Homeomorphic approximation of the intersection curve of two rational surfaces

TL;DR

This paper introduces a method to approximate the intersection curve of two rational surfaces by analyzing the topology of a related algebraic curve and constructing a homeomorphic space graph for accurate approximation.

Contribution

The approach combines implicitization of a projectable surface with topology graph analysis to approximate intersection curves with guaranteed homeomorphism.

Findings

Successfully approximates intersection curves with controlled precision

Provides a topology-preserving approximation method

Applicable to rational surfaces with one being projectable

Abstract

We present an approach of computing the intersection curve $\mathcal{C}$ of two rational parametric surface $\S_1(u,s)$ and $\S_2(v,t)$, one being projectable and hence can easily be implicitized. Plugging the parametric surface to the implicit surface yields a plane algebraic curve $G(v,t)=0$. By analyzing the topology graph $\G$ of $G(v,t)=0$ and the singular points on the intersection curve $\mathcal{C}$ we associate a space topology graph to $\mathcal{C}$, which is homeomorphic to $\mathcal{C}$ and therefore leads us to an approximation for $\mathcal{C}$ in a given precision.