Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 2004 by Xah Lee. (http://xahlee.org/)

Generalized Clothoid

cornuSpiral[n_][
        t_] := {(∫_0^t) Sin[
              ((x^(n + 1))/(n + 1))] \[DifferentialD]x, (∫_0^t) 
              Cos[((x^(n + 1))/(n + 1))] \[DifferentialD]x}

gp1 = 
        ParametricPlot[cornuSpiral[1][t] // Evaluate, {t, 0, 5}, 
          AspectRatio -> Automatic, 
          PlotRange -> {{- .2, 1.5}, {- .2, 1.5}}];

gp2 = 
        ParametricPlot[cornuSpiral[2][t] // Evaluate, {t, 0, 5}, 
          AspectRatio -> Automatic, 
          PlotRange -> {{- .2, 1.5}, {- .2, 1.5}}];

gp3 = 
      ParametricPlot[cornuSpiral[3][t] // Evaluate, {t, 0, 5}, 
        AspectRatio -> Automatic, 
        PlotRange -> {{- .2, 1.5}, {- .2, 1.5}}]

gp4 = 
      ParametricPlot[cornuSpiral[ .5][t] // Evaluate, {t, 0, 5}, 
        AspectRatio -> Automatic, 
        PlotRange -> {{- .2, 1.5}, {- .2, 1.5}}, 
        PlotPoints -> 50]

Show@
      Graphics[{Thickness@ .005, Hue@ .75, First@gp1, Hue@0, First@gp2, 
          Hue@ .34, First@gp3, GrayLevel@ .5, First@gp4}, 
        AspectRatio -> Automatic, Frame -> True, 
        PlotRange -> {{- .1, 1.41}, {- .1, 1.43}}]