Note

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

Packages and Initializations

Needs["ParaPlot`"]
Needs["PlaneCurveFormulas`"]
Needs["PlaneCurveGenerator`"]

Needs["Graphics`ImplicitPlot`"]
(*MathSource Item:0200-529*)

$HistoryLength = 10;

Clothoid

?Clothoid

Clothoid[1]@t

ParaPlot[Clothoid[1]@t // Evaluate, {t, -9, 5}, 
      AspectRatio -> Automatic]

Clothoid is the curve whose curvature varies proportional to its length.

curvature & length

The following shows the length of Clothoid. It shows that the formula for clothoid is unit length. That is, its length is the parameter t.

?ArcLength

Clothoid[1]@t

ArcLength[Clothoid[1]@t,  {t,  0,  b} ]

The following computes the curvature function of clothoid

?Curvature

Curvature[Clothoid[1]@t, t] // Simplify

A look at the integrand

Look at the function inside the integral

g1 = 
      Plot[Sin[(t^(n + 1))/(n + 1)] /. n -> 1, {t, 0, 
          10}]

look at the plot of the integral

g2 = 
      Plot[Evaluate[
Integrate[
            Sin[(t^(n + 1))/(n + 1)] /. n -> 1, {t, 0, 
              tt}]
], {tt, 0, 10}]

Show@
      Graphics[{Hue@ .75, First@g1, Hue@0, First@g2}, 
        Axes -> True]

So we see the x coordinate for the curve vacillates, apparently in sync with the y-coordinate for form the spiral. (the y-coodinate is the same except it uses Cos in the integrand)

Limit[
      Evaluate@Integrate[
          Sin[(t^(n + 1))/(n + 1)] /. n -> 1, {t, 0, tt}], 
      tt -> Infinity]

Actually, the limit is Sqrt[Pi]/2.