Extrinsic Vector Field Processing

TL;DR

This paper introduces a new discretization method for tangent vector fields on triangle meshes, enabling continuous, weakly differentiable vector fields and the computation of covariant derivatives and related operators.

Contribution

It presents a novel extrinsic discretization approach for tangent vector fields on meshes, facilitating the evaluation of covariant derivatives and standard vector field operators.

Findings

Enables point-wise evaluation of covariant derivatives on meshes

Defines standard vector field processing operators like Hodge and Connection Laplacians

Allows computation of Lie brackets for tangent vector fields

Abstract

We propose a novel discretization of tangent vector fields for triangle meshes. Starting with a Phong map continuously assigning normals to all points on the mesh, we define an extrinsic bases for continuous tangent vector fields by using the Rodrigues rotation to transport tangent vectors assigned to vertices to tangent vectors in the interiors of the triangles. As our vector fields are continuous and weakly differentiable, we can use them to define a covariant derivative field that is evaluatable almost-everywhere on the mesh. Decomposing the covariant derivative in terms of diagonal multiple of the identity, anti-symmetric, and trace-less symmetric components, we can define the standard operators used for vector field processing including the Hodge Laplacian energy, Connection Laplacian energy, and Killing energy. Additionally, the ability to perform point-wise evaluation of the covariant derivative also makes it possible for us to define the Lie bracket.