A Condition Number Analysis of a Line-Surface Intersection Algorithm

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

An adaptive iterative/subdivision hybrid algorithm for curve/curve intersection

The behavior of the iterative/subdivision hybrid algorithm for curve/curve intersection proposed in [20] depends on the choice of the domain for their convergence test. Using either too small or too large test domain may cause the test to fail to detect cases where Newton's method in fact converges to a solution, which results in unnecessary additional subdivisions and consequently more computation time. We propose a modification to the algorithm to adaptively adjust the test domain size according to what happens during the test of the parent region. This is in contrast to the original algorithm whose test domain is always a fixed multiple of the input domain under consideration. Computational results show that the proposed algorithm is slightly more efficient than the original algorithm

A condition number analysis of an algorithm for solving a system of polynomial equations with one degree of freedom

This article considers the problem of solving a system of $n$ real polynomial equations in $n+1$ variables. We propose an algorithm based on Newton's method and subdivision for this problem. Our algorithm is intended only for nondegenerate cases, in which case the solution is a 1-dimensional curve. Our first main contribution is a definition of a condition number measuring reciprocal distance to degeneracy that can distinguish poor and well conditioned instances of this problem. (Degenerate problems would be infinitely ill conditioned in our framework.) Our second contribution, which is the main novelty of our algorithm, is an analysis showing that its running time is bounded in terms of the condition number of the problem instance as well as $n$ and the polynomial degrees

A condition number analysis of a line-surface intersection algorithm

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. Our analysis shows that the running time of the algorithm is bounded in terms of the condition number of the polynomial’s zeros. We argue that, in contrast, some other well known algorithms for this problem similarly based on Newton and subdivision may in certain cases require excessive computation even for well conditioned problems.