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# Evolute

Matthen Nov 21st, 2012 126 Never
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1. x[t_] := (1 + Cos[t]) Sin[t];
2. y[t_] := -(1 + Cos[t]) Cos[t];
3. tmin = 0; tmax = 2 Pi;
4. n = 41;
5. notzero[x_] := If[x == 0, 0.001, x];
6. normals = N@Table[{y'[t], -x'[t]}, {t, tmin, tmax, (tmax - tmin)/n}];
7. ps = N@Table[{x[t], y[t]}, {t, tmin, tmax, (tmax - tmin)/n}];
8. ts = N@Table[t, {t, tmin, tmax, (tmax - tmin)/n}];
9. X[t_] := x[t] -
10.    y'[t] ( x'[t]^2 + y'[t]^2)/notzero[x'[t] y''[t] - x''[t] y'[t]];
11. Y[t_] := y[t] +
12.    x'[t] ( x'[t]^2 + y'[t]^2)/notzero[x'[t] y''[t] - x''[t] y'[t]];
13. Manipulate[
14.  Show[
15.   Graphics[{}, PlotRange -> {{-2, 2}, {-3, 1}},
16.    ImageSize -> {300, 300}],
17.   Reverse@Table[
18.     ParametricPlot[
19.      ps[[i]] + t normals[[i]]/notzero@Norm[normals[[i]]], {t, -100,
20.       100},
21.      PlotStyle ->
22.       If[i == \[Tau] || i == \[Tau] - Length@ts,
23.        Directive[Thickness[0.01], RGBColor[0.4, 0.4, 0.9]],
24.        Directive[Thick, Opacity[0.4], RGBColor[0.1, 0.1, 0.1]]]
25.      ], {i, If[\[Tau] > Length[ts], \[Tau] - Length[ts], 1],
26.      If[\[Tau] > Length[ts], Length[ts], \[Tau]]}],
27.   ParametricPlot[{{x[t], y[t]}, {X[t], Y[t]}}, {t, tmin, tmax},
28.    PlotStyle -> {Directive[Thick, RGBColor[0.9, 0.2, 0.3]],
29.      Directive[Thick, RGBColor[1.0, 0.1, 0.2]]}],
30.   Graphics[{PointSize[0.05], RGBColor[0.2, 0.2, 0.6],
31.     Point[ps[[Mod[\[Tau], Length@ts, 1]]]]}]
32.   ],
33.
34.  {\[Tau], 1, 2 Length[ts], 1}]
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