Locally supported tangential vector,n‐vector, and tensor fields

Abstract

We introduce a construction of subspaces of the spaces of tangential vector, n‐vector, and tensor fields on surfaces. The resulting subspaces can be used as the basis of fast approximation algorithms for design and processing problems that involve tangential fields. Important features of our construction are that it is based on a general principle, from which constructions for different types of tangential fields can be derived, and that it is scalable, making it possible to efficiently compute and store large subspace bases for large meshes. Moreover, the construction is adaptive, which allows for controlling the distribution of the degrees of freedom of the subspaces over the surface. We evaluate our construction in several experiments addressing approximation quality, scalability, adaptivity, computation times and memory requirements. Our design choices are justified by comparing our construction to possible alternatives. Finally, we discuss examples of how subspace methods can be used to build interactive tools for tangential field design and processing tasks.

Arbitrary order principal directions and how to compute them

“…The generalization of the Laplace-Beltrami operator to tangential vector fields and the subspace raised by its eigenvectors up to a given order may be used for regularizing vector field design [3]. Following this development of subspace methods for tangent field design [4], Nasikun et al [20] consider tangent vector field design and processing via locally-supported tangential fields leading to fast approximation and design algorithms. Application of Tangent Vector Fields.…”