Symmetries of Canal Surfaces and Dupin Cyclides

TL;DR

This paper characterizes symmetries of canal surfaces and Dupin cyclides, providing algorithms for symmetry computation and applications in surface design.

Contribution

It introduces a new characterization for symmetries of rational canal surfaces and Dupin cyclides, enabling symmetry detection and construction methods.

Findings

Derived an intrinsic description of Dupin cyclide symmetries

Developed an algorithm for computing symmetries of rational canal surfaces

Applied results to surface patch and blend construction with prescribed symmetry

Abstract

We develop a characterization for the existence of symmetries of canal surfaces defined by a rational spine curve and rational radius function. In turn, this characterization inspires an algorithm for computing the symmetries of such canal surfaces. For Dupin cyclides in canonical form, we apply the characterization to derive an intrinsic description of their symmetries and symmetry groups, which gives rise to a method for computing the symmetries of a Dupin cyclide not necessarily in canonical form. As a final application, we discuss the construction of patches and blends of rational canal surfaces with a prescribed symmetry.