Arbitrary order principal directions and how to compute them

TL;DR

This paper extends the concept of principal directions on surfaces to higher orders using symmetric differential tensors, providing explicit approximation methods and demonstrating their usefulness in geometric analysis.

Contribution

It introduces a novel framework for higher-order principal directions on surfaces, expanding beyond traditional second-order curvature analysis.

Findings

Higher-order principal directions can be explicitly approximated on point set surfaces.

Higher-order directions convey valuable geometric information.

The method enhances surface analysis techniques.

Abstract

Curvature principal directions on geometric surfaces are a ubiquitous concept of Geometry Processing techniques. However they only account for order 2 differential quantities, oblivious of higher order differential behaviors. In this paper, we extend the concept of principal directions to higher orders for surfaces in R^3 by considering symmetric differential tensors. We further show how they can be explicitly approximated on point set surfaces and that they convey valuable geometric information, that can help the analysis of 3D surfaces.