A Condition Number Analysis of a Line-Surface Intersection Algorithm

TL;DR

This paper introduces a Newton-based subdivision algorithm for finding line-surface intersections, with performance bounds linked to the polynomial's zero condition number, applicable across various polynomial bases.

Contribution

It presents a novel algorithm with a condition number analysis that bounds its running time, applicable to multiple polynomial bases.

Findings

Algorithm effectively finds all zeros within bounded regions.

Performance is bounded by the condition number of polynomial zeros.

Compatible with various polynomial bases like power, Bernstein, and Chebyshev.

Abstract

We propose an algorithm based on Newton's method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has applications in graphics and computer-aided geometric design. The algorithm can operate on polynomials represented in any basis that satisfies a few conditions. The power basis, the Bernstein basis, and the first-kind Chebyshev basis are among those compatible with the algorithm. The main novelty of our algorithm is an analysis showing that its running is bounded only in terms of the condition number of the polynomial's zeros and a constant depending on the polynomial basis.