Unit\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \ \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"Sphere\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\",\",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"i\",\n\ FontSlant->\"Italic\"]\)ncrease the \!\(\*\nStyleBox[\"b\",\n\ FontColor->RGBColor[0, 0, 1]]\) \!\(\*\nStyleBox[\"Domain\",\n\ FontSlant->\"Italic\"]\) to watch how the spirals winds around the z-axis;\n\ \!\(\*\nStyleBox[\"Do\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"not\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\ \*\nStyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"try\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0.5, 0]]\): Decreasing \!\(\*\nStyleBox[\"a\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\",\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)since (-\[Infinity],\!\(\*FractionBox[\(1\), \ \(4\)]\)] is the natural domain of this parameterization;\n\!\(\*\n\ StyleBox[\"VISTA\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) \ \!\(\*\nStyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]]\),\ \!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[SubscriptBox[\"a\", \"t\"], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\),\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \ \", SubscriptBox[\"a\", \"n\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 1]]\)and\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graphs;"], $CellContext`f8, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := {$CellContext`t, 1/(2 $CellContext`t), 0}; $CellContext`a0 = -3; $CellContext`b0 = 3; $CellContext`Fnumber = 8; $CellContext`ptsize = 0.02; $CellContext`jumps = {0}; $CellContext`asymptote = Graphics[]; $CellContext`LABEL = Text["HYPERBOLA\n\!\(\*\nStyleBox[\"Things\",\nFontVariations->{\ \"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"to\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): t=0 is not in the domain of any of \ the graphs, but assuming \!\(\*\nStyleBox[\"\[Kappa]\",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\)(0)=0 makes \!\(\*\n\ StyleBox[\"\[Kappa]\",\nFontColor->RGBColor[0.6, 0.4, 0.2]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0.6, 0.4, 0.2]]\)graph smooth and \ setting \!\(\*\nStyleBox[SubscriptBox[\"a\", \"n\"],\nFontColor->RGBColor[0, \ 1, 1]]\)(0)=0 makes \!\(\*\nStyleBox[SubscriptBox[\"a\", \"n\"],\n\ FontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 1]]\)graph only continuous (look at the formulas to \ see why there is a difference); \!\(\*\nStyleBox[\"\[Kappa]\",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\) approaches 0 at \[PlusMinus]\!\(\*\n\ StyleBox[\"\[Infinity]\",\nFontSize->16]\), i.e. the curve flats out; Tip of \ \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, \ 1, 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)is always on \ x=1 line, no matter on what hyperbola branch the particle is;\n\!\(\*\n\ StyleBox[\"Things\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"to\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \ \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\): Examine the behavior of \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\), we can \ conveniently assign {0,-1} as its position corresponding t=0, when you move \ right or left the vector approaches {1,0}; Track the movement of \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\) along \ the 8-shape; \!\(\*\nStyleBox[\"2\",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\ \"Italic\"]\)\!\(\*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\);\n\!\(\*\n\ StyleBox[\"VISTA\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) \n1. \ Get \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"t\ \"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\),\!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"n\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 1]]\)and \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)graphs \ and vectors on the screen, deploy \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\) = \ \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\ \[Rule]\)]\)n diagram, go to full screen view, \!\(\*\n\ StyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) graphs and vectors, watch \ in slow motion;\n2. \!\(\*\nStyleBox[\"Translate\",\nFontSlant->\"Italic\"]\)\ \!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\),\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\"a\", \" \"}], \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)graphs;"], $CellContext`f9, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := { Sin[$CellContext`t], Sin[2 $CellContext`t], Cos[$CellContext`t]}; $CellContext`a0 = 0; $CellContext`b0 = 2 Pi; $CellContext`Fnumber = 9; $CellContext`ptsize = 0.02; $CellContext`jumps = None; $CellContext`asymptote = Graphics[]; $CellContext`LABEL = Text["TRIGONOMETRIC SADDLE\n\!\(\*\nStyleBox[\"Things\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"to\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): \!\(\*OverscriptBox[\(r\), \ \(\[Rule]\)]\), \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 0]]\), \!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\ \"a\",\nFontColor->RGBColor[1, 0, 0]], \"\[Rule]\"],\nFontColor->RGBColor[1, \ 0, 0]]\) graphs form three saddles, can you find if they intersect? Can you \ explain why \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", \ SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 0, 1]]\)graph has 8 \ petals?\n\!\(\*\nStyleBox[\"Things\",\nFontVariations->{\"Underline\"->True}]\ \)\!\(\*\nStyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"to\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \ \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\): Use \!\(\*\nStyleBox[\"Origin\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\) and \!\(\*\n\ StyleBox[\"Unit\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"Sphere\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)to \ examine \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \ graphs; Examine how \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\) and \!\(\ \*OverscriptBox[\(r\), \(\[Rule]\)]\) graphs are linked;\n\!\(\*\n\ StyleBox[\"VISTA\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) Get \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\), \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)graphs and vectors and \!\(\*\nStyleBox[\"Unit\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\"Sphere\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)on the screen, adjust opacity, go to full screen \ view, watch in slow motion;"], $CellContext`f10, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := { 2 ArcCos[1 - $CellContext`t/4] - Sin[ 2 ArcCos[ 1 - $CellContext`t/4]], ((-(1/ 8)) (-8 + $CellContext`t)) $CellContext`t, 0}; $CellContext`a0 = 0; $CellContext`b0 = 8; $CellContext`Fnumber = 10; $CellContext`ptsize = 0.015; $CellContext`jumps = None; $CellContext`asymptote = { Graphics[{Dashed, Line[{{0, 0}, {0, 2}}]}], Graphics[{Dashed, Line[{{8, 0}, {8, 2}}]}]}; $CellContext`LABEL = Text["CYCLOID \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP], ONE LINK\n\!\(\*\nStyleBox[\"Curve\",\n\ FontVariations->{\"Underline\"->True}]\): A point on the rim of a circle \ rolling along a straight line forms a cycloid; We deal only with one link \ here; See the next graph for a more informative view;\n\!\(\*\n\ StyleBox[\"Notice\",\nFontVariations->{\"Underline\"->True}]\): \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP]; At \ both endpoints \!\(\*\nStyleBox[\"\[Kappa]\",\nFontColor->RGBColor[0.6, 0.4, \ 0.2]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0.6, 0.4, 0.2]]\)and \!\(\ \*\nStyleBox[OverscriptBox[\n StyleBox[\"a\",\nFontColor->RGBColor[1, 0, \ 0]], \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0, 0]]\)are infinite; Tip of \!\(\*\n\ StyleBox[OverscriptBox[\n StyleBox[\n RowBox[{\" \", \"a\"}],\n\ FontColor->RGBColor[1, 0, 0]], \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\ \!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)is always on \ y=-\!\(\*FractionBox[\(1\), \(4\)]\) line;\n\!\(\*\nStyleBox[\"Things\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"to\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\): Get a good portion of \!\(\*\n\ StyleBox[OverscriptBox[\n StyleBox[\"a\",\nFontColor->RGBColor[1, 0, 0]], \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0, 0]]\)graph in the range (it's an infinite line) and \ examine \!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\"a\",\n\ FontColor->RGBColor[1, 0, 0]], \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\ \!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)vector, as you \ approach the endpoints, use + and - buttons rather than the slider to avoid \ hitting infinity; \!\(\*\nStyleBox[\"2\",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\ \"Italic\"]\)\!\(\*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\);\n\!\(\*\n\ StyleBox[\"Do\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\ \*\nStyleBox[\"not\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\ \*\nStyleBox[\"try\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0.5, 0]]\): Domain increase, since [0,8] is the \ natural domain of this parameterization of the cycloid (revisit the formulas \ to confirm). The next graph will have unrestricted domain, but we will lose \ the \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \ property, can not have them both;\n\!\(\*\nStyleBox[\"VISTA\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\" \",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\"POINT\",\nFontVariations->{\ \"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\ \nFontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) Get \ \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, \ 1, 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 1]]\)and \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)graphs and vectors on the screen, go to full screen \ view, \!\(\*\nStyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) graphs and \ vectors, watch in slow motion;"], $CellContext`f11, $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := {$CellContext`t - Sin[$CellContext`t], 1 - Cos[$CellContext`t], 0}; $CellContext`a0 = 0; $CellContext`b0 = 8 Pi; $CellContext`Fnumber = 11; $CellContext`ptsize = 0.015; $CellContext`jumps = { 0, 2 Pi, 4 Pi, 6 Pi, 8 Pi}; $CellContext`asymptote = { Graphics[{Dashed, Line[{{0, -1}, {0, 3}}]}], Graphics[{Dashed, Line[{{2 Pi, -1}, {2 Pi, 3}}]}], Graphics[{Dashed, Line[{{4 Pi, -1}, {4 Pi, 3}}]}], Graphics[{Dashed, Line[{{6 Pi, -1}, {6 Pi, 3}}]}], Graphics[{Dashed, Line[{{8 Pi, -1}, {8 Pi, 3}}]}]}; $CellContext`LABEL = Text["CYCLOID\n\!\(\*\nStyleBox[\"Curve\",\n\ FontVariations->{\"Underline\"->True}]\): A point on the rim of a circle \ rolling along a straight line forms a cycloid; 4 links are displayed;\n\!\(\*\ \nStyleBox[\"Notice\",\nFontVariations->{\"Underline\"->True}]\): At cusps \!\ \(\*\nStyleBox[\"\[Kappa]\",\nFontColor->RGBColor[0.6, 0.4, 0.2]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0.6, 0.4, 0.2]]\)is infinite; \!\(\*\n\ StyleBox[\"S\",\nFontColor->RGBColor[0, 1, 0]]\) is not smooth (also has \ cusps) and \!\(\*\nStyleBox[SubscriptBox[\"a\", \"t\"],\n\ FontColor->RGBColor[0, 0, 1]]\) is not even continuous (has jump \ discontinuities); \!\(\*\nStyleBox[\"a\",\nFontColor->RGBColor[1, 0, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)and \!\(\*\n\ StyleBox[SubscriptBox[\"a\", \"n\"],\nFontColor->RGBColor[0, 1, 1]]\) are \ constant; Tip of \!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)\ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\) is \ always on y=-\!\(\*FractionBox[\(1\), \(4\)]\) line;\n\!\(\*\n\ StyleBox[\"Things\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"to\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \ \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\): Examine peculiar placement of \ \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", \"v\"}], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 0]]\)and \!\(\*\nStyleBox[OverscriptBox[\n \ StyleBox[\"a\",\nFontColor->RGBColor[1, 0, 0]], \"\[Rule]\"],\n\ FontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0, 0]]\)graphs; \!\(\*\nStyleBox[\"2\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"View\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\";\",\nFontSlant->\"Italic\"]\)\n\!\ \(\*\nStyleBox[\"JUMPS\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0.5, 0]]\) \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)graph is discontinuous at \ cusps, velocity vector makes a sudden U-turn, when crossing a cusp and jumps \ across the graph, remember, it is always a unit vector and all changes occur \ in direction only. These jumps are indicated with straight segment across the \ circle; Get \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\" \",\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\)graph and \ vector on the screen, go to \!\(\*\nStyleBox[\"Origin\",\nFontSlant->\"Italic\ \"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"View\ \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\",\",\n\ FontSlant->\"Italic\"]\) click \!\(\*\nStyleBox[\"Play\",\n\ FontSlant->\"Italic\"]\) on \!\(\*\nStyleBox[\"t\",\nFontColor->RGBColor[0, \ 0, 1]]\) slider and set the animation to very slow, you will see \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\)jump;\n\!\(\*\nStyleBox[\"VISTA\",\nFontVariations->{\ \"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\ \nFontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\ \*\nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) Get \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)graph and vector on the screen, go to full screen \ view, \!\(\*\nStyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) graph and \ vector, watch \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)jump in VERY slow motion;"], $CellContext`f12, \ $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := { Sin[$CellContext`t^2], Cos[$CellContext`t^2], $CellContext`t^2}; $CellContext`a0 = -2; \ $CellContext`b0 = 2; $CellContext`Fnumber = 12; $CellContext`ptsize = 0.035; $CellContext`jumps = {0}; $CellContext`asymptote = Graphics[]; $CellContext`LABEL = Text["DOUBLE-TRACED HELIX\n\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): Geometrically it is a helix, but \ parameterization makes it not smooth at t=0. As the name says, it's double \ traced, \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\) goes down to the endpoint \ {0,1,0} as t approaches 0 from negative infinity, after reaching {0,1,0} \!\(\ \*OverscriptBox[\(r\), \(\[Rule]\)]\) goes back up the curve as t increases \ to positive infinity; \!\(\*\nStyleBox[\"\[Kappa]\",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\)is constant (helix after all); \!\(\*\n\ StyleBox[SubscriptBox[\"a\", \"t\"],\nFontColor->RGBColor[0, 0, 1]]\) graph \ has a jump at t=0, because the particle slows down at constant rate when \ going down and immediately starts to speed up at a constant rate on the way \ back up the graph; Due to fast motion, most graphs go far away from the \ curve, adjust the range to get the complete picture;\n\!\(\*\n\ StyleBox[\"Things\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"to\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \ \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\): Watch an interesting effect of \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\) graph \ splitting into 2 components corresponding to upward and downward movements; \ Increase the \!\(\*\nStyleBox[\"Domain\",\nFontSlant->\"Italic\"]\) gently to \ close \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \ \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)and \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\) graphs;\n\ \!\(\*\nStyleBox[\"JUMPS\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0.5, 0]]\) \ Components of \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, \ 0]]\)graph are connected with a segment indicating the jump from upward to \ downward movement when crossing a cusp;\n\!\(\*\nStyleBox[\"VISTA\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\" \",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\"POINT\",\nFontVariations->{\ \"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\ \nFontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) Get \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)graph, vector and \!\(\*\nStyleBox[\"Unit\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\"Sphere\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)on the screen, adjust opacity, go to full screen \!\(\ \*\nStyleBox[\"Origin\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\), \ watch in slow motion;"]]; $CellContext`\[Lambda] = $CellContext`scaleDomA$$ \ $CellContext`DomainA$$; $CellContext`\[Mu] = $CellContext`scaleDomB$$ \ $CellContext`DomainB$$; $CellContext`a$$ = (($CellContext`\[Lambda] + 1) $CellContext`a0 - ($CellContext`\[Lambda] - 1) $CellContext`b0)/ 2; $CellContext`b$$ = (($CellContext`\[Mu] + 1) $CellContext`b0 - ($CellContext`\[Mu] - 1) $CellContext`a0)/ 2; If[$CellContext`u$$ > $CellContext`b$$, $CellContext`u$$ = \ $CellContext`b$$]; If[$CellContext`u$$ < $CellContext`a$$, $CellContext`u$$ = \ $CellContext`a$$]; If[ MemberQ[$CellContext`VEC$$, $CellContext`RemoveVect], \ $CellContext`VEC$$ = {}]; If[ MemberQ[$CellContext`GRH$$, $CellContext`RemoveGRphs], \ $CellContext`GRH$$ = {}]; If[ MemberQ[$CellContext`VEC$$, $CellContext`ShowVect], \ $CellContext`VEC$$ = {$CellContext`RVect, $CellContext`VVect, \ $CellContext`AVect, $CellContext`AtVect, $CellContext`AnVect, \ $CellContext`ALPVVect, $CellContext`ALPAVect}]; If[ MemberQ[$CellContext`GRH$$, $CellContext`ShowGr], \ $CellContext`GRH$$ = {$CellContext`CURVE, $CellContext`VT, $CellContext`AT, \ $CellContext`AtT, $CellContext`AnT, $CellContext`ALPV, $CellContext`ALPA}]; If[ Not[ And[ MemberQ[$CellContext`VEC$$, $CellContext`AVect], MemberQ[$CellContext`VEC$$, $CellContext`AtVect], MemberQ[$CellContext`VEC$$, $CellContext`AnVect]]], \ $CellContext`add$$ = False]; If[ MemberQ[$CellContext`VEC$$, $CellContext`TRANSLATEVect], \ $CellContext`TV = 1, $CellContext`TV = 0]; If[ MemberQ[$CellContext`GRH$$, $CellContext`TRANSLATEGraphs], \ $CellContext`TG = 1, $CellContext`TG = 0]; $CellContext`RESCALE = $CellContext`rescale$$ \ $CellContext`scaleVct$$; If[$CellContext`text$$, $CellContext`IMSIZE = {485, 565}, $CellContext`IMSIZE = {979, 540}]; If[ MemberQ[$CellContext`VEC$$, $CellContext`RVect], $CellContext`P1 = Graphics3D[{{Black, Arrow[{{0, 0, 0}, $CellContext`RESCALE \ $CellContext`f[$CellContext`u$$]}]}}], $CellContext`P1 = Graphics3D[]]; If[ MemberQ[$CellContext`VEC$$, $CellContext`VVect], $CellContext`P2 = Graphics3D[{{Green, Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`V[$CellContext`u$$]}]}}], $CellContext`P2 = Graphics3D[]]; If[ MemberQ[$CellContext`VEC$$, $CellContext`AVect], $CellContext`P3 = Graphics3D[{{Red, Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`A[$CellContext`u$$]}]}}], $CellContext`P3 = Graphics3D[]]; If[ MemberQ[$CellContext`VEC$$, $CellContext`AtVect], $CellContext`P4 = Graphics3D[{{Blue, Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`At[$CellContext`u$$]}]}}], $CellContext`P4 = Graphics3D[]]; If[ MemberQ[$CellContext`VEC$$, $CellContext`AnVect], $CellContext`P5 = Graphics3D[{{Cyan, Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`An[$CellContext`u$$]}]}}], $CellContext`P5 = Graphics3D[]]; If[ MemberQ[$CellContext`VEC$$, $CellContext`ALPVVect], { If[$CellContext`Fnumber == 1, $CellContext`P7 = Graphics3D[{{ Darker[ Darker[Green]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`Tn1[$CellContext`u$$]}]}}]], If[$CellContext`Fnumber == 6, $CellContext`P7 = Graphics3D[{{ Darker[ Darker[Green]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`Tn6[$CellContext`u$$]}]}}]], If[$CellContext`Fnumber == 9, $CellContext`P7 = Graphics3D[{{ Darker[ Darker[Green]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`Tn9[$CellContext`u$$]}]}}]], If[$CellContext`Fnumber == 11, $CellContext`P7 = Graphics3D[{{ Darker[ Darker[Green]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`Tn11[$CellContext`u$$]}]}}]], If[$CellContext`Fnumber == 12, $CellContext`P7 = Graphics3D[{{ Darker[ Darker[Green]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`Tn12[$CellContext`u$$]}]}}]], If[ Or[$CellContext`Fnumber == 2, $CellContext`Fnumber == 4, $CellContext`Fnumber == 5, $CellContext`Fnumber == 7, $CellContext`Fnumber == 10], $CellContext`P7 = Graphics3D[]], If[ Or[$CellContext`Fnumber == 3, $CellContext`Fnumber == 8], $CellContext`P7 = Graphics3D[{{ Darker[ Darker[Green]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`Tn[$CellContext`u$$]}]}}]]}, $CellContext`P7 = Graphics3D[]]; If[ MemberQ[$CellContext`VEC$$, $CellContext`ALPAVect], { If[$CellContext`Fnumber == 1, $CellContext`P8 = Graphics3D[{{ Darker[ Darker[Red]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`AnALP1[$CellContext`u$$]}]}}]], If[$CellContext`Fnumber == 6, $CellContext`P8 = Graphics3D[{{ Darker[ Darker[Red]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`AnALP6[$CellContext`u$$]}]}}]], If[$CellContext`Fnumber == 9, $CellContext`P8 = Graphics3D[{{ Darker[ Darker[Red]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`AnALP9[$CellContext`u$$]}]}}]], If[$CellContext`Fnumber == 11, $CellContext`P8 = Graphics3D[{{ Darker[ Darker[Red]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`AnALP11[$CellContext`u$$]}]}}]], If[$CellContext`Fnumber == 12, $CellContext`P8 = Graphics3D[{{ Darker[ Darker[Red]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`AnALP12[$CellContext`u$$]}]}}]], If[ Or[$CellContext`Fnumber == 2, $CellContext`Fnumber == 4, $CellContext`Fnumber == 5, $CellContext`Fnumber == 7, $CellContext`Fnumber == 10], $CellContext`P8 = Graphics3D[]], If[ Or[$CellContext`Fnumber == 3, $CellContext`Fnumber == 8], $CellContext`P8 = Quiet[ Graphics3D[{{ Darker[ Darker[Red]], Arrow[{$CellContext`TV $CellContext`f[$CellContext`u$$], \ $CellContext`TV $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`AnALP[$CellContext`u$$]}]}}]]]}, $CellContext`P8 = Graphics3D[]]; If[$CellContext`add$$, {$CellContext`P9 = Graphics3D[{{Blue, Line[{Abs[ Sign[ $CellContext`at[$CellContext`u$$]]] ($CellContext`TV \ $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`An[$CellContext`u$$]), Abs[ Sign[ $CellContext`at[$CellContext`u$$]]] ($CellContext`TV \ $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`An[$CellContext`u$$]) + $CellContext`RESCALE \ $CellContext`At[$CellContext`u$$]}]}}], $CellContext`P10 = Graphics3D[{{Cyan, Line[{Sign[ $CellContext`an[$CellContext`u$$]] ($CellContext`TV \ $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`At[$CellContext`u$$]), Sign[ $CellContext`an[$CellContext`u$$]] ($CellContext`TV \ $CellContext`f[$CellContext`u$$] + $CellContext`RESCALE \ $CellContext`At[$CellContext`u$$]) + $CellContext`RESCALE \ $CellContext`An[$CellContext`u$$]}]}}]}, {$CellContext`P9 = Graphics3D[], $CellContext`P10 = Graphics3D[]}]; If[ MemberQ[$CellContext`GRH$$, $CellContext`CURVE], \ $CellContext`Curve = ParametricPlot3D[ Evaluate[ $CellContext`f[$CellContext`s]], {$CellContext`s, \ $CellContext`a$$, $CellContext`b$$}, PlotStyle -> {Black, Thick}, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed", Exclusions -> $CellContext`jumps, ColorFunction -> If[$CellContext`ColorCode$$, Function[$CellContext`s, Hue[0.75, $CellContext`Cv[$CellContext`s], 1]], Automatic], ColorFunctionScaling -> False], $CellContext`Curve = Graphics3D[]]; If[ MemberQ[$CellContext`GRH$$, $CellContext`VT], $CellContext`Vtrace = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`V[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}, PlotStyle -> {Green, Thick}, Exclusions -> $CellContext`jumps, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"], $CellContext`Vtrace = Graphics3D[]]; If[ MemberQ[$CellContext`GRH$$, $CellContext`AT], $CellContext`Atrace = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`A[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}, PlotStyle -> {Red, Thick}, Exclusions -> $CellContext`jumps, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"], $CellContext`Atrace = Graphics3D[]]; If[ MemberQ[$CellContext`GRH$$, $CellContext`AtT], \ $CellContext`ATtrace = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`At[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}, PlotStyle -> {Blue, Thick}, Exclusions -> $CellContext`jumps, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"], $CellContext`ATtrace = Graphics3D[]]; If[ MemberQ[$CellContext`GRH$$, $CellContext`AnT], \ $CellContext`ANtrace = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`An[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}, PlotStyle -> {Cyan, Thick}, Exclusions -> $CellContext`jumps, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"], $CellContext`ANtrace = Graphics3D[]]; If[ MemberQ[$CellContext`GRH$$, $CellContext`ALPV], { If[$CellContext`Fnumber == 1, $CellContext`ALPv = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`Tn1[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}, PlotStyle -> { Darker[ Darker[Green]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]], If[$CellContext`Fnumber == 6, $CellContext`ALPv = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`Tn6[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$ + 0.1}, PlotStyle -> { Darker[ Darker[Green]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]], If[$CellContext`Fnumber == 9, $CellContext`ALPv = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`Tn9[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}, PlotStyle -> { Darker[ Darker[Green]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]], If[$CellContext`Fnumber == 11, $CellContext`ALPv = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`Tn11[$CellContext`s], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}], PlotStyle -> { Darker[ Darker[Green]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]], If[$CellContext`Fnumber == 12, $CellContext`ALPv = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`Tn12[$CellContext`s], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}], PlotStyle -> { Darker[ Darker[Green]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]], If[ Or[$CellContext`Fnumber == 2, $CellContext`Fnumber == 4, $CellContext`Fnumber == 5, $CellContext`Fnumber == 7, $CellContext`Fnumber == 10], $CellContext`ALPv = Graphics3D[]], If[ Or[$CellContext`Fnumber == 3, $CellContext`Fnumber == 8], $CellContext`ALPv = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`Tn[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}, PlotStyle -> { Darker[ Darker[Green]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]]}, $CellContext`ALPv = Graphics3D[]]; If[ MemberQ[$CellContext`GRH$$, $CellContext`ALPA], { If[$CellContext`Fnumber == 1, $CellContext`ALPa = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`AnALP1[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}, PlotStyle -> { Darker[ Darker[Red]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]], If[$CellContext`Fnumber == 6, $CellContext`ALPa = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`AnALP6[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$ + 0.1}, PlotStyle -> { Darker[ Darker[Red]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]], If[$CellContext`Fnumber == 9, $CellContext`ALPa = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`AnALP9[$CellContext`s], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}], PlotStyle -> { Darker[ Darker[Red]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]], If[$CellContext`Fnumber == 11, $CellContext`ALPa = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`AnALP11[$CellContext`s], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}], PlotStyle -> { Darker[ Darker[Red]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]], If[$CellContext`Fnumber == 12, $CellContext`ALPa = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`AnALP12[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}, PlotStyle -> { Darker[ Darker[Red]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]], If[ Or[$CellContext`Fnumber == 2, $CellContext`Fnumber == 4, $CellContext`Fnumber == 5, $CellContext`Fnumber == 7, $CellContext`Fnumber == 10], $CellContext`ALPa = Graphics3D[]], If[ Or[$CellContext`Fnumber == 3, $CellContext`Fnumber == 8], $CellContext`ALPa = ParametricPlot3D[ Evaluate[$CellContext`TG $CellContext`f[$CellContext`s] + \ $CellContext`AnALP[$CellContext`s]], {$CellContext`s, $CellContext`a$$, \ $CellContext`b$$}, PlotStyle -> { Darker[ Darker[Red]], Thick}, Exclusions -> None, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> $CellContext`PlotPoints3D$$, PerformanceGoal -> "Speed"]]}, $CellContext`ALPa = Graphics3D[]]; If[$CellContext`UnitSphere$$, $CellContext`UnS = Graphics3D[{ Opacity[$CellContext`opacity$$], Sphere[]}], $CellContext`UnS = Graphics3D[]]; If[$CellContext`Fnumber == 8, $CellContext`EXTRA = ParametricPlot3D[{{$CellContext`s, 0, 0}, { 0, $CellContext`s, 0}}, {$CellContext`s, $CellContext`a$$, $CellContext`b$$}, PlotStyle -> Dashed], $CellContext`EXTRA = Graphics3D[]]; $CellContext`L = Line[{{0, 0, 0}, $CellContext`f[$CellContext`u$$]}]; $CellContext`\[Xi] = \ $CellContext`range$$ $CellContext`scaleRg$$; $CellContext`fRange = Options[ ParametricPlot3D[ Evaluate[ $CellContext`f[$CellContext`s]], {$CellContext`s, \ $CellContext`a$$, $CellContext`b$$}], PlotRange]; $CellContext`x = Part[ Part[ Part[ Part[$CellContext`fRange, 1], 2], 1], 1]; $CellContext`X = Part[ Part[ Part[ Part[$CellContext`fRange, 1], 2], 1], 2]; $CellContext`y = Part[ Part[ Part[ Part[$CellContext`fRange, 1], 2], 2], 1]; $CellContext`Y = Part[ Part[ Part[ Part[$CellContext`fRange, 1], 2], 2], 2]; $CellContext`z = Part[ Part[ Part[ Part[$CellContext`fRange, 1], 2], 3], 1]; $CellContext`Z = Part[ Part[ Part[ Part[$CellContext`fRange, 1], 2], 3], 2]; If[$CellContext`OriginZoom$$, {$CellContext`PRange = {{(-1.5) \ $CellContext`\[Xi], 1.5 $CellContext`\[Xi]}, {(-1.5) $CellContext`\[Xi], 1.5 $CellContext`\[Xi]}, {(-1.5) $CellContext`\[Xi], 1.5 $CellContext`\[Xi]}}}, $CellContext`PRange = \ {{($CellContext`x ($CellContext`\[Xi] + 1) - $CellContext`X ($CellContext`\[Xi] - 1))/2 - 1, ($CellContext`X ($CellContext`\[Xi] + 1) - $CellContext`x ($CellContext`\[Xi] - 1))/2 + 1}, {($CellContext`y ($CellContext`\[Xi] + 1) - $CellContext`Y ($CellContext`\[Xi] - 1))/2 - 1, ($CellContext`Y ($CellContext`\[Xi] + 1) - $CellContext`y ($CellContext`\[Xi] - 1))/2 + 1}, {($CellContext`z ($CellContext`\[Xi] + 1) - $CellContext`Z ($CellContext`\[Xi] - 1))/2 - 1, ($CellContext`Z ($CellContext`\[Xi] + 1) - $CellContext`z ($CellContext`\[Xi] - 1))/2 + 1}}]; $CellContext`GRAPH = Show[$CellContext`P1, $CellContext`P2, $CellContext`P3, \ $CellContext`P4, $CellContext`P5, $CellContext`P7, $CellContext`P8, \ $CellContext`P9, $CellContext`P10, $CellContext`Curve, $CellContext`Vtrace, \ $CellContext`Atrace, $CellContext`ANtrace, $CellContext`ATtrace, \ $CellContext`ALPv, $CellContext`ALPa, $CellContext`UnS, $CellContext`EXTRA, Graphics3D[{ PointSize[0.015], Point[{0, 0, 0}]}], Graphics3D[{Dashed, $CellContext`L}], Graphics3D[{ RGBColor[1, 0, 1], PointSize[0.015], Point[ $CellContext`f[$CellContext`u$$]]}], Axes -> True, PlotRange -> $CellContext`PRange, If[$CellContext`twoD$$, ViewPoint -> {0, 0, Infinity}, ViewPoint -> {1.3, -2.4, 2}], AxesLabel -> { "\!\(\*\nStyleBox[\"X\",\nFontColor->GrayLevel[0]]\)", "\!\(\*\nStyleBox[\"Y\",\nFontColor->GrayLevel[0]]\)", "\!\(\*\nStyleBox[\"Z\",\nFontColor->GrayLevel[0]]\)"}, AspectRatio -> Automatic, ImageSize -> $CellContext`IMSIZE]; If[$CellContext`text$$, $CellContext`GRAPHOUT = Column[{ Row[{ PopupWindow[ Button["READ about this GRAPH"], $CellContext`LABEL], " ", " ", " ", " ", " ", " ", Style["r(", Bold], Style[$CellContext`u$$, Blue, Bold], Style[") = ", Bold], Style[ Round[ $CellContext`f[$CellContext`u$$], 0.01], Magenta, Bold]}], $CellContext`GRAPH}, Left, Frame -> None], $CellContext`GRAPHOUT = Column[{ Row[{ PopupWindow[ Button[" READ about this GRAPH"], $CellContext`LABEL], " ", Style["r(t) = ", Bold], Style[ TraditionalForm[ $CellContext`f[$CellContext`t]], Bold], ";", " ", Style[$CellContext`a$$, Blue, Bold], Style["\[LessEqual]", Bold], Style[$CellContext`t, Blue, Bold], Style["\[LessEqual]", Bold], Style[$CellContext`b$$, Blue, Bold], ";", " ", Style["r(", Bold], Style[$CellContext`u$$, Blue, Bold], Style[") = ", Bold], Style[ Round[ $CellContext`f[$CellContext`u$$], 0.01], Magenta, Bold]}], $CellContext`GRAPH}, Left, Frame -> None]]; If[$CellContext`text$$, If[$CellContext`TEXT$$ == 0, $CellContext`TEXTOUTPUT = Style[ Text[ "\n1. \!\(\*\nStyleBox[\"Top\",\nFontVariations->{\"Underline\ \"->True}]\)\!\(\*\nStyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\ \!\(\*\nStyleBox[\"Controls\",\nFontVariations->{\"Underline\"->True}]\). \ Main pop up menu supplies a vector function and its initial domain. First, go \ over the topics in the \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\) column, then \ move to the right, do not clutter the graph with all possible objects, you \ can always arrange a combination that you want to study later. \n\!\(\*\n\ StyleBox[\"Before\",\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \ \",\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"switching\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"to\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"a\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"new\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"vector\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"function\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\",\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"press\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"RESET\",\n\ FontSlant->\"Italic\",\nFontColor->RGBColor[1, 0.5, 0]]\) button in the right \ top corner to return the domain and range to initial positions matching the \ new function. You can rotate space graphs with your mouse.\n\n2. \!\(\*\n\ StyleBox[\"Left\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\ \" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"Controls\",\nFontVariations->{\"Underline\"->True}]\). You added a \ new graph, but nothing happened. Use the slider to increase the \!\(\*\n\ StyleBox[\"Range\",\nFontSlant->\"Italic\"]\) gently, chose 10, 20,... \!\(\*\ \nStyleBox[\"Scale\",\nFontSlant->\"Italic\"]\) for rapid zoom out and 1/10, \ 1/20,... \!\(\*\nStyleBox[\"Scale\",\nFontSlant->\"Italic\"]\) for rapid zoom \ in. Separate \!\(\*\nStyleBox[\"Domain\",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"a\",\n\ FontColor->RGBColor[0, 0, 1]]\) and \!\(\*\nStyleBox[\"b\",\n\ FontColor->RGBColor[0, 0, 1]]\) controls allow you to see the curve for more \ or fewer values of parameter t. Domain expands and shrinks about the midpoint \ of the original [a,b] interval. If vectors you want to see are too small or \ too large use \!\(\*\nStyleBox[\"Vectors\",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\" \",\nFontSlant->\"Italic\"]\)controls, press \!\(\*\n\ StyleBox[\"Reset1\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)to return vectors to their genuine length. \n\n3. \ \!\(\*\nStyleBox[\"Right\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"Controls\",\nFontVariations->{\"Underline\"->True}]\). \!\(\*\n\ StyleBox[\"Speed\",\nFontSlant->\"Italic\"]\) and \!\(\*\n\ StyleBox[\"Quality\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)allow to compromise between the rate of graph \ rendering and its smoothness. \!\(\*\nStyleBox[\"Color\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\"Coded\",\nFontSlant->\"Italic\"]\) provides easier match \ between the space graph and supporting plane graphs, though it makes the \ graphs less readable and the program runs slower. \!\(\*\nStyleBox[\"2\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"View\",\n\ FontSlant->\"Italic\"]\) is useful if you study a flat curve (third component \ of the vector function is 0). \!\(\*\nStyleBox[\"Origin\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\) and \!\(\*\n\ StyleBox[\"Unit\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"Sphere\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)with \ controlled opacity are useful, especially, for \!\(\*\nStyleBox[\"\ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP]\",\n\ FontSize->10]\) \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\). \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\) = \ \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\ \[Rule]\)]\)n checkbox works only, if all three components are displayed. \n\ \!\(\*\nStyleBox[\"To\",\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"see\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"the\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"full\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"screen\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"space\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"graph\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"uncheck\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"Text\",\n\ FontSlant->\"Italic\",\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \ \",\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"button\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\".\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\n\n4. \!\(\*\nStyleBox[\"Bottom\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Control\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)is the way to move along the curve, \ its play feature will animate the graph."], 14 + $CellContext`font$$]]; If[$CellContext`TEXT$$ == 1, $CellContext`TEXTOUTPUT = Style[ Column[{ Row[{"Position:" " ", " ", "\!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\)(t) = ", TraditionalForm[ $CellContext`f[$CellContext`t]]}], Row[{"\!\(\*\nStyleBox[\"Curvature\",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\)\!\(\*\nStyleBox[\":\",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\)" " ", " ", "\!\(\*\nStyleBox[\"\[Kappa]\",\nFontColor->RGBColor[0.6, \ 0.4, 0.2]]\)(t) = ", TraditionalForm[ $CellContext`Cv[$CellContext`t]]}], Framed[ Text[ " By definition, \!\(\*\nStyleBox[\"\[Kappa]\",\n\ FontColor->RGBColor[0.6, 0.4, \ 0.2]]\)(s)=\[LeftDoubleBracketingBar]\!\(\*SuperscriptBox[OverscriptBox[\"r\",\ \"\[Rule]\"], \"\[Prime]\[Prime]\",\nMultilineFunction->None]\)(s)\ \[RightDoubleBracketingBar], where \!\(\*OverscriptBox[\(r\), \ \(\[Rule]\)]\)(s) is an arc length parameterization (\[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]). Usually we avoid finding \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP]\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\) ( more on \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP] in \ later topics) and express \!\(\*\nStyleBox[\"\[Kappa]\",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\)in terms of the original parameter \ \!\(\*\nStyleBox[\"\[Kappa]\",\nFontColor->RGBColor[0.6, 0.4, 0.2]]\)(t)=\!\(\ \*FractionBox[\n RowBox[{\"\[LeftDoubleBracketingBar]\", \n \ RowBox[{SuperscriptBox[OverscriptBox[\"r\", \"\[Rule]\"], \"\[Prime]\",\n\ MultilineFunction->None], \n RowBox[{\n RowBox[{\"(\", \"t\", \")\"}], \ \"\[Cross]\", SuperscriptBox[OverscriptBox[\"r\", \"\[Rule]\"], \"\[Prime]\ \[Prime]\",\nMultilineFunction->None]}], \n RowBox[{\"(\", \"t\", \ \")\"}]}], \"\[RightDoubleBracketingBar]\"}], SuperscriptBox[\n RowBox[{\"\ \[LeftDoubleBracketingBar]\", \n \ RowBox[{SuperscriptBox[OverscriptBox[\"r\", \"\[Rule]\"], \"\[Prime]\",\n\ MultilineFunction->None], \n RowBox[{\"(\", \"t\", \")\"}]}], \"\ \[RightDoubleBracketingBar]\"}], \"3\"]]\). In terms of motion, this can be \ given as \!\(\*\nStyleBox[\"\[Kappa]\",\nFontColor->RGBColor[0.6, 0.4, \ 0.2]]\)(t)=\!\(\*FractionBox[\n RowBox[{\"\[LeftDoubleBracketingBar]\", \n \ RowBox[{\n StyleBox[OverscriptBox[\n StyleBox[\"v\",\n\ FontColor->RGBColor[0, 1, 0]], \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]], \ \n RowBox[{\n RowBox[{\"(\", \"t\", \")\"}], \"\[Cross]\", \n \ StyleBox[OverscriptBox[\n StyleBox[\"a\",\nFontColor->RGBColor[1, 0, \ 0]], \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]}], \n RowBox[{\"(\", \ \"t\", \")\"}]}], \"\[RightDoubleBracketingBar]\"}], SuperscriptBox[\n \ RowBox[{\"\[LeftDoubleBracketingBar]\", \n RowBox[{\n \ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]], \n\ RowBox[{\"(\", \"t\", \")\"}]}], \"\[RightDoubleBracketingBar]\"}], \ \"3\"]]\). Locate points that yield maximum curvature or sharpest bends of \ the curve. Let's say, for an ellipse they are, obviously, vertices on the \ major axis. Turn on \!\(\*\nStyleBox[\"Color\",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"Coded\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)feature to track changes in curvature to points on \ the curve (this will slow down the graph rendering)."]], Show[ Graphics[{ RGBColor[1, 0, 1], PointSize[$CellContext`ptsize], Point[{$CellContext`u$$, $CellContext`Cv[$CellContext`u$$]}]}], \ $CellContext`asymptote, Plot[ Evaluate[ $CellContext`Cv[$CellContext`s]], {$CellContext`s, \ $CellContext`a$$, $CellContext`b$$}, PlotStyle -> {Brown, Thick}, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> 15, PerformanceGoal -> "Speed", ColorFunction -> If[$CellContext`ColorCode$$, Function[$CellContext`s, Hue[0.75, $CellContext`Cv[$CellContext`s], 1]], Automatic], ColorFunctionScaling -> False, Exclusions -> $CellContext`jumps], PlotLabel -> Framed[ Style[ Row[{ Style[$CellContext`a$$, Blue], " ", "\[LessEqual]", " ", Style[$CellContext`t, Blue], " ", "\[LessEqual]", " ", Style[$CellContext`b$$, Blue], " ", " ", " ", " ", " ", " ", "\!\(\*\nStyleBox[\"\[Kappa]\",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\)(", Style[ Round[$CellContext`u$$, 0.01], Blue], ") = ", Round[ Quiet[ $CellContext`Cv[$CellContext`u$$]], 0.01]}], 15, Bold]], AxesLabel -> { "\!\(\*\nStyleBox[\"t\",\nFontSize->16,\n\ FontColor->RGBColor[0, 0, 1]]\)", "\!\(\*\nStyleBox[\"\[Kappa]\",\nFontSize->16,\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\)"}, AxesStyle -> {Blue, Black}, ImageSize -> {475, 240}, PlotRange -> All, AspectRatio -> Automatic, Axes -> True, AxesOrigin -> {Automatic, 0}]}, Center], 15 + $CellContext`font$$]]; If[$CellContext`TEXT$$ == 2, $CellContext`TEXTOUTPUT = Style[ Column[{ Row[{"\!\(\*\nStyleBox[\"Velocity\",\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontColor->RGBColor[0, 1, 0]]\)", " ", "\!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 0]]\)(t) = ", TraditionalForm[ $CellContext`V[$CellContext`t]], ", ", " ", Round[ $CellContext`V[$CellContext`u$$], 0.01]}], Row[{"\!\(\*\nStyleBox[\"Speed\",\nFontColor->RGBColor[0, \ 1, 0]]\)\!\(\*\nStyleBox[\":\",\nFontColor->RGBColor[0, 1, 0]]\)", " ", "\!\(\*\nStyleBox[\"S\",\nFontColor->RGBColor[0, 1, \ 0]]\)(t) = ", TraditionalForm[ $CellContext`S[$CellContext`t]]}], Framed[ Text[ " \!\(\*OverscriptBox[\n StyleBox[\"v\",\n\ FontColor->RGBColor[0, 1, 0]], \n StyleBox[\"\[Rule]\",\n\ FontColor->RGBColor[0, 1, 0]]]\)(t)=\!\(\*SuperscriptBox[OverscriptBox[\"r\", \ \"\[Rule]\"], \"\[Prime]\",\nMultilineFunction->None]\)(t) and \!\(\*\n\ StyleBox[\"S\",\nFontColor->RGBColor[0, 1, \ 0]]\)(t)=\[LeftDoubleBracketingBar]\!\(\*OverscriptBox[\n StyleBox[\"v\",\n\ FontColor->RGBColor[0, 1, 0]], \n StyleBox[\"\[Rule]\",\n\ FontColor->RGBColor[0, 1, 0]]]\)(t)\[RightDoubleBracketingBar]. Velocity \ vector is always tangent to the curve. Locate points where the particle (pink \ dot) moves fastest along the curve. Of special interest are points where \!\(\ \*OverscriptBox[\n StyleBox[\"v\",\nFontColor->RGBColor[0, 1, 0]], \n \ StyleBox[\"\[Rule]\",\nFontColor->RGBColor[0, 1, \ 0]]]\)(t)=\!\(\*OverscriptBox[\(0\), \(\[Rule]\)]\) (or equivalently \!\(\*\n\ StyleBox[\"S\",\nFontColor->RGBColor[0, 1, 0]]\)(t)=0). Remember, the curve \ is not smooth at these points, for ex., it may have a cusp or particle may be \ reversing direction. \!\(\*\nStyleBox[\"S\",\nFontColor->RGBColor[0, 1, \ 0]]\)(t) can effectively be used for arc length parameterization (\ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP]) test, \ \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\)(t) is arc length parameterized, if \ and only if, \!\(\*\nStyleBox[\"S\",\nFontColor->RGBColor[0, 1, 0]]\)(t)\ \[Congruent]1. If you can see the dashed horizontal line, \ \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\)(t) failed. Use \!\(\*\n\ StyleBox[\"Color\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"Coded\",\nFontSlant->\"Italic\"]\) \ feature to track the motion speed."]], Show[ Graphics[{ RGBColor[1, 0, 1], PointSize[$CellContext`ptsize], Point[{$CellContext`u$$, $CellContext`S[$CellContext`u$$]}]}], \ $CellContext`asymptote, Plot[ 1, {$CellContext`s, $CellContext`a$$, $CellContext`b$$}, PlotStyle -> {Red, Dashed}], Plot[ Evaluate[ $CellContext`S[$CellContext`s]], {$CellContext`s, \ $CellContext`a$$, $CellContext`b$$}, PlotStyle -> {{Green, Thick}, {Red, Thick}}, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> 10, PerformanceGoal -> "Speed", ColorFunction -> If[$CellContext`ColorCode$$, Function[$CellContext`s, Hue[0.75, $CellContext`S[$CellContext`s], 1]], Automatic], ColorFunctionScaling -> False, Exclusions -> $CellContext`jumps], PlotLabel -> Framed[ Style[ Row[{ Style[$CellContext`a$$, Blue], " ", "\[LessEqual]", " ", Style[$CellContext`t, Blue], " ", "\[LessEqual]", " ", Style[$CellContext`b$$, Blue], " ", " ", " ", " ", " ", " ", "\!\(\*\nStyleBox[\"S\",\nFontSize->16,\n\ FontColor->RGBColor[0, 1, 0]]\)(", Style[ Round[$CellContext`u$$, 0.01], Blue], ") = ", Round[ $CellContext`S[$CellContext`u$$], 0.01]}], 15, Bold]], AxesLabel -> { "\!\(\*\nStyleBox[\"t\",\nFontSize->16,\n\ FontColor->RGBColor[0, 0, 1]]\)", "\!\(\*\nStyleBox[\"S\",\nFontSize->16,\n\ FontColor->RGBColor[0, 1, 0]]\)"}, AxesStyle -> {Blue, Black}, ImageSize -> {475, 240}, PlotRange -> All, AspectRatio -> Automatic, Axes -> True, AxesOrigin -> {Automatic, 0}]}, Center], 15 + $CellContext`font$$]]; If[$CellContext`TEXT$$ == 3, $CellContext`TEXTOUTPUT = Style[ Column[{ Row[{"\!\(\*\nStyleBox[\"Acceleration\",\n\ FontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontColor->RGBColor[1, 0, 0]]\)", " ", "\!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[1, 0, 0]]\)(t) = ", TraditionalForm[ $CellContext`A[$CellContext`t]], ", ", " ", Round[ $CellContext`A[$CellContext`u$$], 0.01]}], Row[{"\!\(\*\nStyleBox[\"Norm\",\nFontColor->RGBColor[1, 0, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\n\ StyleBox[\"of\",\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\"Acceleration\",\n\ FontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontColor->RGBColor[1, 0, 0]]\)", " ", "\!\(\*\nStyleBox[\"a\",\nFontColor->RGBColor[1, 0, \ 0]]\)(t) = ", TraditionalForm[ $CellContext`ANorm[$CellContext`t]]}], Framed[ Text[ "\!\(\*OverscriptBox[\n StyleBox[\n RowBox[{\" \", \ \"a\"}],\nFontColor->RGBColor[1, 0, 0]], \n StyleBox[\n RowBox[{\" \", \"\ \[Rule]\"}],\nFontColor->RGBColor[1, 0, \ 0]]]\)(t)=\!\(\*SuperscriptBox[OverscriptBox[\"r\", \"\[Rule]\"], \"\[Prime]\ \[Prime]\",\nMultilineFunction->None]\)(t) and \!\(\*\nStyleBox[\"a\",\n\ FontColor->RGBColor[1, 0, \ 0]]\)(t)=\[LeftDoubleBracketingBar]\!\(\*OverscriptBox[\n StyleBox[\"a\",\n\ FontColor->RGBColor[1, 0, 0]], \n StyleBox[\"\[Rule]\",\n\ FontColor->RGBColor[1, 0, 0]]]\)(t)\[RightDoubleBracketingBar]. Acceleration \ vector can always be decomposed into sum of two orthogonal vectors, of which \ the first is tangent to the curve and the second is orthogonal. The first \ component is called vector tangential acceleration,\!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"t\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\)(t) and the second vector normal \ acceleration,\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\ \"a\", \"n\"]}], \n RowBox[{\" \", \"\[Rule]\"}]],\nFontColor->RGBColor[0, \ 1, 1]]\)(t). Use\!\(\*OverscriptBox[\(\(\\ \)\(A\)\), \(\[Rule]\)]\) = \ \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\ \[Rule]\)]\)n \!\(\*\nStyleBox[\"Checkbox\",\nFontSlant->\"Italic\"]\) to \ verify the decomposition (it only works, if all three vectors are displayed). \ Notice, that by parallelogram rule \!\(\*OverscriptBox[\n StyleBox[\"a\",\n\ FontColor->RGBColor[1, 0, 0]], \n StyleBox[\"\[Rule]\",\n\ FontColor->RGBColor[1, 0, 0]]]\) is the sum of two components and, moreover, \ you see a rectangle, which means that components are orthogonal. Read more \ about \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \ \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\)(t) and\!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"n\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)(t) under their radio \ buttons.\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)Locate points where \ the changes in velocity are most rapid. At points where \!\(\*OverscriptBox[\n\ StyleBox[\n RowBox[{\" \", \"a\"}],\nFontColor->RGBColor[1, 0, 0]], \n \ StyleBox[\"\[Rule]\",\nFontColor->RGBColor[1, 0, \ 0]]]\)(t)=\!\(\*OverscriptBox[\(0\), \(\[Rule]\)]\) velocity vector is \ instantaneously constant."]], Show[ Graphics[{ RGBColor[1, 0, 1], PointSize[$CellContext`ptsize], Point[{$CellContext`u$$, $CellContext`ANorm[$CellContext`u$$]}]}], \ $CellContext`asymptote, Plot[ Evaluate[ $CellContext`ANorm[$CellContext`s]], {$CellContext`s, \ $CellContext`a$$, $CellContext`b$$}, PlotStyle -> {Red, Thick}, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> 10, PerformanceGoal -> "Speed", ColorFunction -> If[$CellContext`ColorCode$$, Function[$CellContext`s, Hue[0.75, $CellContext`ANorm[$CellContext`s], 1]], Automatic], ColorFunctionScaling -> False, Exclusions -> $CellContext`jumps], PlotLabel -> Framed[ Style[ Row[{ Style[$CellContext`a$$, Blue], " ", "\[LessEqual]", " ", Style[$CellContext`t, Blue], " ", "\[LessEqual]", " ", Style[$CellContext`b$$, Blue], " ", " ", " ", " ", " ", " ", "\!\(\*\nStyleBox[\"a\",\nFontSize->16,\n\ FontColor->RGBColor[1, 0, 0]]\)(", Style[ Round[$CellContext`u$$, 0.01], Blue], ") = ", Round[ $CellContext`ANorm[$CellContext`u$$], 0.01]}], 15, Bold]], AxesLabel -> { "\!\(\*\nStyleBox[\"t\",\nFontSize->16,\n\ FontColor->RGBColor[0, 0, 1]]\)", "\!\(\*\nStyleBox[\"a\",\nFontSize->16,\n\ FontColor->RGBColor[1, 0, 0]]\)"}, AxesStyle -> {Blue, Black}, ImageSize -> {475, 240}, PlotRange -> All, AspectRatio -> Automatic, Axes -> True, AxesOrigin -> {Automatic, 0}]}, Center], 15 + $CellContext`font$$]]; If[$CellContext`TEXT$$ == 4, $CellContext`TEXTOUTPUT = Style[ Column[{ Row[{"\!\(\*\nStyleBox[\"Vector\",\nFontColor->RGBColor[0, \ 0, 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\n\ StyleBox[\"Tangential\",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \ \",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Acceleration\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\":\",\n\ FontColor->RGBColor[0, 0, 1]]\)", " ", "\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", \ SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)(t) = ", TraditionalForm[ $CellContext`At[$CellContext`t]], ", ", " ", Round[ $CellContext`At[$CellContext`u$$], 0.01]}], Row[{"\!\(\*\nStyleBox[\"Scalar\",\nFontColor->RGBColor[0, \ 0, 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\n\ StyleBox[\"Tangential\",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \ \",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Acceleration\",\n\ FontColor->RGBColor[0, 0, 1]]\)", " ", "\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 0, 1]]\)\ \!\(\*\nStyleBox[SubscriptBox[\"a\", \"t\"],\nFontSize->14,\n\ FontColor->RGBColor[0, 0, 1]]\)(t) = ", TraditionalForm[ $CellContext`at[$CellContext`t]]}], Framed[ Text[ " Norm of \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \ \", SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)(t) is called scalar tangential acceleration \!\(\*\n\ StyleBox[SubscriptBox[\"a\", \"t\"],\nFontSize->14,\nFontColor->RGBColor[0, \ 0, 1]]\)(t) and it can be found by \!\(\*\nStyleBox[SubscriptBox[\"a\", \ \"t\"],\nFontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)(t)=\!\(\*FractionBox[\ \n RowBox[{\n StyleBox[OverscriptBox[\n StyleBox[\"v\",\n\ FontColor->RGBColor[0, 1, 0]], \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]], \ \n RowBox[{\n RowBox[{\"(\", \"t\", \")\"}], \"\[CenterDot]\", \n \ StyleBox[OverscriptBox[\n StyleBox[\"a\",\nFontColor->RGBColor[1, 0, 0]], \ \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]}], \n RowBox[{\"(\", \"t\", \")\ \"}]}], \n RowBox[{\"\[LeftDoubleBracketingBar]\", \n RowBox[{\n \ StyleBox[OverscriptBox[\n StyleBox[\"v\",\nFontColor->RGBColor[0, 1, 0]], \ \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]], \n RowBox[{\"(\", \"t\", \ \")\"}]}], \"\[RightDoubleBracketingBar]\"}]]\). Vector tangential \ acceleration is tangent to the curve, thus \!\(\*\nStyleBox[OverscriptBox[\n \ RowBox[{\" \", SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)(t)=\!\(\*\nStyleBox[SubscriptBox[\"a\", \ \"t\"],\nFontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)(t)\!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", \"T\"}], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)(t), where \!\(\*\nStyleBox[OverscriptBox[\n \ StyleBox[\"T\",\nFontColor->RGBColor[0, 0, 1]], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)(t) is the unit tangent vector. Since \!\(\*\n\ StyleBox[OverscriptBox[\n StyleBox[\"T\",\nFontColor->RGBColor[0, 0, 1]], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\)(t)= \!\(\*FractionBox[\n \ RowBox[{\n StyleBox[OverscriptBox[\n StyleBox[\"v\",\n\ FontColor->RGBColor[0, 1, 0]], \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]], \ \n RowBox[{\"(\", \"t\", \")\"}]}], \n \ RowBox[{\"\[LeftDoubleBracketingBar]\", \n RowBox[{\n \ StyleBox[OverscriptBox[\n StyleBox[\"v\",\nFontColor->RGBColor[0, 1, 0]], \ \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]], \n RowBox[{\"(\", \"t\", \ \")\"}]}], \"\[RightDoubleBracketingBar]\"}]]\), \!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"t\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\)(t)= \!\(\*FractionBox[\n \ RowBox[{\n StyleBox[SubscriptBox[\"a\", \"t\"],\nFontSize->14,\n\ FontColor->RGBColor[0, 0, 1]], \n RowBox[{\"(\", \"t\", \")\"}], \n \ StyleBox[OverscriptBox[\n StyleBox[\"v\",\nFontColor->RGBColor[0, 1, 0]], \ \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]], \n RowBox[{\"(\", \"t\", \ \")\"}]}], \n RowBox[{\"\[LeftDoubleBracketingBar]\", \n RowBox[{\n \ StyleBox[OverscriptBox[\n StyleBox[\"v\",\nFontColor->RGBColor[0, 1, 0]], \ \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]], \n RowBox[{\"(\", \"t\", \ \")\"}]}], \"\[RightDoubleBracketingBar]\"}]]\) . Scalar tangential \ acceleration can have any sign, or \!\(\*\nStyleBox[OverscriptBox[\n \ RowBox[{\" \", SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)(t) can point opposite to or in the same \ direction as \!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\"v\",\n\ FontColor->RGBColor[0, 1, 0]], \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]]\)\ \!\(\*\nStyleBox[\"(\",\nFontColor->GrayLevel[0]]\)\!\(\*\nStyleBox[\"t\",\n\ FontColor->GrayLevel[0]]\)\!\(\*\nStyleBox[\")\",\nFontColor->GrayLevel[0]]\)\ \!\(\*\nStyleBox[\".\",\nFontColor->GrayLevel[0]]\) Locate points where the \ particle speeds up or slows down the fastest along the curve. Points \ where\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \ \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)(t)=\!\(\*OverscriptBox[\(0\), \(\[Rule]\)]\) likely indicate that the \ particle starts slowing down or speeding up."]], Show[ Graphics[{ RGBColor[1, 0, 1], PointSize[$CellContext`ptsize], Point[{$CellContext`u$$, $CellContext`at[$CellContext`u$$]}]}], \ $CellContext`asymptote, Plot[ Evaluate[ $CellContext`at[$CellContext`s]], {$CellContext`s, \ $CellContext`a$$, $CellContext`b$$}, PlotStyle -> {Blue, Thick}, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> 10, PerformanceGoal -> "Speed", ColorFunction -> If[$CellContext`ColorCode$$, Function[$CellContext`s, Hue[0.75, $CellContext`at[$CellContext`s], 1]], Automatic], ColorFunctionScaling -> False, Exclusions -> $CellContext`jumps], PlotLabel -> Framed[ Style[ Row[{ Style[$CellContext`a$$, Blue], " ", "\[LessEqual]", " ", Style[$CellContext`t, Blue], " ", "\[LessEqual]", " ", Style[$CellContext`b$$, Blue], " ", " ", " ", " ", " ", " ", "\!\(\*\nStyleBox[SubscriptBox[\"a\", \"t\"],\n\ FontSize->16,\nFontColor->RGBColor[0, 0, 1]]\)(", Style[ Round[$CellContext`u$$, 0.01], Blue], ") = ", Round[ $CellContext`at[$CellContext`u$$], 0.01]}], 15, Bold]], AxesLabel -> { "\!\(\*\nStyleBox[\"t\",\nFontSize->16,\n\ FontColor->RGBColor[0, 0, 1]]\)", "\!\(\*\nStyleBox[SubscriptBox[\"a\", \"t\"],\n\ FontSize->18,\nFontColor->RGBColor[0, 0, 1]]\)"}, AxesStyle -> {Blue, Black}, ImageSize -> {475, 240}, PlotRange -> All, AspectRatio -> Automatic, Axes -> True, AxesOrigin -> {Automatic, 0}]}, Center], 14 + $CellContext`font$$]]; If[$CellContext`TEXT$$ == 5, $CellContext`TEXTOUTPUT = Style[ Column[{ Row[{"\!\(\*\nStyleBox[\"Vector\",\nFontColor->RGBColor[0, \ 1, 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\n\ StyleBox[\"Normal\",\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\"Acceleration\",\n\ FontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\":\",\n\ FontColor->RGBColor[0, 1, 1]]\)", " ", "\!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \ \"n\"], \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)(t) = ", TraditionalForm[ $CellContext`An[$CellContext`t]], ", ", " ", Round[ $CellContext`An[$CellContext`u$$], 0.01]}], Row[{"\!\(\*\nStyleBox[\"Scalar\",\nFontColor->RGBColor[0, \ 1, 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\n\ StyleBox[\"Normal\",\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\"Acceleration\",\n\ FontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\":\",\n\ FontColor->RGBColor[0, 1, 1]]\)", " ", "\!\(\*\nStyleBox[SubscriptBox[\"a\", \"n\"],\n\ FontColor->RGBColor[0, 1, 1]]\)(t) = ", TraditionalForm[ $CellContext`an[$CellContext`t]]}], Framed[ Text[ " Norm of \!\(\*\n\ StyleBox[OverscriptBox[SubscriptBox[\"a\", \"n\"], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 1]]\)(t) is called scalar normal acceleration \ \!\(\*\nStyleBox[SubscriptBox[\"a\", \"n\"],\nFontSize->14,\n\ FontColor->RGBColor[0, 1, 1]]\)(t) and it can be found by \!\(\*\n\ StyleBox[SubscriptBox[\"a\", \"n\"],\nFontSize->14,\nFontColor->RGBColor[0, \ 1, 1]]\)(t)=\!\(\*FractionBox[\n RowBox[{\"\[LeftDoubleBracketingBar]\", \n \ RowBox[{\n StyleBox[OverscriptBox[\n StyleBox[\"v\",\n\ FontColor->RGBColor[0, 1, 0]], \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 0]], \ \n RowBox[{\n RowBox[{\"(\", \"t\", \")\"}], \"\[Cross]\", \n \ StyleBox[OverscriptBox[\n StyleBox[\"a\",\nFontColor->RGBColor[1, 0, \ 0]], \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]}], \n RowBox[{\"(\", \ \"t\", \")\"}]}], \"\[RightDoubleBracketingBar]\"}], \n RowBox[{\"\ \[LeftDoubleBracketingBar]\", \n RowBox[{\n StyleBox[OverscriptBox[\n \ StyleBox[\"v\",\nFontColor->RGBColor[0, 1, 0]], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 0]], \n RowBox[{\"(\", \"t\", \")\"}]}], \"\ \[RightDoubleBracketingBar]\"}]]\). Vector normal acceleration is orthogonal \ to the curve, thus \!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"n\"], \ \"\[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)(t)=\!\(\*\n\ StyleBox[SubscriptBox[\"a\", \"n\"],\nFontSize->14,\nFontColor->RGBColor[0, \ 1, 1]]\)(t)\!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\"N\",\n\ FontColor->RGBColor[0, 0, 1]], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)(t), where \!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\"N\",\n\ FontColor->RGBColor[0, 0, 1]], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)(t) is the unit normal vector. Luckily, we can avoid often cumbersome \ calculations of \!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\"N\",\n\ FontColor->RGBColor[0, 0, 1]], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)(t) and use basic formula\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \ \", SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)(t)+\!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"n\"], \"\[Rule]\ \"],\nFontColor->RGBColor[0, 1, 1]]\)(t)=\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, \ 0]]\)(t). Thus, \!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"n\"], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)(t)=\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, \ 0]]\)(t)-\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", \ SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)(t). Scalar normal acceleration is always nonnegative and translated \ \!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"n\"], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 1]]\)(t) always points in the same direction as \ \!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\"N\",\nFontColor->RGBColor[0, 0, \ 1]], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\)(t) and, therefore, always \ points towards the concave side inwards the graph\!\(\*\nStyleBox[\".\",\n\ FontColor->GrayLevel[0]]\) The consequence of the basic formula and \ Pythogoras is a useful relation \!\(\*SuperscriptBox[\n \ StyleBox[SubscriptBox[\"a\", \"t\"],\nFontSize->14,\nFontColor->RGBColor[0, \ 0, 1]], \"2\"]\)+\!\(\*SuperscriptBox[\n StyleBox[SubscriptBox[\"a\", \"n\"],\ \nFontSize->14,\nFontColor->RGBColor[0, 1, 1]], \ \"2\"]\)=\!\(\*SuperscriptBox[\n StyleBox[\"a\",\nFontColor->RGBColor[1, 0, \ 0]], \"2\"]\). Locate points where the particle speeds up or slows down the \ fastest across the curve. Points where\!\(\*\nStyleBox[OverscriptBox[\n \ RowBox[{\" \", SubscriptBox[\"a\", \"n\"]}], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 1]]\)(t)=\!\(\*OverscriptBox[\(0\), \(\[Rule]\)]\) \ indicate that the instantaneous movement is fully along the curve."]], Show[ Graphics[{ RGBColor[1, 0, 1], PointSize[$CellContext`ptsize], Point[{$CellContext`u$$, $CellContext`an[$CellContext`u$$]}]}], \ $CellContext`asymptote, Plot[ Evaluate[ $CellContext`an[$CellContext`s]], {$CellContext`s, \ $CellContext`a$$, $CellContext`b$$}, PlotStyle -> {Cyan, Thick}, MaxRecursion -> $CellContext`Recursion$$, PlotPoints -> 10, PerformanceGoal -> "Speed", ColorFunction -> If[$CellContext`ColorCode$$, Function[$CellContext`s, Hue[0.75, $CellContext`an[$CellContext`s], 1]], Automatic], ColorFunctionScaling -> False, Exclusions -> $CellContext`jumps], PlotLabel -> Framed[ Style[ Row[{ Style[$CellContext`a$$, Blue], " ", "\[LessEqual]", " ", Style[$CellContext`t, Blue], " ", "\[LessEqual]", " ", Style[$CellContext`b$$, Blue], " ", " ", " ", " ", " ", " ", "\!\(\*\nStyleBox[SubscriptBox[\"a\", \"n\"],\n\ FontSize->14,\nFontColor->RGBColor[0, 1, 1]]\)(", Style[ Round[$CellContext`u$$, 0.01], Blue], ") = ", Round[ $CellContext`an[$CellContext`u$$], 0.01]}], 15, Bold]], AxesLabel -> { "\!\(\*\nStyleBox[\"t\",\nFontSize->16,\n\ FontColor->RGBColor[0, 0, 1]]\)", "\!\(\*\nStyleBox[SubscriptBox[\"a\", \"n\"],\n\ FontSize->14,\nFontColor->RGBColor[0, 1, 1]]\)"}, AxesStyle -> {Blue, Black}, ImageSize -> {475, 240}, PlotRange -> All, AspectRatio -> Automatic, Axes -> True, AxesOrigin -> {Automatic, 0}]}, Center], 12 + $CellContext`font$$]]; If[$CellContext`TEXT$$ == 6, $CellContext`TEXTOUTPUT = Style[ Column[{ Row[{ Style[ "\[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] Velocity", Darker[ Darker[Green]]], TraditionalForm[ $CellContext`Tn[$CellContext`s]]}], Framed[ Text[ " If \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\)(t) is \ already arc length parameterized (\[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]), you can not add anything new \ on this screen. If it's not, we may still want to see how the velocity graph \ would look like for an arc length parameterization of the same curve. For \ simple cases we could find an arc length parameterization \ \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\)(s) and use it to produce the graph, \ but, in general, it is impossible to find \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP] explicitly. Such basic curves, \ as ellipses or parabolas do not admit explicit \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]. Anyway, we can get the \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP]\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)graphs without knowing the \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP]! Since \ \[LeftDoubleBracketingBar]\!\(\*SuperscriptBox[OverscriptBox[\"r\", \"\[Rule]\ \"], \"\[Prime]\",\n\ MultilineFunction->None]\)(t)\[RightDoubleBracketingBar]=1 and velocity is \ always tangent to the curve, \!\(\*\nStyleBox[OverscriptBox[\n \ StyleBox[\"v\",\nFontColor->RGBColor[0, 1, 0]], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 1, 0]]\)(s)=\!\(\*\nStyleBox[OverscriptBox[\n \ StyleBox[\"T\",\nFontColor->RGBColor[0, 0, 1]], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)(s) (\!\(\*\nStyleBox[OverscriptBox[\n \ StyleBox[\"T\",\nFontColor->RGBColor[0, 0, 1]], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)(s) is unit tangent vector). This graph is \ uniquely related to the curve (recall evolute) and does not depend on \ parameterization. Word of caution though, we are only saying that the \ formulas for \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\) produce the graph we are interested in. \!\(\*\n\ StyleBox[OverscriptBox[\n StyleBox[\"v\",\nFontColor->RGBColor[0, 1, 0]], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 0]]\)(s) graph is always located on \ the unit sphere, turn \!\(\*\nStyleBox[\"Unit\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\"Sphere\",\nFontSlant->\"Italic\"]\) and \!\(\*\n\ StyleBox[\"Origin\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"Zoom\",\nFontSlant->\"Italic\"]\) \ on and set \!\(\*\nStyleBox[\"Domain\",\nFontSlant->\"Italic\"]\) to see the \ full shape of the graph."]]}, Center], 18 + $CellContext`font$$]]; If[$CellContext`TEXT$$ == 7, $CellContext`TEXTOUTPUT = Style[ Column[{ Row[{ Style[ "\[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] Acceleration", Darker[ Darker[Red]]], TraditionalForm[ If[$CellContext`Fnumber == 6, $CellContext`AnALP6[$CellContext`s], $CellContext`AnALP[$CellContext`s]]]}], Framed[ Text[(" If \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\)(t) is \ arc length parameterized (\[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP]), you can not add anything new on this screen. If \ it's not, we may still want to see how the acceleration graph would look like \ for an arc length parameterization of the same curve. For simple cases we \ could find an arc length parameterization \!\(\*OverscriptBox[\(r\), \ \(\[Rule]\)]\)(s) and use it to produce the graph, but, in general, it is \ impossible to find \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] explicitly. Such basic curves, as ellipses or \ parabolas do not admit explicit \ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP]. \ Anyway, we can get the \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] graphs without knowing the \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]! First of all, for \ \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\)(s) the tangential acceleration is \ zero, since the velocity does not change in tangential direction. Since \ \!\(\*\nStyleBox[SubscriptBox[\"a\", \"n\"],\nFontSize->14,\n\ FontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\"=\",\nFontSize->14,\n\ FontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[" $CellContext`\[Kappa]) ",\nFontColor->RGBColor[0.6, 0.4, \ 0.2]]\)\!\(\*SuperscriptBox[\n StyleBox[\"s\",\nFontColor->RGBColor[0, 1, \ 0]], \"2\"]\) and acceleration is always orthogonal to the curve (remember, \ \!\(\*\nStyleBox[SubscriptBox[\"a\", \"t\"],\nFontSize->14,\n\ FontColor->RGBColor[0, 0, 1]]\)=0 here), \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, \ 0]]\)(s)=\!\(\*\nStyleBox[\"\[Kappa]\",\nFontColor->RGBColor[0.6, 0.4, \ 0.2]]\)\!\(\*SuperscriptBox[\n StyleBox[\"s\",\nFontColor->RGBColor[0, 1, \ 0]], \"2\"]\)\!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\"N\",\n\ FontColor->RGBColor[0, 0, 1]], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, \ 1]]\)(s) (\!\(\*\nStyleBox[OverscriptBox[\n StyleBox[\"N\",\n\ FontColor->RGBColor[0, 0, 1]], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\)\ \!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 0, 1]]\)is unit normal \ vector). Finally, \!\(\*\nStyleBox[\"s\",\nFontColor->RGBColor[0, 1, 0]]\)=1 \ and \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[1, 0, 0]]\)(s)=\!\(\*\nStyleBox[\"\[Kappa]\",\n\ FontColor->RGBColor[0.6, 0.4, 0.2]]\)\!\(\*\nStyleBox[OverscriptBox[\n \ StyleBox[\"N\",\nFontColor->RGBColor[0, 0, 1]], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)(s). This formula was used to obtain the \ graph, which is also uniquely related to the curve and does not depend on \ parameterization. Word of caution though, we are only saying that the \ formulas for \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP] \!\(\*\nStyleBox[OverscriptBox[\"a\", \n RowBox[{\" \ \", \"\[Rule]\"}]],\nFontSize->14,\nFontColor->RGBColor[0.5019607843137255, \ 0., 0.25098039215686274`]]\) produce the graph we are interested in."]]}, Center], 18 + $CellContext`font$$]]]; If[$CellContext`text$$, $CellContext`OUTPUT = Text[ Grid[{{ Item[$CellContext`TEXTOUTPUT, Alignment -> Center], Item[$CellContext`GRAPHOUT, Alignment -> Center]}}, Dividers -> Center, Alignment -> {Center, Top}, ItemSize -> {{40, 40}, {{48, 48}}}]], $CellContext`OUTPUT = $CellContext`GRAPHOUT]; \ $CellContext`OUTPUT, {970, 580}, Alignment -> {Center, Center}, ImageSizeAction -> Automatic]]), "Specifications" :> { " \!\(\*\nStyleBox[\"Range\",\nFontSize->14,\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0, 0]]\)", " Scale", {{$CellContext`scaleRg$$, 1, ""}, { Rational[1, 90], Rational[1, 80], Rational[1, 70], Rational[1, 60], Rational[1, 50], Rational[1, 40], Rational[1, 30], Rational[1, 20], Rational[1, 10], 1 -> "\!\(\*\nStyleBox[\"1\",\nFontColor->RGBColor[0, 0, 1]]\)", 10, 20, 30, 40, 50, 60, 70, 80, 90}, ControlType -> PopupMenu}, {{$CellContext`range$$, 1, ""}, 1, 10, ImageSize -> Tiny, AutoAction -> True, ControlType -> VerticalSlider}, Delimiter, " \!\(\*\nStyleBox[\"Domain\",\nFontSize->14,\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0, 0]]\)", " Scale", {{$CellContext`scaleDomB$$, 1, ""}, { Rational[1, 90], Rational[1, 80], Rational[1, 70], Rational[1, 60], Rational[1, 50], Rational[1, 40], Rational[1, 30], Rational[1, 20], Rational[1, 10], 1 -> "\!\(\*\nStyleBox[\"1\",\nFontColor->RGBColor[0, 0, 1]]\)", 10, 20, 30, 40, 50, 60, 70, 80, 90}, ControlType -> PopupMenu}, " \!\(\*\nStyleBox[\"b\",\nFontColor->RGBColor[0, 0, 1]]\)", \ {{$CellContext`DomainB$$, 1, ""}, 1, 10, ImageSize -> Tiny, AutoAction -> True, ControlType -> VerticalSlider}, " Scale", {{$CellContext`scaleDomA$$, 1, ""}, { Rational[1, 90], Rational[1, 80], Rational[1, 70], Rational[1, 60], Rational[1, 50], Rational[1, 40], Rational[1, 30], Rational[1, 20], Rational[1, 10], 1 -> "\!\(\*\nStyleBox[\"1\",\nFontColor->RGBColor[0, 0, 1]]\)", 10, 20, 30, 40, 50, 60, 70, 80, 90}, ControlType -> PopupMenu}, " \!\(\*\nStyleBox[\"a\",\nFontColor->RGBColor[0, 0, 1]]\)", \ {{$CellContext`DomainA$$, 1, ""}, 10, 1, ImageSize -> Tiny, AutoAction -> True, ControlType -> VerticalSlider}, Delimiter, " \!\(\*\nStyleBox[\"Vectors\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0, 0]]\)", \ {{$CellContext`rescale$$, 1, ""}, 0.5, 2, AppearanceElements -> {"StepRightButton", "StepLeftButton"}, AutoAction -> False, ImageSize -> Tiny, ControlType -> Trigger}, " Scale", {{$CellContext`scaleVct$$, 1, ""}, { Rational[1, 90], Rational[1, 80], Rational[1, 70], Rational[1, 60], Rational[1, 50], Rational[1, 40], Rational[1, 30], Rational[1, 20], Rational[1, 10], 1 -> "\!\(\*\nStyleBox[\"1\",\nFontColor->RGBColor[0, 0, 1]]\)", 10, 20, 30, 40, 50, 60, 70, 80, 90}, ControlType -> PopupMenu}, " Reset", {{$CellContext`rescale$$, 1, ""}, {1}, ControlType -> Setter}, {{$CellContext`fcn$$, $CellContext`f1, Style[ Row[{ Spacer[50], "r(\!\(\*\nStyleBox[\"t\",\nFontColor->RGBColor[0, 0, 1]]\)) = \ "}], 20, Bold]}, {$CellContext`f1 -> Row[{ Style[ TraditionalForm[{2 Sin[$CellContext`t], Cos[$CellContext`t], 0}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=2\[Pi], Ellipse", 15]}], $CellContext`f2 -> Row[{ Style[ TraditionalForm[{ Cos[2^Rational[-1, 2] $CellContext`t], Sin[2^Rational[-1, 2] $CellContext`t], 2^Rational[-1, 2] $CellContext`t}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=4\!\(\*SqrtBox[\(2\)]\)\ \[Pi], Circilar Helix \!\(\*\nStyleBox[\"\[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]\",\nFontSize->14]\) ", 15]}], $CellContext`f3 -> Row[{ Style[ TraditionalForm[{$CellContext`t, $CellContext`t^2, 0}], 15], Style[ ", a=-2 \[LessEqual] t \[LessEqual] b=2, Parabola", 15]}], $CellContext`f4 -> Row[{ Style[ TraditionalForm[{ Sin[ Sin[$CellContext`t]], Cos[ Sin[$CellContext`t]], Sin[$CellContext`t]}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=2\[Pi], Trigonometric \ Curve \!\(\*\nStyleBox[\"\[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP]\",\nFontSize->14]\) ", 15]}], $CellContext`f5 -> Row[{ Style[ TraditionalForm[{(Rational[1, 3] 3^Rational[-1, 2]) (3 - 2 $CellContext`t)^Rational[3, 2], ( Rational[2, 3] Rational[2, 3]^Rational[1, 2]) $CellContext`t^ Rational[3, 2], 0}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=\!\(\*FractionBox[\(3\), \ \(2\)]\), One Link of Astroid \!\(\*\nStyleBox[\"\[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP]\",\nFontSize->14]\) ", 15]}], $CellContext`f6 -> Row[{ Style[ TraditionalForm[{ Cos[$CellContext`t]^3, Sin[$CellContext`t]^3, 0}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=2\[Pi], Astroid ", 15]}], $CellContext`f7 -> Row[{ Style[ TraditionalForm[{ Sin[$CellContext`t^Rational[1, 2]], Cos[$CellContext`t^ Rational[1, 2]], ( Rational[1, 2] (4 - $CellContext`t^(-1))^ Rational[1, 2]) $CellContext`t + Rational[-1, 8] Log[-1 + (8 + 4 (4 - $CellContext`t^(-1))^ Rational[1, 2]) $CellContext`t]}], 15], Style[ ", a=1 \[LessEqual] t \[LessEqual] b=50, Root Helix \!\(\*\n\ StyleBox[\"\[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP]\",\nFontSize->14]\) ", 15]}], $CellContext`f8 -> Row[{ Style[ TraditionalForm[{$CellContext`t, Rational[1, 2]/$CellContext`t, 0}], 15], Style[ ", a=-3 \[LessEqual] t \[LessEqual] b=3, Hyperbola ", 15]}], $CellContext`f9 -> Row[{ Style[ TraditionalForm[{ Sin[$CellContext`t], Cos[2 $CellContext`t], Cos[$CellContext`t]}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=2\[Pi], Trigonometric \ Saddle", 15]}], $CellContext`f10 -> Row[{ Style[ TraditionalForm[{ 2 ArcCos[1 + Rational[-1, 4] $CellContext`t] - Sin[ 2 ArcCos[1 + Rational[-1, 4] $CellContext`t]], ( Rational[-1, 8] (-8 + $CellContext`t)) $CellContext`t, 0}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=8, One Link of Cycloid \ \!\(\*\nStyleBox[\"\[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\ \[DoubleStruckCapitalP]\",\nFontSize->14]\) ", 15]}], $CellContext`f11 -> Row[{ Style[ TraditionalForm[{$CellContext`t - Sin[$CellContext`t], 1 - Cos[$CellContext`t], 0}], 15], Style[ ", a=0 \[LessEqual] t \[LessEqual] b=8\[Pi], Cycloid ", 15]}], $CellContext`f12 -> Row[{ Style[ TraditionalForm[{ Sin[$CellContext`t^2], Cos[$CellContext`t^2], $CellContext`t^2}], 15], Style[ ", a=-2 \[LessEqual] t \[LessEqual] b=2, Double Traced Helix", 15]}]}, ControlType -> PopupMenu, FieldSize -> { 95, 1.5}}, {{$CellContext`VEC$$, {$CellContext`ShowVect}, Row[{ Spacer[60], Style[ "\!\(\*\nStyleBox[\"VECTORS\",\n\ FontVariations->{\"Underline\"->True}]\)", 12, Bold], Spacer[19]}]}, {$CellContext`RVect -> Style["\!\(\*\nStyleBox[OverscriptBox[\"r\", \"\[Rule]\"],\n\ FontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`VVect -> Style["\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\"v\", \" \"}], \ \"\[Rule]\"],\nFontSize->14,\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`AVect -> Style["\!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[1, 0, 0]]\) ", { GrayLevel[0], 10, Bold}], $CellContext`AtVect -> Style["\!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"t\"], \ \"\[Rule]\"],\nFontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\n\ StyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`AnVect -> Style["\!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"n\"], \ \"\[Rule]\"],\nFontSize->14,\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\n\ StyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ALPVVect -> Style["ALP \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ALPAVect -> Style["ALP \!\(\*\nStyleBox[OverscriptBox[\"a\", \n RowBox[{\" \", \ \"\[Rule]\"}]],\nFontSize->14,\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ShowVect -> Style["\!\(\*\nStyleBox[\"All\",\nFontColor->GrayLevel[0]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)", { GrayLevel[0], 10, Bold}], $CellContext`RemoveVect -> Style["\!\(\*\nStyleBox[\"None\",\nFontColor->GrayLevel[0]]\)\!\(\*\ \nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)", { GrayLevel[0], 10, Bold}], $CellContext`TRANSLATEVect -> Style["Translate Vectors", { GrayLevel[0], 10, Bold}]}, ControlType -> CheckboxBar}, {{$CellContext`GRH$$, {$CellContext`CURVE}, Row[{ Spacer[60], Style[ "\!\(\*\nStyleBox[\"GRAPHS\",\n\ FontVariations->{\"Underline\"->True}]\)", 12, Bold], Spacer[19]}]}, {$CellContext`CURVE -> Style["\!\(\*\nStyleBox[OverscriptBox[\"r\", \"\[Rule]\"],\n\ FontSize->14]\)\!\(\*\nStyleBox[\"(\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\"t\",\nFontSize->14]\)\!\(\*\nStyleBox[\")\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`VT -> Style["\!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\"(\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\"t\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\")\",\nFontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`AT -> Style["\!\(\*\nStyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\"(\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\"t\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\")\",\nFontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`AtT -> Style["\!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"t\"], \ \"\[Rule]\"],\nFontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\n\ StyleBox[\"(\",\nFontSize->14]\)\!\(\*\nStyleBox[\"t\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\")\",\nFontSize->14]\)\!\(\*\nStyleBox[\" \ \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`AnT -> Style["\!\(\*\nStyleBox[OverscriptBox[SubscriptBox[\"a\", \"n\"], \ \"\[Rule]\"],\nFontSize->14,\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\n\ StyleBox[\"(\",\nFontSize->14]\)\!\(\*\nStyleBox[\"t\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\")\",\nFontSize->14]\)\!\(\*\nStyleBox[\" \ \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ALPV -> Style["ALP \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\"(\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\"t\",\nFontSize->14]\)\!\(\*\nStyleBox[\")\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ALPA -> Style["ALP \!\(\*\nStyleBox[OverscriptBox[\"a\", \n RowBox[{\" \", \ \"\[Rule]\"}]],\nFontSize->14,\nFontColor->RGBColor[0.5019607843137255, 0., \ 0.25098039215686274`]]\)\!\(\*\nStyleBox[\"(\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\"t\",\nFontSize->14]\)\!\(\*\nStyleBox[\")\",\n\ FontSize->14]\)\!\(\*\nStyleBox[\" \",\nFontSize->14]\)", { GrayLevel[0], 10, Bold}], $CellContext`ShowGr -> Style["\!\(\*\nStyleBox[\"All\",\nFontColor->GrayLevel[0]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)", { GrayLevel[0], 10, Bold}], $CellContext`RemoveGRphs -> Style["\!\(\*\nStyleBox[\"None\",\nFontColor->GrayLevel[0]]\)\!\(\*\ \nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)", { GrayLevel[0], 10, Bold}], $CellContext`TRANSLATEGraphs -> Style["Translate Graphs \!\(\*\nStyleBox[\"MOTION\",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"ALONG\",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"THE\",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"CURVE\",\n\ FontSize->14,\nFontColor->RGBColor[0, 0, 1]]\) ", { GrayLevel[0], 10, Bold}]}, ControlType -> CheckboxBar}, {{$CellContext`TEXT$$, 1, Row[{ Spacer[60], Style[ "\!\(\*\nStyleBox[\"CONTENT\",\n\ FontVariations->{\"Underline\"->True}]\)", 12, Bold], Spacer[19]}]}, { 1 -> Invisible["1234567"], 2 -> Invisible["123456"], 3 -> Invisible["1234567"], 4 -> Invisible["1234567"], 5 -> Invisible["12345678"], 6 -> Invisible["123456789101"], 7 -> Invisible["123456789101"], 0 -> Row[{"\!\(\*\nStyleBox[\"HELP\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0.5, 0]]\)", Invisible["123456789101213141516171819202122232425262728"]}]}, Enabled -> Dynamic[$CellContext`text$$], ControlType -> RadioButtonBar}, " \!\(\*\nStyleBox[\"IMAGE\",\nFontSize->16,\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0, 0]]\)", " Quality", {{$CellContext`PlotPoints3D$$, 80, ""}, { 1, 2, 3, 4, 5, 10, 20, 30, 40, 50, 60, 70, 80 -> "\!\(\*\nStyleBox[\"80\",\nFontColor->RGBColor[0, 0, 1]]\)", 90, 100, 110, 120, 130, 140, 150}, ControlType -> PopupMenu}, " Speed", {{$CellContext`Recursion$$, Automatic, ""}, { 2 -> "Fast", Automatic -> "\!\(\*\nStyleBox[\"Auto\",\nFontColor->RGBColor[0, 0, 1]]\)", 15 -> "Slow"}, ControlType -> PopupMenu}, "", " Color", " Coded", {{$CellContext`ColorCode$$, False, ""}, {True, False}}, "", " 2D View", {{$CellContext`twoD$$, False, ""}, {True, False}}, "", " \!\(\*\nStyleBox[\"Origin\",\nFontSize->14,\n\ FontColor->GrayLevel[0]]\)", " \!\(\*\nStyleBox[\"View\",\nFontSize->14,\n\ FontColor->GrayLevel[0]]\)", {{$CellContext`OriginZoom$$, False, ""}, { True, False}}, "", " \!\(\*\nStyleBox[\"Unit\",\nFontColor->GrayLevel[0]]\)", "\!\(\*\nStyleBox[\" \",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\"Sphere\",\n\ FontColor->GrayLevel[0]]\)", {{$CellContext`UnitSphere$$, False, ""}, { True, False}}, " Opacity", {{$CellContext`opacity$$, 0.5, ""}, 0, 1, AppearanceElements -> {"PlayButton", "PauseButton"}, AutoAction -> True, AnimationRate -> 0.1, ControlType -> Trigger}, "", Style[ "\!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\) = \ \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\ \[Rule]\)]\)n", 13, RGBColor[1, 0, 0], Bold, Background -> RGBColor[0.87, 0.94, 1]], {{$CellContext`add$$, True, ""}, { True, False}, Enabled -> Dynamic[ And[ MemberQ[$CellContext`VEC$$, $CellContext`AVect], MemberQ[$CellContext`VEC$$, $CellContext`AtVect], MemberQ[$CellContext`VEC$$, $CellContext`AnVect]]]}, "", Delimiter, "", {{$CellContext`text$$, False, ""}, {True, False}, Appearance -> Large}, " \!\(\*\nStyleBox[\"TEXT\",\nFontSize->16,\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[1, 0, 0]]\)", Dynamic[ Row[{ Spacer[11], Framed[ Style[$CellContext`font$$, Bold, 15]]}]], {{$CellContext`font$$, 0, ""}, -10, 10, 1, AutoAction -> False, Enabled -> Dynamic[$CellContext`text$$], AppearanceElements -> {"StepRightButton", "StepLeftButton"}, ControlType -> Trigger}, {{$CellContext`u$$, 1, Row[{"\!\(\*\nStyleBox[\"\[Copyright]\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"N\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\".\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Bykov\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\",\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"SJ\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Delta\",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"College\",\n\ FontColor->RGBColor[0, 0, 1]]\)", Spacer[300], "\!\(\*\nStyleBox[\"t\",\nFontSize->24,\nFontColor->RGBColor[0, \ 0, 1]]\)"}]}, Dynamic[$CellContext`a$$], Dynamic[$CellContext`b$$], 0.001, Appearance -> "Open", ImageSize -> Large, AutoAction -> True}, {$CellContext`a$$, 0, 1, ControlType -> None}, {$CellContext`b$$, 0, 1, ControlType -> None}}, "Options" :> { ControlPlacement -> { Left, Left, Left, Left, Left, Left, Left, Left, Left, Left, Left, Left, Left, Left, Left, Left, Left, Left, Left, Top, Top, Top, Top, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Right, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom, Bottom}, Alignment -> {Center, Center}, FrameMargins -> 0, PreserveImageOptions -> True, AppearanceElements -> All, ContinuousAction -> True, SynchronousUpdating -> True, TrackedSymbols -> Manipulate}, "DefaultOptions" :> {}], ImageSizeCache->{1130., {394., 399.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({{$CellContext`f[ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`y, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]], Pattern[$CellContext`c, Blank[]]] := $CellContext`a $CellContext`x^2 + ($CellContext`b \ $CellContext`x) $CellContext`y + $CellContext`c $CellContext`y^2, \ $CellContext`f[ Pattern[$CellContext`t, Blank[]]] := {2 Sin[$CellContext`t], Cos[$CellContext`t], 0}, $CellContext`x = -1.999999718426341, $CellContext`y = \ -0.9999998830731719, $CellContext`a0 = 0, $CellContext`b0 = 2 Pi, $CellContext`Fnumber = 1, $CellContext`ptsize = 0.015, $CellContext`jumps = None, $CellContext`asymptote = Graphics[{}], $CellContext`LABEL = Text["ELLIPSE\n\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)and \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graphs coincide with the curve; \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) is the opposite of \ \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\); \!\(\*\nStyleBox[OverscriptBox[\n \ RowBox[{\" \", SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 0]]\)and\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\ \" \", SubscriptBox[\"a\", \"n\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, \ 1, 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 1]]\)graphs turn \ into each other under translation;\n\!\(\*\nStyleBox[\"Things\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"to\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\): Compare \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\),\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\) and \!\(\ \*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 0]]\),\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graphs; Increase the \!\(\*\nStyleBox[\"Domain\",\nFontSlant->\"Italic\"]\)\!\ \(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)to maximum, due to rendering \ errors the \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\) graph will fill in the \ ellipse; \!\(\*\nStyleBox[\"2\",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\ \"Italic\"]\)\!\(\*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\);\n\!\(\*\n\ StyleBox[\"VISTA\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) Get \ \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \ \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\),\!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"n\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 1]]\)and \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)graphs \ and vectors on the screen, deploy \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\) = \ \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\ \[Rule]\)]\)n diagram, go to full screen view, \!\(\*\n\ StyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) graphs and vectors, watch \ in slow motion;"], $CellContext`\[Lambda] = 1, $CellContext`\[Mu] = 1, $CellContext`TV = 0, $CellContext`TG = 0, $CellContext`RESCALE = 1, $CellContext`IMSIZE = {979, 540}, $CellContext`P1 = Graphics3D[{{ GrayLevel[0], Arrow[{{0, 0, 0}, {2 Sin[1], Cos[1], 0}}]}}], $CellContext`P2 = Graphics3D[{{ RGBColor[0, 1, 0], Arrow[{{0, 0, 0}, {2 Cos[1], -Sin[1], 0}}]}}], $CellContext`V[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Derivative[1][$CellContext`f][$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Attributes[Derivative] = {NHoldAll, ReadProtected}, $CellContext`P3 = Graphics3D[{{ RGBColor[1, 0, 0], Arrow[{{0, 0, 0}, {(-2) Sin[1], -Cos[1], 0}}]}}], $CellContext`A[ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]], Pattern[$CellContext`c, Blank[]]] := {{ 2 $CellContext`a, $CellContext`b}, {$CellContext`b, 2 $CellContext`c}}, $CellContext`A[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Derivative[1][$CellContext`V][$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`P4 = Graphics3D[{{ RGBColor[0, 0, 1], Arrow[{{0, 0, 0}, {(((-6) Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}], $CellContext`At[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`at[$CellContext`t] \ $CellContext`Tn[$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`at[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Refine[ Factor[(1/$CellContext`S[$CellContext`t]) (Part[ $CellContext`V[$CellContext`t], 1] Part[ $CellContext`A[$CellContext`t], 1] + Part[ $CellContext`V[$CellContext`t], 2] Part[ $CellContext`A[$CellContext`t], 2] + Part[ $CellContext`V[$CellContext`t], 3] Part[ $CellContext`A[$CellContext`t], 3])], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Element[$CellContext`t, Reals]], $CellContext`S[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Refine[ Sqrt[ Factor[Part[ $CellContext`V[$CellContext`t], 1]^2 + Part[ $CellContext`V[$CellContext`t], 2]^2 + Part[ $CellContext`V[$CellContext`t], 3]^2]], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`Tn[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`V[$CellContext`t]/$CellContext`S[$CellContext`\ t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`P5 = Graphics3D[{{ RGBColor[0, 1, 1], Arrow[{{0, 0, 0}, {((-4) Sin[1])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]), 0}}]}}], $CellContext`An[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`A[$CellContext`t] - \ $CellContext`At[$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`P7 = Graphics3D[{{ RGBColor[0, 4/9, 0], Arrow[{{0, 0, 0}, {(2 Cos[1])/Sqrt[ 4 Cos[1]^2 + Sin[1]^2], -(Sin[1]/Sqrt[4 Cos[1]^2 + Sin[1]^2]), 0}}]}}], $CellContext`Tn1[ Pattern[$CellContext`t, Blank[]]] := {(2 Cos[$CellContext`t])/Sqrt[ 4 Cos[$CellContext`t]^2 + Sin[$CellContext`t]^2], -( Sin[$CellContext`t]/Sqrt[ 4 Cos[$CellContext`t]^2 + Sin[$CellContext`t]^2]), 0}, $CellContext`Tn6[ Pattern[$CellContext`t, Blank[]]] := {-((Cos[$CellContext`t]^2 Sin[$CellContext`t])/(Abs[ Cos[$CellContext`t]] Abs[ Sin[$CellContext`t]])), (Cos[$CellContext`t] Sin[$CellContext`t]^2)/(Abs[ Cos[$CellContext`t]] Abs[ Sin[$CellContext`t]]), 0}, $CellContext`Tn9[ Pattern[$CellContext`t, Blank[]]] := { Cos[$CellContext`t]/Sqrt[ 3 + 2 Cos[4 $CellContext`t]], (2 Cos[2 $CellContext`t])/Sqrt[ 3 + 2 Cos[4 $CellContext`t]], -(Sin[$CellContext`t]/Sqrt[ 3 + 2 Cos[4 $CellContext`t]])}, $CellContext`Tn11[ Pattern[$CellContext`t, Blank[]]] := {1/Abs[ Csc[$CellContext`t/2]], Sin[$CellContext`t]/Sqrt[ 2 - 2 Cos[$CellContext`t]], 0}, $CellContext`Tn12[ Pattern[$CellContext`t, Blank[]]] := {($CellContext`t Cos[$CellContext`t^2])/(Sqrt[2] Abs[$CellContext`t]), -(($CellContext`t Sin[$CellContext`t^2])/( Sqrt[2] Abs[$CellContext`t])), $CellContext`t/(Sqrt[2] Abs[$CellContext`t])}, $CellContext`P8 = Graphics3D[{{ RGBColor[4/9, 0, 0], Arrow[{{0, 0, 0}, {((-8) Sin[1])/(5 + 3 Cos[2])^2, ((-16) Cos[1])/(5 + 3 Cos[2])^2, 0}}]}}], $CellContext`AnALP1[ Pattern[$CellContext`t, Blank[]]] := {-((8 Sin[$CellContext`t])/(5 + 3 Cos[2 $CellContext`t])^2), -((16 Cos[$CellContext`t])/(5 + 3 Cos[2 $CellContext`t])^2), 0}, $CellContext`AnALP6[ Pattern[$CellContext`t, Blank[]]] := {((Abs[ Tan[$CellContext`t]] Cos[$CellContext`t]^3) Sin[$CellContext`t]^2)/(((3 Abs[ Cos[$CellContext`t]]^3) Abs[ Sin[$CellContext`t]]^3) Piecewise[{{-1, (Cos[$CellContext`t] Sin[$CellContext`t]) Sin[2 $CellContext`t] < 0}}, 1]), ((Abs[ Cot[$CellContext`t]] Cos[$CellContext`t]^2) Sin[$CellContext`t]^3)/(((3 Abs[ Cos[$CellContext`t]]^3) Abs[ Sin[$CellContext`t]]^3) Piecewise[{{-1, (Cos[$CellContext`t] Sin[$CellContext`t]) Sin[2 $CellContext`t] < 0}}, 1]), 0}, $CellContext`AnALP9[ Pattern[$CellContext`t, Blank[]]] := {((-3) Sin[$CellContext`t] + 3 Sin[3 $CellContext`t] + Sin[5 $CellContext`t])/(3 + 2 Cos[4 $CellContext`t])^2, -((4 Sin[2 $CellContext`t])/(3 + 2 Cos[4 $CellContext`t])^2), ((-3) (Cos[$CellContext`t] + Cos[3 $CellContext`t]) + Cos[5 $CellContext`t])/(3 + 2 Cos[4 $CellContext`t])^2}, $CellContext`AnALP11[ Pattern[$CellContext`t, Blank[]]] := {-((( Cos[$CellContext`t/2] (-1 + Cos[$CellContext`t])) Sign[ Csc[$CellContext`t/2]])/(8 Abs[ Sin[$CellContext`t/2]]^3)), -(1/4), 0}, $CellContext`AnALP12[ Pattern[$CellContext`t, Blank[]]] := {(-(1/2)) Sin[$CellContext`t^2], (-(1/2)) Cos[$CellContext`t^2], 0}, $CellContext`AnALP[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`Cv[$CellContext`t] \ $CellContext`Nr[$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`Cv[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`NCross[$CellContext`t]/$CellContext`S[$\ CellContext`t]^3, Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`NCross[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Refine[ Sqrt[ Factor[Part[ Cross[ $CellContext`V[$CellContext`t], Derivative[1][$CellContext`V][$CellContext`t]], 1]^2 + Part[ Cross[ $CellContext`V[$CellContext`t], Derivative[1][$CellContext`V][$CellContext`t]], 2]^2 + Part[ Cross[ $CellContext`V[$CellContext`t], Derivative[1][$CellContext`V][$CellContext`t]], 3]^2]], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Element[$CellContext`t, Reals]], $CellContext`Nr[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`DTn[$CellContext`t]/$CellContext`DTnNorm[$\ CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`DTn[ Pattern[$CellContext`t, Blank[]]] := FullSimplify[ Derivative[1][$CellContext`Tn][$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`DTnNorm[ Pattern[$CellContext`t, Blank[]]] := FullSimplify[ Refine[ Sqrt[ Factor[Part[ Derivative[1][$CellContext`Tn][$CellContext`t], 1]^2 + Part[ Derivative[1][$CellContext`Tn][$CellContext`t], 2]^2 + Part[ Derivative[1][$CellContext`Tn][$CellContext`t], 3]^2]], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`P9 = Graphics3D[{{ RGBColor[0, 0, 1], Line[{{((-4) Sin[1])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]), 0}, {((-4) Sin[1])/(5 + 3 Cos[2]) - ((6 Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]) + ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}], $CellContext`P10 = Graphics3D[{{ RGBColor[0, 1, 1], Line[{{(((-6) Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}, {((-4) Sin[1])/(5 + 3 Cos[2]) - ((6 Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]) + ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}], $CellContext`an[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`NCross[$CellContext`t]/$CellContext`S[$\ CellContext`t]^2, Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`Curve = Graphics3D[{{{}, {}, { GrayLevel[0], Thickness[Large], Line[CompressedData[" 1:eJxd13k4lGsfB3CSLS1eS7JFcSprRkyF0x0S6kRJTgsqSZRIG07pCFkOQttR WdMyVMq+Nd1lHyM7M4QiNJOtsTPMc+Zc1/vet+t9ruv55/PHM/dzP/fvN7/v Ohdvu9NLBAQEXvJvIf69/9yuivAjt3b0Ef9eo0Dgv9fLPEs58v27ID9ITzVl ka9RJMtO3UsDlLjG4EAedqYUu/qz82vAjUoipc5jv8l9BsXu5QD7uHqDJXPY h8xC9+j6FYKbBvKJaVPYs0ha5I/HqODExlc3bo1hH7OfXeIUWwY0HTpkJFjY 5eLsLa4aVoJl7qEnlHuw7x0TUR1vrwYUOkPDth27jHa2/ifVOiB501dOqgq7 Sv7doIUDLaDBVmtAKR67RXLoOkXxbsCq1eyzUMT+sdDX/3TWN+DmbFgSeXEE eZftlqyUJyywx+naKlr4MPI/FfKbROWHQeIVw6ZjgUPIyZ/2tj3u/glUaBOX yaRB5IwUalyZ2DhgmcSmkovYeJ1PTqS6/pwAA/Ynwkr1Wcj7lZiQ2TUFUs8r xr9zGUDO+dL1++XQGZBSEMhSvt2HPHTr0SD3JXMg7bxCsNnTXuSjXUqVVy9x AWXG15jj8xX55/R7bPXUedDl49u48e9O5OU7w2jBBQtAQs144i2VgTwD7kyq 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$CellContext`PRange = {{-2.999999718426341, 2.9999997564224232`}, {-1.9999998830731718`, 1.9999999999999918`}, {-2., 2.}}, $CellContext`GRAPH = Graphics3D[{{{ GrayLevel[0], Arrow[{{0, 0, 0}, {2 Sin[1], Cos[1], 0}}]}}, {{ RGBColor[0, 1, 0], Arrow[{{0, 0, 0}, {2 Cos[1], -Sin[1], 0}}]}}, {{ RGBColor[1, 0, 0], Arrow[{{0, 0, 0}, {(-2) Sin[1], -Cos[1], 0}}]}}, {{ RGBColor[0, 0, 1], Arrow[{{0, 0, 0}, {(((-6) Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}, {{ RGBColor[0, 1, 1], Arrow[{{0, 0, 0}, {((-4) Sin[1])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]), 0}}]}}, {{ RGBColor[0, 4/9, 0], Arrow[{{0, 0, 0}, {(2 Cos[1])/Sqrt[ 4 Cos[1]^2 + Sin[1]^2], -(Sin[1]/Sqrt[ 4 Cos[1]^2 + Sin[1]^2]), 0}}]}}, {{ RGBColor[4/9, 0, 0], Arrow[{{0, 0, 0}, {((-8) Sin[1])/(5 + 3 Cos[2])^2, ((-16) Cos[1])/(5 + 3 Cos[2])^2, 0}}]}}, {{ RGBColor[0, 0, 1], Line[{{((-4) Sin[1])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]), 0}, {((-4) Sin[1])/(5 + 3 Cos[2]) - ((6 Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]) + ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}, {{ RGBColor[0, 1, 1], Line[{{(((-6) Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}, {((-4) Sin[1])/(5 + 3 Cos[2]) - ((6 Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]) + ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}, {{{}, {}, { GrayLevel[0], Thickness[Large], Line[CompressedData[" 1:eJxd13k4lGsfB3CSLS1eS7JFcSprRkyF0x0S6kRJTgsqSZRIG07pCFkOQttR WdMyVMq+Nd1lHyM7M4QiNJOtsTPMc+Zc1/vet+t9ruv55/PHM/dzP/fvN7/v Ohdvu9NLBAQEXvJvIf69/9yuivAjt3b0Ef9eo0Dgv9fLPEs58v27ID9ITzVl ka9RJMtO3UsDlLjG4EAedqYUu/qz82vAjUoipc5jv8l9BsXu5QD7uHqDJXPY h8xC9+j6FYKbBvKJaVPYs0ha5I/HqODExlc3bo1hH7OfXeIUWwY0HTpkJFjY 5eLsLa4aVoJl7qEnlHuw7x0TUR1vrwYUOkPDth27jHa2/ifVOiB501dOqgq7 Sv7doIUDLaDBVmtAKR67RXLoOkXxbsCq1eyzUMT+sdDX/3TWN+DmbFgSeXEE eZftlqyUJyywx+naKlr4MPI/FfKbROWHQeIVw6ZjgUPIyZ/2tj3u/glUaBOX yaRB5IwUalyZ2DhgmcSmkovYeJ1PTqS6/pwAA/Ynwkr1Wcj7lZiQ2TUFUs8r 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Internal`PopupWindowNotebook[ Text[ "ELLIPSE\n\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)and \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graphs coincide with the curve; \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) is the opposite of \ \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\); \!\(\*\nStyleBox[OverscriptBox[\n \ RowBox[{\" \", SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 0]]\)and\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\ \" \", SubscriptBox[\"a\", \"n\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, \ 1, 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 1]]\)graphs turn \ into each other under translation;\n\!\(\*\nStyleBox[\"Things\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"to\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\): Compare \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\),\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\) and \!\(\ \*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 0]]\),\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graphs; Increase the \!\(\*\nStyleBox[\"Domain\",\nFontSlant->\"Italic\"]\)\!\ \(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)to maximum, due to rendering \ errors the \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\) graph will fill in the \ ellipse; \!\(\*\nStyleBox[\"2\",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\ \"Italic\"]\)\!\(\*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\);\n\!\(\*\n\ StyleBox[\"VISTA\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) Get \ \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \ \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\),\!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"n\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 1]]\)and \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)graphs \ and vectors on the screen, deploy \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\) = \ \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\ \[Rule]\)]\)n diagram, go to full screen view, \!\(\*\n\ StyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) graphs and vectors, watch \ in slow motion;"]]], Appearance -> None, BaseStyle -> {}, DefaultBaseStyle -> {}], " ", Style["r(t) = ", Bold], Style[ TraditionalForm[{2 Sin[$CellContext`t], Cos[$CellContext`t], 0}], Bold], ";", " ", Style[0, RGBColor[0, 0, 1], Bold], Style["\[LessEqual]", Bold], Style[$CellContext`t, RGBColor[0, 0, 1], Bold], Style["\[LessEqual]", Bold], Style[2 Pi, RGBColor[0, 0, 1], Bold], ";", " ", Style["r(", Bold], Style[1, RGBColor[0, 0, 1], Bold], Style[") = ", Bold], Style[{1.68, 0.54, 0}, RGBColor[1, 0, 1], Bold]}], Graphics3D[{{{ GrayLevel[0], Arrow[{{0, 0, 0}, {2 Sin[1], Cos[1], 0}}]}}, {{ RGBColor[0, 1, 0], Arrow[{{0, 0, 0}, {2 Cos[1], -Sin[1], 0}}]}}, {{ RGBColor[1, 0, 0], Arrow[{{0, 0, 0}, {(-2) Sin[1], -Cos[1], 0}}]}}, {{ RGBColor[0, 0, 1], Arrow[{{0, 0, 0}, {(((-6) Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}, {{ RGBColor[0, 1, 1], Arrow[{{0, 0, 0}, {((-4) Sin[1])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]), 0}}]}}, {{ RGBColor[0, 4/9, 0], Arrow[{{0, 0, 0}, {(2 Cos[1])/Sqrt[ 4 Cos[1]^2 + Sin[1]^2], -(Sin[1]/Sqrt[ 4 Cos[1]^2 + Sin[1]^2]), 0}}]}}, {{ RGBColor[4/9, 0, 0], Arrow[{{0, 0, 0}, {((-8) Sin[1])/(5 + 3 Cos[2])^2, ((-16) Cos[1])/(5 + 3 Cos[2])^2, 0}}]}}, {{ RGBColor[0, 0, 1], Line[{{((-4) Sin[1])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]), 0}, {((-4) Sin[1])/(5 + 3 Cos[2]) - ((6 Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]) + ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}, {{ RGBColor[0, 1, 1], Line[{{(((-6) Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}, {((-4) Sin[1])/(5 + 3 Cos[2]) - ((6 Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]) + ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}, {{{}, {}, { GrayLevel[0], Thickness[Large], Line[CompressedData[" 1:eJxd13k4lGsfB3CSLS1eS7JFcSprRkyF0x0S6kRJTgsqSZRIG07pCFkOQttR WdMyVMq+Nd1lHyM7M4QiNJOtsTPMc+Zc1/vet+t9ruv55/PHM/dzP/fvN7/v Ohdvu9NLBAQEXvJvIf69/9yuivAjt3b0Ef9eo0Dgv9fLPEs58v27ID9ITzVl ka9RJMtO3UsDlLjG4EAedqYUu/qz82vAjUoipc5jv8l9BsXu5QD7uHqDJXPY h8xC9+j6FYKbBvKJaVPYs0ha5I/HqODExlc3bo1hH7OfXeIUWwY0HTpkJFjY 5eLsLa4aVoJl7qEnlHuw7x0TUR1vrwYUOkPDth27jHa2/ifVOiB501dOqgq7 Sv7doIUDLaDBVmtAKR67RXLoOkXxbsCq1eyzUMT+sdDX/3TWN+DmbFgSeXEE eZftlqyUJyywx+naKlr4MPI/FfKbROWHQeIVw6ZjgUPIyZ/2tj3u/glUaBOX yaRB5IwUalyZ2DhgmcSmkovYeJ1PTqS6/pwAA/Ynwkr1Wcj7lZiQ2TUFUs8r 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FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Controls\",\n\ FontVariations->{\"Underline\"->True}]\). Main pop up menu supplies a vector \ function and its initial domain. First, go over the topics in the \ \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\) column, then move to the right, do \ not clutter the graph with all possible objects, you can always arrange a \ combination that you want to study later. \n\!\(\*\nStyleBox[\"Before\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"switching\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"to\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"a\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"new\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"vector\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"function\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\",\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"press\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"RESET\",\n\ FontSlant->\"Italic\",\nFontColor->RGBColor[1, 0.5, 0]]\) button in the right \ top corner to return the domain and range to initial positions matching the \ new function. You can rotate space graphs with your mouse.\n\n2. \!\(\*\n\ StyleBox[\"Left\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\ \" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"Controls\",\nFontVariations->{\"Underline\"->True}]\). You added a \ new graph, but nothing happened. Use the slider to increase the \!\(\*\n\ StyleBox[\"Range\",\nFontSlant->\"Italic\"]\) gently, chose 10, 20,... \!\(\*\ \nStyleBox[\"Scale\",\nFontSlant->\"Italic\"]\) for rapid zoom out and 1/10, \ 1/20,... \!\(\*\nStyleBox[\"Scale\",\nFontSlant->\"Italic\"]\) for rapid zoom \ in. Separate \!\(\*\nStyleBox[\"Domain\",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"a\",\n\ FontColor->RGBColor[0, 0, 1]]\) and \!\(\*\nStyleBox[\"b\",\n\ FontColor->RGBColor[0, 0, 1]]\) controls allow you to see the curve for more \ or fewer values of parameter t. Domain expands and shrinks about the midpoint \ of the original [a,b] interval. If vectors you want to see are too small or \ too large use \!\(\*\nStyleBox[\"Vectors\",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\" \",\nFontSlant->\"Italic\"]\)controls, press \!\(\*\n\ StyleBox[\"Reset1\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)to return vectors to their genuine length. \n\n3. \ \!\(\*\nStyleBox[\"Right\",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\" \",\nFontVariations->{\"Underline\"->True}]\)\!\(\*\n\ StyleBox[\"Controls\",\nFontVariations->{\"Underline\"->True}]\). \!\(\*\n\ StyleBox[\"Speed\",\nFontSlant->\"Italic\"]\) and \!\(\*\n\ StyleBox[\"Quality\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)allow to compromise between the rate of graph \ rendering and its smoothness. \!\(\*\nStyleBox[\"Color\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\"Coded\",\nFontSlant->\"Italic\"]\) provides easier match \ between the space graph and supporting plane graphs, though it makes the \ graphs less readable and the program runs slower. \!\(\*\nStyleBox[\"2\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"View\",\n\ FontSlant->\"Italic\"]\) is useful if you study a flat curve (third component \ of the vector function is 0). \!\(\*\nStyleBox[\"Origin\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\ \*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\) and \!\(\*\n\ StyleBox[\"Unit\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\"Sphere\",\n\ FontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)with \ controlled opacity are useful, especially, for \!\(\*\nStyleBox[\"\ \[DoubleStruckCapitalA]\[DoubleStruckCapitalL]\[DoubleStruckCapitalP]\",\n\ FontSize->10]\) \!\(\*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\n\ FontSize->14,\nFontColor->RGBColor[0., 0.5019607843137255, \ 0.25098039215686274`]]\). \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\) = \ \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\ \[Rule]\)]\)n checkbox works only, if all three components are displayed. \n\ \!\(\*\nStyleBox[\"To\",\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"see\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"the\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"full\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"screen\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"space\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"graph\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"uncheck\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"Text\",\n\ FontSlant->\"Italic\",\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\" \ \",\nFontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\"button\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\!\(\*\nStyleBox[\".\",\n\ FontColor->RGBColor[1, 0.5, 0]]\)\n\n4. \!\(\*\nStyleBox[\"Bottom\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Control\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)is the way to move along the curve, \ its play feature will animate the graph."], 14], $CellContext`ANorm[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Refine[ Sqrt[ Factor[Part[ $CellContext`A[$CellContext`t], 1]^2 + Part[ $CellContext`A[$CellContext`t], 2]^2 + Part[ $CellContext`A[$CellContext`t], 3]^2]], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], $CellContext`OUTPUT = Column[{ Row[{ Button[ Button[" READ about this GRAPH"], CreateDocument[ Internal`PopupWindowNotebook[ Text[ "ELLIPSE\n\!\(\*\nStyleBox[\"Notice\",\n\ FontVariations->{\"Underline\"->True}]\): \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)and \!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graphs coincide with the curve; \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) is the opposite of \ \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\); \!\(\*\nStyleBox[OverscriptBox[\n \ RowBox[{\" \", SubscriptBox[\"a\", \"t\"]}], \"\[Rule]\"],\n\ FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 0]]\)and\!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\ \" \", SubscriptBox[\"a\", \"n\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, \ 1, 1]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 1]]\)graphs turn \ into each other under translation;\n\!\(\*\nStyleBox[\"Things\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"to\",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True}]\)\!\(\*\nStyleBox[\"Do\",\n\ FontVariations->{\"Underline\"->True}]\): Compare \[DoubleStruckCapitalA]\ \[DoubleStruckCapitalL]\[DoubleStruckCapitalP] \!\(\*\n\ StyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontSize->14,\n\ FontColor->RGBColor[0., 0.5019607843137255, 0.25098039215686274`]]\),\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\) and \!\(\ \*\nStyleBox[OverscriptBox[\"v\", \"\[Rule]\"],\nFontColor->RGBColor[0, 1, \ 0]]\),\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\n\ StyleBox[OverscriptBox[\"a\", \"\[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\) \ graphs; Increase the \!\(\*\nStyleBox[\"Domain\",\nFontSlant->\"Italic\"]\)\!\ \(\*\nStyleBox[\" \",\nFontSlant->\"Italic\"]\)to maximum, due to rendering \ errors the \!\(\*OverscriptBox[\(r\), \(\[Rule]\)]\) graph will fill in the \ ellipse; \!\(\*\nStyleBox[\"2\",\nFontSlant->\"Italic\"]\)\!\(\*\n\ StyleBox[\"D\",\nFontSlant->\"Italic\"]\)\!\(\*\nStyleBox[\" \",\nFontSlant->\ \"Italic\"]\)\!\(\*\nStyleBox[\"View\",\nFontSlant->\"Italic\"]\);\n\!\(\*\n\ StyleBox[\"VISTA\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\)\!\(\*\ \nStyleBox[\"POINT\",\nFontVariations->{\"Underline\"->True},\n\ FontColor->RGBColor[0, 1, 0]]\)\!\(\*\nStyleBox[\":\",\n\ FontVariations->{\"Underline\"->True},\nFontColor->RGBColor[0, 1, 0]]\) Get \ \!\(\*\nStyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \ \"t\"]}], \"\[Rule]\"],\nFontColor->RGBColor[0, 0, 1]]\),\!\(\*\n\ StyleBox[OverscriptBox[\n RowBox[{\" \", SubscriptBox[\"a\", \"n\"]}], \"\ \[Rule]\"],\nFontColor->RGBColor[0, 1, 1]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0, 1, 1]]\)and \!\(\*\nStyleBox[OverscriptBox[\"a\", \"\ \[Rule]\"],\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\ FontColor->RGBColor[0.5019607843137255, 0., 0.25098039215686274`]]\)graphs \ and vectors on the screen, deploy \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\) = \ \!\(\*OverscriptBox[\(A\), \(\[Rule]\)]\)t+\!\(\*OverscriptBox[\(A\), \(\ \[Rule]\)]\)n diagram, go to full screen view, \!\(\*\n\ StyleBox[\"Translate\",\nFontSlant->\"Italic\"]\) graphs and vectors, watch \ in slow motion;"]]], Appearance -> None, BaseStyle -> {}, DefaultBaseStyle -> {}], " ", Style["r(t) = ", Bold], Style[ TraditionalForm[{2 Sin[$CellContext`t], Cos[$CellContext`t], 0}], Bold], ";", " ", Style[0, RGBColor[0, 0, 1], Bold], Style["\[LessEqual]", Bold], Style[$CellContext`t, RGBColor[0, 0, 1], Bold], Style["\[LessEqual]", Bold], Style[2 Pi, RGBColor[0, 0, 1], Bold], ";", " ", Style["r(", Bold], Style[1, RGBColor[0, 0, 1], Bold], Style[") = ", Bold], Style[{1.68, 0.54, 0}, RGBColor[1, 0, 1], Bold]}], Graphics3D[{{{ GrayLevel[0], Arrow[{{0, 0, 0}, {2 Sin[1], Cos[1], 0}}]}}, {{ RGBColor[0, 1, 0], Arrow[{{0, 0, 0}, {2 Cos[1], -Sin[1], 0}}]}}, {{ RGBColor[1, 0, 0], Arrow[{{0, 0, 0}, {(-2) Sin[1], -Cos[1], 0}}]}}, {{ RGBColor[0, 0, 1], Arrow[{{0, 0, 0}, {(((-6) Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}, {{ RGBColor[0, 1, 1], Arrow[{{0, 0, 0}, {((-4) Sin[1])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]), 0}}]}}, {{ RGBColor[0, 4/9, 0], Arrow[{{0, 0, 0}, {(2 Cos[1])/Sqrt[ 4 Cos[1]^2 + Sin[1]^2], -(Sin[1]/Sqrt[ 4 Cos[1]^2 + Sin[1]^2]), 0}}]}}, {{ RGBColor[4/9, 0, 0], Arrow[{{0, 0, 0}, {((-8) Sin[1])/(5 + 3 Cos[2])^2, ((-16) Cos[1])/(5 + 3 Cos[2])^2, 0}}]}}, {{ RGBColor[0, 0, 1], Line[{{((-4) Sin[1])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]), 0}, {((-4) Sin[1])/(5 + 3 Cos[2]) - ((6 Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]) + ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}, {{ RGBColor[0, 1, 1], Line[{{(((-6) Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}, {((-4) Sin[1])/(5 + 3 Cos[2]) - ((6 Cos[1]) Sin[2])/(5 + 3 Cos[2]), ((-8) Cos[1])/(5 + 3 Cos[2]) + ((3 Sin[1]) Sin[2])/(5 + 3 Cos[2]), 0}}]}}, {{{}, {}, { GrayLevel[0], Thickness[Large], Line[CompressedData[" 1:eJxd13k4lGsfB3CSLS1eS7JFcSprRkyF0x0S6kRJTgsqSZRIG07pCFkOQttR WdMyVMq+Nd1lHyM7M4QiNJOtsTPMc+Zc1/vet+t9ruv55/PHM/dzP/fvN7/v 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{-1.9999998830731718`, 1.9999999999999918`}, {-2., 2.}}, ViewPoint -> {1.3, -2.4, 2}, AxesLabel -> { "\!\(\*\nStyleBox[\"X\",\nFontColor->GrayLevel[0]]\)", "\!\(\*\nStyleBox[\"Y\",\nFontColor->GrayLevel[0]]\)", "\!\(\*\nStyleBox[\"Z\",\nFontColor->GrayLevel[0]]\)"}, AspectRatio -> Automatic, ImageSize -> {979, 540}}]}, Left, Frame -> None]}; {$CellContext`st1 = { Black, 10, Bold}; $CellContext`st2 = 15; $CellContext`V[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Derivative[1][$CellContext`f][$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`S[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Refine[Factor[Part[ $CellContext`V[$CellContext`t], 1]^2 + Part[ $CellContext`V[$CellContext`t], 2]^2 + Part[ $CellContext`V[$CellContext`t], 3]^2]^Rational[1, 2], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`A[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Derivative[1][$CellContext`V][$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`ANorm[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Refine[Factor[Part[ $CellContext`A[$CellContext`t], 1]^2 + Part[ $CellContext`A[$CellContext`t], 2]^2 + Part[ $CellContext`A[$CellContext`t], 3]^2]^Rational[1, 2], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`Tn[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`V[$CellContext`t]/$CellContext`S[$CellContext`\ t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`DTn[ Pattern[$CellContext`t, Blank[]]] := FullSimplify[ Derivative[1][$CellContext`Tn][$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`DTnNorm[ Pattern[$CellContext`t, Blank[]]] := FullSimplify[ Refine[Factor[Part[ Derivative[1][$CellContext`Tn][$CellContext`t], 1]^2 + Part[ Derivative[1][$CellContext`Tn][$CellContext`t], 2]^2 + Part[ Derivative[1][$CellContext`Tn][$CellContext`t], 3]^2]^ Rational[1, 2], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`Nr[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`DTn[$CellContext`t]/$CellContext`DTnNorm[$\ CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`NCross[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Refine[Factor[Part[ Cross[ $CellContext`V[$CellContext`t], Derivative[1][$CellContext`V][$CellContext`t]], 1]^2 + Part[ Cross[ $CellContext`V[$CellContext`t], Derivative[1][$CellContext`V][$CellContext`t]], 2]^2 + Part[ Cross[ $CellContext`V[$CellContext`t], Derivative[1][$CellContext`V][$CellContext`t]], 3]^2]^ Rational[1, 2], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Element[$CellContext`t, Reals]]; $CellContext`Cv[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`NCross[$CellContext`t]/$CellContext`S[$\ CellContext`t]^3, Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`at[ Pattern[$CellContext`t, Blank[]]] := Simplify[ Refine[ Factor[(1/$CellContext`S[$CellContext`t]) (Part[ $CellContext`V[$CellContext`t], 1] Part[ $CellContext`A[$CellContext`t], 1] + Part[ $CellContext`V[$CellContext`t], 2] Part[ $CellContext`A[$CellContext`t], 2] + Part[ $CellContext`V[$CellContext`t], 3] Part[ $CellContext`A[$CellContext`t], 3])], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}], Element[$CellContext`t, Reals]]; $CellContext`an[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`NCross[$CellContext`t]/$CellContext`S[$\ CellContext`t]^2, Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`At[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`at[$CellContext`t] \ $CellContext`Tn[$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`An[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`A[$CellContext`t] - \ $CellContext`At[$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`AnALP[ Pattern[$CellContext`t, Blank[]]] := Simplify[$CellContext`Cv[$CellContext`t] \ $CellContext`Nr[$CellContext`t], Element[$CellContext`t, Reals], Assumptions -> {$CellContext`a0 <= $CellContext`t <= \ $CellContext`b0}]; $CellContext`Tn1[ Pattern[$CellContext`t, Blank[]]] := { 2 (Cos[$CellContext`t]/(4 Cos[$CellContext`t]^2 + Sin[$CellContext`t]^2)^Rational[1, 2]), -( Sin[$CellContext`t]/(4 Cos[$CellContext`t]^2 + Sin[$CellContext`t]^2)^Rational[1, 2]), 0}; $CellContext`AnALP1[ Pattern[$CellContext`t, Blank[]]] := {-( 8 (Sin[$CellContext`t]/(5 + 3 Cos[2 $CellContext`t])^2)), -( 16 (Cos[$CellContext`t]/(5 + 3 Cos[2 $CellContext`t])^2)), 0}; $CellContext`Tn6[ Pattern[$CellContext`t, Blank[]]] := {-(Cos[$CellContext`t]^2 (Sin[$CellContext`t]/(Abs[ Cos[$CellContext`t]] Abs[ Sin[$CellContext`t]]))), Cos[$CellContext`t] (Sin[$CellContext`t]^2/(Abs[ Cos[$CellContext`t]] Abs[ Sin[$CellContext`t]])), 0}; $CellContext`AnALP6[ Pattern[$CellContext`t, Blank[]]] := {(Abs[ Tan[$CellContext`t]] Cos[$CellContext`t]^3) ( Sin[$CellContext`t]^2/(((3 Abs[ Cos[$CellContext`t]]^3) Abs[ Sin[$CellContext`t]]^3) Piecewise[{{-1, (Cos[$CellContext`t] Sin[$CellContext`t]) Sin[2 $CellContext`t] < 0}}, 1])), (Abs[ Cot[$CellContext`t]] Cos[$CellContext`t]^2) ( Sin[$CellContext`t]^3/(((3 Abs[ Cos[$CellContext`t]]^3) Abs[ Sin[$CellContext`t]]^3) Piecewise[{{-1, (Cos[$CellContext`t] Sin[$CellContext`t]) Sin[2 $CellContext`t] < 0}}, 1])), 0}; $CellContext`Tn9[ Pattern[$CellContext`t, Blank[]]] := { Cos[$CellContext`t]/(3 + 2 Cos[4 $CellContext`t])^Rational[1, 2], 2 (Cos[2 $CellContext`t]/(3 + 2 Cos[4 $CellContext`t])^ Rational[1, 2]), -( Sin[$CellContext`t]/(3 + 2 Cos[4 $CellContext`t])^ Rational[1, 2])}; $CellContext`AnALP9[ Pattern[$CellContext`t, Blank[]]] := {((-3) Sin[$CellContext`t] + 3 Sin[3 $CellContext`t] + Sin[5 $CellContext`t])/(3 + 2 Cos[4 $CellContext`t])^2, -( 4 (Sin[2 $CellContext`t]/(3 + 2 Cos[4 $CellContext`t])^2)), ((-3) (Cos[$CellContext`t] + Cos[3 $CellContext`t]) + Cos[5 $CellContext`t])/(3 + 2 Cos[4 $CellContext`t])^2}; $CellContext`Tn11[ Pattern[$CellContext`t, Blank[]]] := {1/Abs[ Csc[$CellContext`t/2]], Sin[$CellContext`t]/(2 - 2 Cos[$CellContext`t])^Rational[1, 2], 0}; $CellContext`AnALP11[ Pattern[$CellContext`t, Blank[]]] := {-(( Cos[$CellContext`t/2] (-1 + Cos[$CellContext`t])) (Sign[ Csc[$CellContext`t/2]]/(8 Abs[ Sin[$CellContext`t/2]]^3))), -(1/4), 0}; $CellContext`Tn12[ Pattern[$CellContext`t, Blank[]]] := {$CellContext`t (Cos[$CellContext`t^2]/( 2^Rational[1, 2] Abs[$CellContext`t])), -($CellContext`t (Sin[$CellContext`t^2]/( 2^Rational[1, 2] Abs[$CellContext`t]))), $CellContext`t/( 2^Rational[1, 2] Abs[$CellContext`t])}; $CellContext`AnALP12[ Pattern[$CellContext`t, Blank[]]] := {(-(1/2)) Sin[$CellContext`t^2], (-(1/2)) Cos[$CellContext`t^2], 0}; Null}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{3.469658846125*^9}] }, WindowSize->{1272, 922}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, 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