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A Condition Number Analysis of a Line-Surface Intersection Algorithm

Gun Srijuntongsiri, Stephen A. Vavasis

TL;DR

The paper introduces the Kantorovich-Test Subdivision (KTS) algorithm to compute all intersections between a line and a parametric surface by reducing the problem to a 2D polynomial system and exploiting a Kantorovich-based convergence test together with subdivision. A problem-dependent condition number cond$(f)$ is defined to bound the algorithm’s running time, and the analysis shows how basis choice (power, Bernstein, Chebyshev) affects bounding polygons and Lipschitz constants, ultimately influencing subdivision depth. The method is affine invariant and combines Newton’s quadratic convergence with safe region identification to avoid divergence and missed roots, with implementation details for multiple bases and demonstrated computational results. The work offers a unifying framework for line–surface intersection that links conditioning, convergence tests, and geometric bounding, opening avenues for extensions to more general surface representations and tighter conditioning analyses.

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.

A Condition Number Analysis of a Line-Surface Intersection Algorithm

TL;DR

The paper introduces the Kantorovich-Test Subdivision (KTS) algorithm to compute all intersections between a line and a parametric surface by reducing the problem to a 2D polynomial system and exploiting a Kantorovich-based convergence test together with subdivision. A problem-dependent condition number cond is defined to bound the algorithm’s running time, and the analysis shows how basis choice (power, Bernstein, Chebyshev) affects bounding polygons and Lipschitz constants, ultimately influencing subdivision depth. The method is affine invariant and combines Newton’s quadratic convergence with safe region identification to avoid divergence and missed roots, with implementation details for multiple bases and demonstrated computational results. The work offers a unifying framework for line–surface intersection that links conditioning, convergence tests, and geometric bounding, opening avenues for extensions to more general surface representations and tighter conditioning analyses.

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.

Paper Structure

This paper contains 13 sections, 5 theorems, 63 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The theorem of Kantorovich
  3. Formulation and representation of the line-surface intersection problem
  4. The Kantorovich-Test Subdivision algorithm
  5. Implementation details when using power, Bernstein, or Chebyshev bases
  6. Bounding polygons
  7. Computation of Lipschitz constant
  8. Significance of our condition number
  9. Condition number and the Kantorovich test
  10. Condition number and the exclusion test
  11. Time complexity analysis
  12. Computational results
  13. Conclusion and future directions

Key Result

Theorem 2.1

Let $f : D \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^n$ be differentiable in the open convex set $D$. Assume that for some point $x^0 \in D$, the Jacobian $f'(x^0)$ is invertible with Let there be a Lipschitz constant $\omega > 0$ for $f'(x^0)^{-1} f'$ such that If $h = \eta\omega \leq 1/2$ and $\bar{B}(x^0,\rho_-) \subseteq D$, where then $f$ has a zero $x^*$ in $\bar{B}(x^0,\rho_-)$. More

Figures (1)

  • Figure 1: The circular arc $A$ centered at $(u^0,-\epsilon)$ that goes from $(u^0+r,v^0-r)$ to $(u^0-r,v^0-r)$ and its range $f(A)$ where $f$ is as in (\ref{['fexc']}). Figure \ref{['f2b']} shows that the bounding convex polygon of $f(A)$ contains the origin, and therefore $\bar{B}(x^0,r)$ fails the convex bounding polygon test.

Theorems & Definitions (9)

  • Theorem 2.1: Kantorovich, affine invariant form deuflhardkantorovich
  • Theorem 7.1
  • proof
  • Lemma 7.2
  • proof
  • Theorem 7.3
  • proof
  • Theorem 7.4
  • proof