Convergence Analysis and Comparison for Geometric Interval Clipping

In order to develop a simple, yet fast and robust algorithm to accelerate the intersection calculation of ray tracing, we make a deep study on the convergence of geometric interval clipping.We formally prove its 3rd order convergence for computing all roots of a given univariate polynomial and for calculating the intersections of two plane curves, which guarantees its better performance than Bézier clipping algorithm.We also provide the comparison against quadratic clipping.Although both algorithms exhibit the 3rd order convergence, the geometric interval clipping algorithm is about 30% faster than quadratic clipping algorithm.