Ambient Isotopic Meshing of Implicit Algebraic Surface with Singularities

TL;DR

This paper introduces a novel symbolic-numeric method for computing certified, topologically accurate meshes of implicit algebraic surfaces with singularities, ensuring correctness within specified geometric precision.

Contribution

It presents the first method capable of producing certified meshes for algebraic surfaces with singularities, combining symbolic-numeric techniques with a modified marching cube approach.

Findings

Successfully handles surfaces with singularities

Produces meshes with guaranteed topology and precision

Demonstrates effectiveness on complex examples

Abstract

A complete method is proposed to compute a certified, or ambient isotopic, meshing for an implicit algebraic surface with singularities. By certified, we mean a meshing with correct topology and any given geometric precision. We propose a symbolic-numeric method to compute a certified meshing for the surface inside a box containing singularities and use a modified Plantinga-Vegter marching cube method to compute a certified meshing for the surface inside a box without singularities. Nontrivial examples are given to show the effectiveness of the algorithm. To our knowledge, this is the first method to compute a certified meshing for surfaces with singularities.