Shape - preserving functions

Transcription

1 Shape - preserving functions Introduction This Mathematica notebook is licensed under a Creative Commons Attribution - ShareAlike 3.0 License It creates the demonstrations used in my post Maps. Graphs created using this notebook may be found at I hope anyone interested will feel free to improve this work and to use it in their own publications and coursework. Preliminaries Colors c = RGBColor[.3,.,.3]; c = RGBColor[.7,.,.]; c3 = RGBColor[.5,.3,.5] Sample matrix to work with mat := {{-.,.3}, {-,.5}} TableForm[{{-0.,.3}, {-, 0.5}}] mat // Det. imat = mat // Inverse {{0.5, -.3}, {., -0.}}

2 Functions and shapes.nb Circle cc[c_, r_, col_] := ParametricPlot[{ r Cos[t], r Sin[t]} + c, {t, 0, Pi}, PlotStyle {Thick, col}, ImageSize 00] cc[{, },., Red] Straight line ln[p_, q_, color_] := ParametricPlotp x + q - x, {x, 0, }, PlotStyle {Thick, color}, ImageSize 00 ln[{.9,.5}, {-, }, c] Rectangle rect[p_, q_, color_] := Show[ ] { } ln[p, {p[[]], q[[]]}, color], ln[{p[[]], q[[]]}, q, color], ln[q, {q[[]], p[[]]}, color], ln[{q[[]], p[[]]}, p, color] rect[{-, }, {-.7,.5}, c] Special triangle tr[p_, color_] := {ln[p, {p[[]] -., p[[]] +.}, color], ln[{p[[]] -., p[[]] +.}, {p[[]] +., p[[]] +.}, color], ln[{p[[]] +., p[[]] +.}, p, color]}

3 Functions and shapes.nb 3 Show[tr[{, }, c3], ImageSize 00, PlotRange {{0.5,.3}, {.5, }}, AspectRatio -> Automatic] Display shapes Show[ {cc[{, },., c], rect[{.3, }, {.7,.}, c], tr[{, 3}, c3] }, Axes -> True, AxesOrigin {0, 0}, PlotRange {{-, }, {0, }}, ImageSize 00 ]

4 Functions and shapes.nb Image of circle ParametricPlot[mat.# &[{. Cos[t],. Sin[t]} + 0], {t, 0, Pi}, PlotStyle {Thick, c}, ImageSize 00] apcc[c_, r_, col_, ap_] := ParametricPlot[ap[{ r Cos[t], r Sin[t]} + c], {t, 0, Pi}, PlotStyle {Thick, col}, ImageSize 00] apcc[{, },., c, mat.# &] apcc[{0, 0},., c, mat.# &] Image of rectangle ln[p_, q_, color_] := ParametricPlotp x + q - x, {x, 0, }, PlotStyle {Thick, color}, ImageSize 00 apln[p_, q_, color_, ap_] := ParametricPlotapp x + q - x, {x, 0, }, PlotStyle {Thick, color}, ImageSize 00

5 Functions and shapes.nb 5 ln[{, }, {, }, c] Show[apln[{, }, {, }, c, mat.# &], PlotRange -> {{-, }, {-, 0}}, AspectRatio Automatic, ImageSize 00] aprect[p_, q_, color_, ap_] := Show[ ] { } apln[p, {p[[]], q[[]]}, color, ap], apln[{p[[]], q[[]]}, q, color, ap], apln[q, {q[[]], p[[]]}, color, ap], apln[{q[[]], p[[]]}, p, color, ap] rect[{-, }, {-.7,.5}, c] Show[aprect[{-, }, {-.7,.5}, c, mat.# &], PlotRange Automatic, AspectRatio Automatic, ImageSize 00]

6 Functions and shapes.nb Image of triangle tr[p_, color_] := {ln[p, {p[[]] -., p[[]] +.}, color], ln[{p[[]] -., p[[]] +.}, {p[[]] +., p[[]] +.}, color], ln[{p[[]] +., p[[]] +.}, p, color]} aptr[p_, color_, ap_] := {apln[p, {p[[]] -., p[[]] +.}, color, ap], apln[{p[[]] -., p[[]] +.}, {p[[]] +., p[[]] +.}, color, ap], apln[{p[[]] +., p[[]] +.}, p, color, ap]} Show[aptr[{, }, c3, mat.# &]] Variable matrix with det df[a_, b_, c_, d_] := -b c + a d Solve[-b c + a d, {a}] a + b c d Solve[-b c + a, {a}] {{a + b c}}

7 Functions and shapes.nb 7 Display area-preserving functions Manipulate Module {mat}, mat[b_, c_, d_] := + b c d, b, {c, d}; ShowGraphics[Text[mat[b, c, d] // MatrixForm, {7, }]], cc[{, },., c], apcc[{, },., c, mat[b, c, d].# &], rect[{.3, }, {.7,.}, c], aprect[{.3, }, {.7,.}, c, mat[b, c, d].# &], tr[{, 3}, c3], aptr[{, 3}, c3, mat[b, c, d].# &], Axes -> True, AxesOrigin {0, 0}, PlotRange {{-, 9}, {-, }}, AspectRatio 7 3, ImageSize 00, {{b,.7}, -,, Appearance "Open"}, {{c,.}, -,, Appearance "Open"}, {{d, }, 0.5,.5, Appearance "Open"}, SaveDefinitions True b c d

8 Functions and shapes.nb b c d

9 Functions and shapes.nb 9 b c d Examples apf[b_, c_, d_] := Module {mat}, mat[b, c, d] = + b c d, b, {c, d}; ShowGraphics[Text[mat[b, c, d] // MatrixForm, {7, }]], cc[{, },., c], apcc[{, },., c, mat[b, c, d].# &], rect[{.3, }, {.7,.}, c], aprect[{.3, }, {.7,.}, c, mat[b, c, d].# &], tr[{, 3}, c3], aptr[{, 3}, c3, mat[b, c, d].# &], Axes -> True, AxesOrigin {0, 0}, PlotRange {{-, 9}, {-, }}, AspectRatio 7 3, ImageSize 00

10 0 Functions and shapes.nb apf[.5,.5, ] Show[apf[.5,.5, ], Graphics[{Arrowheads[0.0], c3, Arrow[{{, 3.5}, {3, }}, 0], c, Arrow[{{,.5}, {.5,.5}}, 0], c, Arrow[{{, }, {.75,.5}}, 0]}]] Show[apf[0,.,.5], Graphics[{Arrowheads[0.0], c3, Arrow[{{, 3.5}, {,.}}, 0], c, Arrow[{{,.5}, {,.5}}, 0], c, Arrow[{{, }, {, 0.5}}, 0]}]]

11 Functions and shapes.nb Show[apf[-,.5,.5], Graphics[{Arrowheads[0.0], c3, Arrow[{{, 3.5}, {-.,.3}}, 0], c, Arrow[{{,.5}, {-.3,.5}}, 0], c, Arrow[{{, }, {0, }}, 0]}]] Show[apf[-, -,.5], Graphics[{Arrowheads[0.0], c3, Arrow[{{, 3.5}, {.5,.75}}, 0], c, Arrow[{{,.5}, {.5,.5}}, 0], c, Arrow[{{, }, {3, -.5}}, 0]}]] Angle-preserving functions expmat = {{Cos[t], -Sin[t]}, {Sin[t], Cos[t]}} {{Cos[t], -Sin[t]}, {Sin[t], Cos[t]}} expmat // MatrixForm Cos[t] -Sin[t] Sin[t] Cos[t] rnd[x_] := Round[x,.0]

12 Functions and shapes.nb Manipulate Module {mat}, mat[a_, t_] := a {{Cos[t], -Sin[t]}, {Sin[t], Cos[t]}}; ShowGraphics[Text[rnd[mat[a, t]] // MatrixForm, {5, }]], cc[{, },., c], apcc[{, },., c, mat[a, t].# &], rect[{.3, }, {.7,.}, c], aprect[{.3, }, {.7,.}, c, mat[a, t].# &], tr[{, 3}, c3], aptr[{, 3}, c3, mat[a, t].# &], Axes -> True, AxesOrigin {0, 0}, PlotRange {{-, }, {-3, }}, AspectRatio 7, ImageSize 00, {t,.7}, -Pi, Pi, -., Appearance "Open", {{a, }, 0,, Appearance "Open"} t a rnd

13 Functions and shapes.nb 3 epf[t_, a_] := Module {mat}, mat[a, t] = a {{Cos[t], -Sin[t]}, {Sin[t], Cos[t]}}; ShowGraphics[Text[rnd[mat[a, t]] // MatrixForm, {5, }]], cc[{, },., c], apcc[{, },., c, mat[a, t].# &], rect[{.3, }, {.7,.}, c], aprect[{.3, }, {.7,.}, c, mat[a, t].# &], tr[{, 3}, c3], aptr[{, 3}, c3, mat[a, t].# &], Axes -> True, AxesOrigin {0, 0}, PlotRange {{-, }, {-3, }}, AspectRatio 7, ImageSize 00 Show[epf[.7,.5], Graphics[{Arrowheads[0.0], c3, Arrow[{{, 3.5}, {-.3, 5}}, 0], c, Arrow[{{,.5}, {-., 3.}}, 0], c, Arrow[{{, }, {.,.}}, 0]}]] {{.5, -0.97}, {0.97,.5}} {{.5, -0.97}, {0.97,.5}} Pi // N.570

14 Functions and shapes.nb Show[epf[-.,.7], Graphics[{Arrowheads[0.0], c3, Arrow[{{, 3.5}, {, -}}, 0], c, Arrow[{{,.5}, {, -}}, 0], c, Arrow[{{, }, {.75, -}}, 0]}]] {{-.05,.7}, {-.7, -.05}} {{-0.05,.7}, {-.7, -0.05}} Show[epf[.,.], Graphics[{Arrowheads[0.0], c3, Arrow[{{, 3.5}, {-.75,.3}}, 0], c, Arrow[{{,.5}, {-.5,.05}}, 0], c, Arrow[{{, }, {-.,.75}}, 0]}]] {{., -.5}, {.5,.}} {{0., -0.5}, {0.5, 0.}}

15 Functions and shapes.nb 5 Some endographs matr := {{, 0}, {.,.5}} matr.{x, y} { x, 0. x y} bigrg := {{-0, 0}, {-5, 0}} latt := Flatten[Table[{i, j}, {i, -5, 5, }, {j, -, 5, }], ] littlelatt := Flatten[Table[{i, j}, {i, -,,.5}, {j, -,,.5}], ] littlelatt {{-., -.}, {-., -.5}, {-., -.}, {-., -0.5}, {-., 0.}, {-., 0.5}, {-.,.}, {-0.5, -.}, {-0.5, -.5}, {-0.5, -.}, {-0.5, -0.5}, {-0.5, 0.}, {-0.5, 0.5}, {-0.5,.}, {0., -.}, {0., -.5}, {0., -.}, {0., -0.5}, {0., 0.}, {0., 0.5}, {0.,.}, {0.5, -.}, {0.5, -.5}, {0.5, -.}, {0.5, -0.5}, {0.5, 0.}, {0.5, 0.5}, {0.5,.}, {., -.}, {., -.5}, {., -.}, {., -0.5}, {., 0.}, {., 0.5}, {.,.}, {.5, -.}, {.5, -.5}, {.5, -.}, {.5, -0.5}, {.5, 0.}, {.5, 0.5}, {.5,.}, {., -.}, {., -.5}, {., -.}, {., -0.5}, {., 0.}, {., 0.5}, {.,.}} Point[#] & /@ {{, }, {, 3}} {Point[{, }], Point[{, 3}]} Point[matr.#] & /@ {{, }, {, 3}} {Point[{.,.}], Point[{.,.3}]} Point[#] & /@ littlelatt;

16 Functions and shapes.nb Show[ Graphics[ Point[#] & latt ], PlotRange bigrg, Axes True] Show[ Graphics[ {Point[#] & /@ latt, Red, Point[matr.#] & /@ latt }], PlotRange bigrg, Axes True] plotboth[matrix_, arr_, rg_] := Show[ Graphics[ {Point[#] & /@ arr, Red, Point[matrix.#] & /@ arr }], PlotRange rg, Axes True]

17 Functions and shapes.nb 7 plotboth[matr, latt, bigrg] plotboth[{{.5,.5}, {.5, }}, latt, bigrg] plotbotharrs[matrix_, arr_, rg_, arrowheadsize_] := Show[ Graphics[ {Arrowheads[arrowheadsize], Point[#] & /@ arr, Red, Point[matrix.#] & /@ arr, Arrow[{#, matrix.#}, 0] & /@ arr }], PlotRange rg, Axes True]

18 Functions and shapes.nb plotbotharrs[{{.5,.5}, {.5, }}, latt, bigrg,.0] plotbotharrs[{{, -}, {.5,.5}}, latt, bigrg,.0] {{.5, -0.97}, {0.97,.5}} {{.5, -0.97}, {0.97,.5}}

19 Functions and shapes.nb 9 plotbotharrs[{{.5, }, {0.97,.5}}, latt, bigrg,.0] {{., -.5}, {.5,.}} {{0., -0.5}, {0.5, 0.}} plotbotharrs[{{., -.5}, {.5,.}}, latt, bigrg,.0] {{-.05,.7}, {-.7, -.05}} {{-0.05,.7}, {-.7, -0.05}}

20 0 Functions and shapes.nb plotbotharrs[{{-.05,.7}, {-.7, -.05}}, latt, bigrg,.0]