Calculating arc length of Bezier curves by hand

Question

I was wondering if there was a viable method to calculate arc length of a quadratic Bezier curve without using coding. If I know the points of P0, P1 and P2, would it be possible to calculate the arc length of quadratic Bezier curve?

Answer 1

Yes, you can do this calculation "by hand", without using computer numerical methods.

First, convert your quadratic Bezier curve into polynomial form, $P(t) = At^2 + Bt + C$. If the curve's control points are $P_0$, $P_1$, $P_2$, then $A = P_0 -2P_1 + P_2$, $B= 2P_1 - 2P_0$, and $C=P_0$.

Now you have to calculate the integral that represents arclength. This integral is of the form $\int{\sqrt{t^2 + pt + q}\,dt}$, so it can be computed by hand, though it's fairly complicated. The details are given here.

For further info, just do a Google search for "arclength of quadratic Bezier curve".