Extending the A-patch single sheet conditions to enable the tessellation of algebraics

Abstract

A-patches are a form of representation of an algebraic curve or surface over a simplex. The A-patch conditions can be used as the basis for an adaptive subdivision style marching tetrahedra algorithm whose advantage is that it guarantees that we do not miss features of the algebraic: singularities are localized, and in regions around nearby multiple sheets, the subdivision process continues until the sheets are separated. Unfortunately, the A-patch single sheet conditions are too strict: for some algebraics, the subdivision process converges slowly or fails to converge. In this paper, I give an additional single sheet condition for curves that allows for convergence of this process. I also give additional conditions for surfaces that trades off some of the single sheet guarantees for improved convergence.