# Calculate Points on a Clothoid

• Jul 30th 2011, 12:05 PM
schmimona
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?
• Feb 5th 2015, 12:30 AM
Rebesques
Re: Calculate Points on a Clothoid
You can do that for any planar curve with known curvature, $k(s)$.
Let the curve be $a(s)=(x(s),y(s)),s\in [0,1]$, parametrized for arclength.
Then $|a'(s)|=1$, which implies there exists a function $\theta(s),s\in [0,1]$ such that
$x(s)=\cos\theta(s),y(s)=\sin\theta(s)$.
From the formula for curvature, $k(s)=x'(s)y''(s)-y'(s)x''(s)=\theta'(s)$,
and so the curve is $a(s)=(\int \cos(\int k(u)du+c_1)ds+c_2, \int \sin(\int k(u)du+c_3)ds+c_4)$.
Now use $a(0)=p,a(1)=q$ to determine the constants.