Study of Homogeneous Geometric Newton Method for Dealing with Rational Curved Surfaces

Abstract

Although rational polynomial curves and surfaces have become standard forms in computer-aided design, they have many problems. For example, a Newton-Raphson algorithm for dealing with rational polynomial curves tends to be non-robust. This is a fatal problem. In the past, we proposed a new Newton-Raphson method for dealing with the rational polynomial curves. In the method, we homogenize the coordinates of a rational curve when it is applied to the Newton-Raphson algorithm. Then it becomes very robust.This paper proposes a method of adapting the algorithm to the Newton-Raphson method for dealing with the surface-surface intersection problem. Like the case of a curve, the Newton-Raphson algorithm was found to be very robust when the coordinates of a curved surface are homogenized. In this paper, we discuss the reason why the conventional method tends to be non-robust while the new method is robust. The conventional method deals with a rational function that always has a horizontal asymptote as well as vertical ones while the new method deals with a non-rational function and thus has no asymptote. The result of our analysis points out that the cause of the non-robustness lies in the horizontal asymptote of the rational function inherent in the conventional Newton-Raphson method. Our numerical experiments show that our method is very robust.