On the Efficiency of Strategies for Subdividing Polynomial Triangular Surface Patches

TL;DR

This paper evaluates and compares the efficiency of various subdivision strategies for polynomial triangular surface patches, introducing optimal and alternative algorithms with different subdivision patterns and call counts to the de Casteljau algorithm.

Contribution

It presents a simple, optimal subdivision algorithm requiring four calls, and introduces alternative schemes with fewer calls and different subdivision patterns.

Findings

The simple algorithm is optimal with four calls.

A three-call diamond-like subdivision algorithm is proposed.

A spider-like scheme producing six subtriangles is introduced.

Abstract

In this paper, we investigate the efficiency of various strategies for subdividing polynomial triangular surface patches. We give a simple algorithm performing a regular subdivision in four calls to the standard de Casteljau algorithm (in its subdivision version). A naive version uses twelve calls. We also show that any method for obtaining a regular subdivision using the standard de Casteljau algorithm requires at least 4 calls. Thus, our method is optimal. We give another subdivision algorithm using only three calls to the de Casteljau algorithm. Instead of being regular, the subdivision pattern is diamond-like. Finally, we present a ``spider-like'' subdivision scheme producing six subtriangles in four calls to the de Casteljau algorithm.