An Operator Approach to Tangent Vector Field Processing
Article first published online: 19 AUG 2013
© 2013 The Author(s) Computer Graphics Forum © 2013 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum
Volume 32, Issue 5, pages 73–82, August 2013
How to Cite
Azencot, O., Ben-Chen, M., Chazal, F. and Ovsjanikov, M. (2013), An Operator Approach to Tangent Vector Field Processing. Computer Graphics Forum, 32: 73–82. doi: 10.1111/cgf.12174
- Issue published online: 19 AUG 2013
- Article first published online: 19 AUG 2013
- I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling
In this paper, we introduce a novel coordinate-free method for manipulating and analyzing vector fields on discrete surfaces. Unlike the commonly used representations of a vector field as an assignment of vectors to the faces of the mesh, or as real values on edges, we argue that vector fields can also be naturally viewed as operators whose domain and range are functions defined on the mesh. Although this point of view is common in differential geometry it has so far not been adopted in geometry processing applications. We recall the theoretical properties of vector fields represented as operators, and show that composition of vector fields with other functional operators is natural in this setup. This leads to the characterization of vector field properties through commutativity with other operators such as the Laplace-Beltrami and symmetry operators, as well as to a straight-forward definition of differential properties such as the Lie derivative. Finally, we demonstrate a range of applications, such as Killing vector field design, symmetric vector field estimation and joint design on multiple surfaces.