Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides

TL;DR

This paper introduces cyclidic nets as discrete models of curvature line parametrized surfaces and orthogonal coordinate systems, using Dupin cyclides, with explicit formulas and computational implementation.

Contribution

It presents a novel discretization framework for curvature line surfaces and orthogonal coordinate systems based on Dupin cyclides, including explicit formulas and 3D generalizations.

Findings

Explicit formulas for cyclidic nets are derived.

Cyclidic nets are related to existing discretizations like circular nets.

Implementation of the formulas in a computer program is demonstrated.

Abstract

Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth $C^1$-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit description of cyclidic nets is given and their relation to the established discretizations of curvature line parametrized surfaces as circular, conical and principal contact element nets is explained. We introduce 3-dimensional cyclidic nets as discrete analogs of triply-orthogonal coordinate systems and investigate them in detail. Our considerations are based on the Lie geometric description of Dupin cyclides. Explicit formulas are derived and implemented in a computer program.