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Thread: Calculate Points on a Clothoid

  1. #1
    Jul 2011

    Calculate Points on a Clothoid

    Is there a way to find out the points on a clothoid, i.e. their coordinates, if one knows the start point and the end point of the clothoid and its curvature?
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  2. #2
    Super Member Rebesques's Avatar
    Jul 2005
    My house.

    Re: Calculate Points on a Clothoid

    You can do that for any planar curve with known curvature, $\displaystyle k(s)$.
    Let the curve be $\displaystyle a(s)=(x(s),y(s)),s\in [0,1]$, parametrized for arclength.
    Then $\displaystyle |a'(s)|=1$, which implies there exists a function $\displaystyle \theta(s),s\in [0,1]$ such that
    $\displaystyle x(s)=\cos\theta(s),y(s)=\sin\theta(s)$.
    From the formula for curvature, $\displaystyle k(s)=x'(s)y''(s)-y'(s)x''(s)=\theta'(s)$,
    and so the curve is $\displaystyle a(s)=(\int \cos(\int k(u)du+c_1)ds+c_2, \int \sin(\int k(u)du+c_3)ds+c_4)$.
    Now use $\displaystyle a(0)=p,a(1)=q$ to determine the constants.
    Last edited by Rebesques; Feb 5th 2015 at 12:33 AM.
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