Add derivatives of path segments.

Previously, computing x'(t) was inaccurate and depended on
the step size of the iteration scheme. We now implement
derivatives of path segments for accuracy. However, now
the results are off by an unknown factor. This is a bug.
Correct path derivative computation.

There was an inconsistency in the value of the a- and b-
cycles in a previous commit. This was due to an
incorrect computation of the derivative of the path
parameterization. This has been corrected.