(* 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[ 48447, 1041] NotebookOptionsPosition[ 31593, 740] NotebookOutlinePosition[ 48533, 1043] CellTagsIndexPosition[ 48490, 1040] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`d$$ = 1, $CellContext`w$$ = 0, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`d$$], 1, Row[{ Spacer[25], Style[ " One to One Transformations of the Unit Square into a Triangle", 15, Bold], Spacer[140]}]}, { 1 -> Style["Differentiable", 15, Bold], 0 -> Style["Piecewise Linear", 15, Bold]}}, {{ 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Style["\!\(\*\nStyleBox[\"S\",\nFontColor->RGBColor[0, 1, \ 0]]\)=T(\!\(\*\nStyleBox[\"R\",\nFontColor->RGBColor[0, 0, 1]]\)), \ (0,0)=T(0,0), \!\(\*\nStyleBox[SubscriptBox[\"Q\", \"i\"],\n\ FontColor->RGBColor[0, 1, 0]]\)=T(\!\(\*\nStyleBox[SubscriptBox[\"P\", \ \"i\"],\nFontColor->RGBColor[0, 0, 1]]\)) , \!\(\*\nStyleBox[\"Q\",\n\ FontColor->RGBColor[0, 1, 0]]\)=T(\!\(\*\nStyleBox[\"P\",\n\ FontColor->RGBColor[0, 0, 1]]\))", Bold, 20]]]], {{0, 0}, {5, 5}}]}]}, Center], {1050, 600}, Alignment -> {Center, Center}]]]], "Specifications" :> {{{$CellContext`d$$, 1, Row[{ Spacer[25], Style[ " One to One Transformations of the Unit Square into a Triangle", 15, Bold], Spacer[140]}]}, { 1 -> Style["Differentiable", 15, Bold], 0 -> Style["Piecewise Linear", 15, Bold]}, ControlType -> RadioButtonBar}, {{$CellContext`w$$, 0, Row[{"\!\(\*\nStyleBox[\"\[Copyright]\",\nFontSize->14]\)\!\(\*\n\ StyleBox[\" \",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"N\",\n\ FontColor->RGBColor[0, 0, 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{}], ImageSizeCache->{1105., {359., 364.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({{$CellContext`M[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Join[ Transpose[{{$CellContext`a, $CellContext`b}}], Transpose[{{$CellContext`c, $CellContext`d}}], 2], $CellContext`f[{ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`y, Blank[]]}, Pattern[$CellContext`w, Blank[]]] := (1/ 2) {$CellContext`w ($CellContext`y^2 - $CellContext`x^2) + \ ($CellContext`w - 1) $CellContext`x + ( 1 - $CellContext`w) $CellContext`y - $CellContext`x \ $CellContext`y + 1, $CellContext`w ($CellContext`x^2 - $CellContext`y^2) + ( 1 - $CellContext`w) $CellContext`x + ($CellContext`w - 1) $CellContext`y - $CellContext`x $CellContext`y + 1}, $CellContext`f[{ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`y, Blank[]]}, { Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Piecewise[{{ Dot[ Join[ Transpose[{ 0.5 ({$CellContext`a, $CellContext`b} - {$CellContext`c, \ $CellContext`d})}], Transpose[{{$CellContext`c, $CellContext`d}}], 2], {$CellContext`x, $CellContext`y}], $CellContext`x <= \ $CellContext`y}, { Dot[ Join[ Transpose[{{$CellContext`a, $CellContext`b}}], Transpose[{ 0.5 ({$CellContext`c, $CellContext`d} - {$CellContext`a, \ $CellContext`b})}], 2], {$CellContext`x, $CellContext`y}], $CellContext`x >= \ $CellContext`y}}], $CellContext`Br[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Round[ Join[ Transpose[{{$CellContext`a, $CellContext`b}}], Transpose[{ 0.5 ({$CellContext`c, $CellContext`d} - {$CellContext`a, \ $CellContext`b})}], 2], 0.1], $CellContext`Ar[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Round[ Join[ Transpose[{ 0.5 ({$CellContext`a, $CellContext`b} - {$CellContext`c, \ $CellContext`d})}], Transpose[{{$CellContext`c, $CellContext`d}}], 2], 0.1], $CellContext`A[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Join[ Transpose[{ 0.5 ({$CellContext`a, $CellContext`b} - {$CellContext`c, \ $CellContext`d})}], Transpose[{{$CellContext`c, $CellContext`d}}], 2], $CellContext`B[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Join[ Transpose[{{$CellContext`a, $CellContext`b}}], Transpose[{ 0.5 ({$CellContext`c, $CellContext`d} - {$CellContext`a, \ $CellContext`b})}], 2], $CellContext`g[{ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`y, Blank[]]}, { Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Piecewise[{{ Dot[ Join[ Transpose[{ 0.5 ({$CellContext`a, $CellContext`b} - {$CellContext`c, \ $CellContext`d})}], Transpose[{{$CellContext`c, $CellContext`d}}], 2], {$CellContext`x, $CellContext`y}], $CellContext`x <= \ $CellContext`y}, { Dot[ Join[ Transpose[{{$CellContext`a, $CellContext`b}}], Transpose[{ 0.5 ({$CellContext`c, $CellContext`d} - {$CellContext`a, \ $CellContext`b})}], 2], {$CellContext`x, $CellContext`y}], $CellContext`x >= \ $CellContext`y}}]}; {$CellContext`f[{ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`y, Blank[]]}, Pattern[$CellContext`w, Blank[]]] := (1/ 2) {$CellContext`w ($CellContext`y^2 - $CellContext`x^2) + \ ($CellContext`w - 1) $CellContext`x + ( 1 - $CellContext`w) $CellContext`y - $CellContext`x \ $CellContext`y + 1, $CellContext`w ($CellContext`x^2 - $CellContext`y^2) + ( 1 - $CellContext`w) $CellContext`x + ($CellContext`w - 1) $CellContext`y - $CellContext`x $CellContext`y + 1}; $CellContext`M[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Join[ Transpose[{{$CellContext`a, $CellContext`b}}], Transpose[{{$CellContext`c, $CellContext`d}}], 2]; $CellContext`g[{ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`y, Blank[]]}, { Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Piecewise[{{ Dot[ Join[ Transpose[{ 0.5 ({$CellContext`a, $CellContext`b} - {$CellContext`c, \ $CellContext`d})}], Transpose[{{$CellContext`c, $CellContext`d}}], 2], {$CellContext`x, $CellContext`y}], $CellContext`x <= \ $CellContext`y}, { Dot[ Join[ Transpose[{{$CellContext`a, $CellContext`b}}], Transpose[{ 0.5 ({$CellContext`c, $CellContext`d} - {$CellContext`a, \ $CellContext`b})}], 2], {$CellContext`x, $CellContext`y}], $CellContext`x >= \ $CellContext`y}}]; $CellContext`A[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Join[ Transpose[{ 0.5 ({$CellContext`a, $CellContext`b} - {$CellContext`c, \ $CellContext`d})}], Transpose[{{$CellContext`c, $CellContext`d}}], 2]; $CellContext`B[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Join[ Transpose[{{$CellContext`a, $CellContext`b}}], Transpose[{ 0.5 ({$CellContext`c, $CellContext`d} - {$CellContext`a, \ $CellContext`b})}], 2]; $CellContext`Ar[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, { Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}] := Round[ Join[ Transpose[{ 0.5 ({$CellContext`a, $CellContext`b} - {$CellContext`c, \ $CellContext`d})}], Transpose[{{$CellContext`c, $CellContext`d}}], 2], 0.1]; $CellContext`Br[{ Pattern[$CellContext`a, Blank[]], 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