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Given that I have the start, end and control points for a linear Bézier curve, I am trying to find the arc-distance between the start and end points. Google seems to be failing me this morning; can anyone point me in the right direction?

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If you have a parametric curve $\begin{align*}x&=f(t)\\y&=g(t)\end{align*}$ for $a \leq t \leq b$, then the arclength you want is $\int_a^b \sqrt{f^\prime(t)^2+g^\prime(t)^2}\mathrm dt$. – J. M. Dec 31 '11 at 17:26
For the non-linear case, possible duplicate of… – lhf Dec 31 '11 at 20:49

A linear Bézier curve is just a straight line segment and so the answer is given by the Pythagorean theorem. For non-linear curves, see Arc Length of Bézier Curves.

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