Results 1 to 2 of 2

Thread: Calculate Points on a Clothoid

  1. #1
    Newbie
    Joined
    Jul 2011
    Posts
    2

    Calculate Points on a Clothoid

    Hi,
    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    My house.
    Posts
    658
    Thanks
    42

    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Aug 24th 2011, 11:35 AM
  2. Build a Clothoid segment
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Jul 30th 2011, 10:44 AM
  3. Replies: 1
    Last Post: Feb 9th 2011, 10:52 AM
  4. Replies: 3
    Last Post: Jan 27th 2011, 12:17 PM
  5. Replies: 8
    Last Post: Aug 24th 2010, 09:18 AM

Search Tags


/mathhelpforum @mathhelpforum