The Study of Coincidence or Partial Coincidence Condition for Two Rational Cubic Bézier Curves

When two curves are coincident or almost coincident, the corresponding intersection algorithm based on curve splitting will either run out of memory and lead to system crash because of too many times of division, or the final results can not meet the accuracy requirement due to insufficient divisions. Taking two rational cubic Bézier curves for instance, this paper proposes and proves the coincidence condition based on the two control polygons. Firstly, it judges whether or not the two Bézier curves can be degenerated into rational Bézier curves of degree 1 or 2. If both of the two curves are not degenerated, they are represented in the form that their first and last weights are equal to 1; and then deciding whether their control polygons are coincident and their corresponding weights are the same. Finally, it discusses the coincidence condition that two rational cubic Bézier curves are partially coincident, and gives a simple method to determine the corresponding coincidence position, which converts the partially coincidence detection problem into the complete coincidence detection. Numerical examples demonstrate the effectiveness and validity of the proposed algorithm.