Differential and Topology Computation on NURBS

by
Xianming Chen

Advised by
Rich Riesenfeld

My research focus is the differential and topological computation on manifolds with NURBS representation.

The topological computation of orthographic silhouette curves is implemented, and the topological transition is used to solve the visibility and 2-piece mold manufacturing problem.

The rational forward difference operator on NURBS control meshes is defined, and the differential computation for rational Bsplines is carried out subsequently.

NURBS Symbolic computation has major problem of degree explosion when applied to rational BSplines. it is shown that the higher degree terms due to the derivative to the denominator of a rational BSpline can be ignored in most differential computation. Also, by simply carrying out all the differential formulation in terms of (polynomial) numerators and (polynomial) denominators, and canceling out the very likely common factors, another significant degree reduction is possible. Using these two strategies together, the evolute of a rational quadratic curve is degree reduced from 42 to 16!

NURBS multiplication is the most essential operation for NURBS symbolic computation (even the addition of two rational BSplines will result in three multiplications). A very simple direct multiplication scheme is found for two BSplines.

Accompanying the above works, is a software package for NURBS differential and topological computation.