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On Discrete Killing Vector Fields and Patterns on Surfaces
Author(s) -
BenChen Mirela,
Butscher Adrian,
Solomon Justin,
Guibas Leonidas
Publication year - 2010
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/j.1467-8659.2010.01779.x
Subject(s) - symmetry (geometry) , vector field , homogeneous space , killing vector field , complex lamellar vector field , tangent vector , discretization , infinitesimal , tangent , fundamental vector field , surface (topology) , rotational symmetry , mathematics , pure mathematics , geometry , mathematical analysis , algebra over a field , solenoidal vector field , adjoint representation of a lie algebra , lie conformal algebra
Symmetry is one of the most important properties of a shape, unifying form and function. It encodes semantic information on one hand, and affects the shape's aesthetic value on the other. Symmetry comes in many flavors, amongst the most interesting being intrinsic symmetry, which is defined only in terms of the intrinsic geometry of the shape. Continuous intrinsic symmetries can be represented using infinitesimal rigid transformations, which are given as tangent vector fields on the surface – known as Killing Vector Fields. As exact symmetries are quite rare, especially when considering noisy sampled surfaces, we propose a method for relaxing the exact symmetry constraint to allow for approximate symmetries and approximate Killing Vector Fields, and show how to discretize these concepts for generating such vector fields on a triangulated mesh. We discuss the properties of approximate Killing Vector Fields, and propose an application to utilize them for texture and geometry synthesis.