# Approximating arc length of Bezier by another Bezier

(Note: cross posted to http://www.devmaster.net/forums/showthread.php?t=16227 [Broken])

Hey everyone,

As we know, the arc length of a cubic Bezier spline is kinda hard to calculate. There's no closed-form mathematical expression, so most people just subdivide it into a bunch of line segments and add those up. This is slow, but you can dress it up with adaptive subdivision and things like that to make it a bit faster.

I'd like to approximate the parameter-to-arc-length mapping of a Bezier with another Bezier. That is, given some 3D Bezier B(t) I'd like to compute a 1D Bezier s(t) that gives approximately the arc length of B from B(0) to B(t).

I could use the line segment method to get a bunch of "data points" on the arc length of B and then fit the 1D Bezier to it with least-squares or something. But I'm hoping (perhaps vainly) for a faster way...ideally something to compute the control points of s directly from the control points of B. I've googled around a bit but not found anything addressing this specific issue...anyone here happen to have heard of something like this before? :)

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## Answers and Replies

Just to follow up, I did eventually find this paper which does a pretty good job of solving my problem.