Some Nasty Problems

There are some nasty problems with the arc-length parameterization approach:

1. The arc-length s = A(u) is given by the integral

   

   which has no analytic solution if the motion path is a cubic spline.

2. Since A(u) has no analytic form, has no analytic form.

We have to use numerical techniques:

1. Find A(u) with numerical quadrature. Precompute some values of s, at least at the boundaries of segments, to make this easier.
2. Since s is a monotonic function of u, for a given we can find the solution to by bisection (binary search).