Discrete Derivatives of Vector Fields on Surfaces -- An Operator Approach

Abstract

Vector fields on surfaces are fundamental in various applications in computer graphics and geometry processing. In many cases, in addition to representing vector fields, the need arises to compute their derivatives, for example, for solving partial differential equations on surfaces or for designing vector fields with prescribed smoothness properties. In this work, we consider the problem of computing the Levi-Civita covariant derivative, that is, the tangential component of the standard directional derivative, on triangle meshes. This problem is challenging since, formally, tangent vector fields on polygonal meshes are often viewed as being discontinuous, hence it is not obvious what a good derivative formulation would be. We leverage the relationship between the Levi-Civita covariant derivative of a vector field and the directional derivative of its component functions to provide a simple, easy-to-implement discretization for which we demonstrate experimental convergence. In addition, we introduce two linear operators which provide access to additional constructs in Riemannian geometry that are not easy to discretize otherwise, including the parallel transport operator which can be seen simply as a certain matrix exponential. Finally, we show the applicability of our operator to various tasks, such as fluid simulation on curved surfaces and vector field design, by posing algebraic constraints on the covariant derivative operator.

Directional Field Synthesis, Design, and Processing

“…As KVFs are generators of isometries, [GRK12] used Generalized Multi-Dimensional Scaling, a tool which has been previously used for computing approximate isometries between surfaces, for computing AKVFs. Finally, [dGLB * 14] used their general tensor decomposition, and [AOCBC15] used equation (15) by directly discretizing the covariant derivative. This property was leveraged in [ABCCO13], using the operator representation of vector fields, for computing AKVFs.…”

“…On the other hand, [AOCBC15] discretize directly the covariant derivative through its extrinsic representation as the directional derivative of the coordinates of the vector field, projected on the surface. In [dGLB * 14], the authors propose a general discretization of tensor fields on triangulations, using a decomposition which represents such tensors using five scalar functions.…”

“…Fieldguided texture synthesis on triangulated surface has been proposed [PFH00] (using RBF interpolation) and [Tur01] (using a diffusion equation). More recent approaches [AOCBC15] Table 1: Summary of field synthesis algorithms. A similar system focusing on fast, interactive manipulation is proposed in [ZMT06], where singularities can be manually introduced and moved over the surface.…”

Vector field processing on triangle meshes

“…Other ways to think of vector fields on manifolds in the continuous setting can also be leveraged to derive discrete representations. For instance, vector fields can be characterized as derivations of smooth functions on a manifold using directional derivative; this has led to an operator representation of vector fields, used first in fluid animation [Mullen et al 2009;Pavlov et al 2011;Gawlik et al 2011], and more recently in geometry processing [Azencot et al 2013;Azencot et al 2015].…”

“…In the discrete setting, Pavlov et al [2011] showed that if one defines a discrete notion of diffeormorphism that transfers scalar values between dual cells, then the resulting Lie derivative turns out to be (the Hodge star of) the discrete 1-form we used above, since it represents fluxes across dual cells. Since then, this functional point of view was shown to be very relevant to, e.g., the problem of finding correspondences between meshes [Azencot et al 2013;Azencot et al 2015]. We refer the reader to [Gawlik et al 2011] for a complete treatment of the dynamics of fluids using discrete diffeomorphisms.…”