Closed-form Quadrangulation of N-Sided Patches

TL;DR

This paper provides a closed-form analytical solution for quadrangulating n-sided patches with specific boundary subdivisions, introducing conditions and descriptions for such tessellations with minimal irregular vertices.

Contribution

It derives necessary and sufficient conditions for quadrangulating n-sided patches with a single irregular vertex, offering a complete analytical description of the tessellations.

Findings

Derived closed-form conditions for quadrangulation.

Provided a full description of resulting tessellations.

Identified when such quadrangulations are possible.

Abstract

We analyze the problem of quadrangulating a $n$-sided patch, each side at its boundary subdivided into a given number of edges, using a single irregular vertex (or none, when $n = 4$) that breaks the otherwise fully regular lattice. We derive, in an analytical closed-form, (1) the necessary and sufficient conditions that a patch must meet to admit this quadrangulation, and (2) a full description of the resulting tessellation(s).