Harmonic Functions for Rotational Symmetry Vector Fields
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Date
2016
Journal Title
Journal ISSN
Volume Title
Publisher
The Eurographics Association and John Wiley & Sons Ltd.
Abstract
Representing rotational symmetry vector as a set of vectors is not suitable for design due to lacking of a consistent ordering for measurement. In this paper we introduce a spectral method to find rotation invariant harmonic functions for symmetry vector field design. This method is developed for 3D vector fields, but it is applicable in 2D. Given the finite symmetry group G of a symmetry vector field v(x) on a 3D domain W, we formulate the harmonic function h(s) as a stationary point of group G. Using the real spherical harmonic (SH) bases, we showed the coefficients of the harmonic functions are an eigenvector of the SH rotation matrices corresponding to group G. Instead of solving eigen problems to obtain the eigenvector, we developed a forward constructive method based on orthogonal group theory. The harmonic function found by our method is not only invariant under G, but also expressive and can distinguish different rotations with respect to G. At last, we demonstrate some vector field design results with tetrahedron-symmetry, cube-symmetry and dodecahedron-symmetry groups.
Description
@article{10.1111:cgf.13047,
journal = {Computer Graphics Forum},
title = {{Harmonic Functions for Rotational Symmetry Vector Fields}},
author = {Shen, Zhongwei and Fang, Xianzhong and Liu, Xinguo and Bao, Hujun and Huang, Jin},
year = {2016},
publisher = {The Eurographics Association and John Wiley & Sons Ltd.},
ISSN = {1467-8659},
DOI = {10.1111/cgf.13047}
}