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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).

CS488/688: Introduction to Interactive Computer Graphics
University of Waterloo
Computer Graphics Lab