Consistent Computation of First- and Second-Order Differential Quantities for Surface Meshes

TL;DR

This paper introduces a consistent method for computing differential quantities on surface meshes using height function derivatives, providing explicit formulas and a flexible numerical framework to improve accuracy and stability.

Contribution

It derives explicit formulas linking surface differential quantities to height function derivatives and proposes an iterative polynomial fitting framework for improved estimation.

Findings

Explicit formulas for differential quantities from height function derivatives

A flexible polynomial fitting framework with improved accuracy

Theoretical analysis and experiments demonstrating stability and precision

Abstract

Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce inconsistent results and require ad hoc fixes. In this paper, we show that the computation of the gradient and Hessian of a height function provides the foundation for consistently computing the differential quantities. We derive simple, explicit formulas for the transformations between the first- and second-order differential quantities (i.e., normal vector and principal curvature tensor) of a smooth surface and the first- and second-order derivatives (i.e., gradient and Hessian) of its corresponding height function. We then investigate a general, flexible numerical framework to estimate the derivatives of the height function based on local polynomial fittings formulated as weighted least squares approximations. We also propose an iterative fitting scheme to improve accuracy. This framework generalizes polynomial fitting and addresses some of its accuracy and stability issues, as demonstrated by our theoretical analysis as well as experimental results.