(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 176967, 3544] NotebookOptionsPosition[ 175793, 3500] NotebookOutlinePosition[ 176133, 3515] CellTagsIndexPosition[ 176090, 3512] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ \[OpenCurlyDoubleQuote]Transformations and Projections in Computer Graphics,\ \[CloseCurlyDoubleQuote] 2006, by David Salomon.\ \>", "Title", CellChangeTimes->{ 3.432822199015625*^9, 3.432822257640625*^9, {3.432822297265625*^9, 3.432822325765625*^9}, {3.432847615484375*^9, 3.432847616234375*^9}, { 3.432878457188719*^9, 3.43287845748559*^9}, {3.432923988671875*^9, 3.432923988953125*^9}, {3.433638011296875*^9, 3.4336380116875*^9}, { 3.433639751078125*^9, 3.433639751390625*^9}, {3.4336445585625*^9, 3.43364455890625*^9}, {3.4347006283023496`*^9, 3.4347006557869005`*^9}}, Background->RGBColor[0, 1, 1]], Cell["\<\ Based on: Circle Inversion Transformation, Section 4.3, page 162.\ \>", "Subtitle", CellChangeTimes->{{3.43292540534375*^9, 3.432925415703125*^9}, { 3.432925447546875*^9, 3.4329254600625*^9}, {3.434700440159568*^9, 3.434700504363515*^9}, {3.4347006655682135`*^9, 3.4347007321467643`*^9}, { 3.4347007676938667`*^9, 3.43470079322528*^9}, 3.4347783708061233`*^9}, FontColor->GrayLevel[0], Background->RGBColor[1, 0, 1]], Cell[TextData[{ "Some material is from the book, \[OpenCurlyDoubleQuote]Curves and Surfaces \ for Computer Graphics,\[CloseCurlyDoubleQuote] 2006, by David Salomon, and \ the ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook named \"Cubic Bezier Curve.nb\" found at the website:\n\n", Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"http://www.davidsalomon.name/\"\>", True-> StyleBox["\<\"http://www.davidsalomon.name/\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.davidsalomon.name/"], None}, ButtonNote->"http://www.davidsalomon.name/"], Annotation[#, "http://www.davidsalomon.name/", "Hyperlink"]& ]], CellChangeTimes->{{3.434564354453125*^9, 3.43456437728125*^9}}], "\n" }], "Subsubtitle", CellChangeTimes->{{3.434699742103758*^9, 3.4346997584945927`*^9}, { 3.434699803745172*^9, 3.4346998407456455`*^9}, {3.4346998941369543`*^9, 3.434699953278336*^9}, {3.4346999919194555`*^9, 3.434700086170662*^9}, { 3.434700976617079*^9, 3.434701076539593*^9}, 3.434701471620247*^9, { 3.434921335249549*^9, 3.434921354090548*^9}, {3.435188916625*^9, 3.435188917234375*^9}}, FontWeight->"Bold", Background->RGBColor[1, 1, 0]], Cell[CellGroupData[{ Cell["Illustrating the Nonlinearity Nature of a Transformation", "Section", CellChangeTimes->{{3.434778426935793*^9, 3.434778443734065*^9}, { 3.434921279866064*^9, 3.434921314434066*^9}, {3.43492137345457*^9, 3.434921377101781*^9}}], Cell[CellGroupData[{ Cell["\<\ Circle inversion is an example of a nonlinear transfromation. The inverse of \ a point P outside the circle is a point Q inside the circle, located on the \ line that connects P with the origin. Given the four 2 D points in array pts, \ we compute and plot (in red) the cubic Lagrange polynomial that connects them \ (Eq. 3.17, page 82 in the book on curves and surfaces). We then invert points \ pts into qts and compute and plot the same curve (in green) through qts. The \ two sets of points are also plotted. It is easy to see that the inverse of \ each point pts that is outside the circle is inside the circle and vice \ versa, but the green curve as a whole does not satisfy this property (e.g., \ the first section of the original, red curve is completely outside the \ circle, but the first section of the green curve is not completely inside \ the circle). This behavior is the result of the nonlinearity of circle \ inversion. We then compute about 50 points of the original curve into rts, and invert \ each, placing it in array sts. When this array is plotted (with a \ ListLinePlot, as a blue curve) it is easy to see that in spite of its strange \ shape it constitutes the true inverse of the original, red curve.\ \>", "Subsection", CellChangeTimes->{{3.43498326978125*^9, 3.43498329628125*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"pts", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.5"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "0.7"}], ",", " ", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0.5", ",", " ", ".3"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"1.5", ",", " ", "0"}], "}"}]}], "}"}]}], ";"}], " ", RowBox[{"(*", " ", RowBox[{"Four", " ", "2", "D", " ", "points"}], " ", "*)"}], "\n"}], "\n", RowBox[{ RowBox[{"TensMat", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "9"}], "/", "2."}], ",", " ", RowBox[{"27", "/", "2"}], ",", " ", RowBox[{ RowBox[{"-", "27"}], "/", "2"}], ",", " ", RowBox[{"9", 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