On Discrete Killing Vector Fields and Patterns on Surfaces

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.

[1]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[2]  S. Myers.  Isometries of 2-Dimensional Riemannian Manifolds into Themselves. , 1936, Proceedings of the National Academy of Sciences of the United States of America.

[3]  K. Yano.  On Harmonic and Killing Vector Fields , 1952 .

[4]  R. Matzner.  Almost symmetric spaces and gravitational radiation. , 1968 .

[5]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[6]  D. Raine.  General relativity , 1980, Nature.

[7]  R. Zalaletdinov.  Approximate Symmetries in General Relativity , 1999, gr-qc/9912021.

[8]  Anil N. Hirani,et al.  Discrete exterior calculus , 2005, math/0508341.

[9]  G. Hall.  Symmetries and Curvature Structure in General Relativity , 2004 .

[10]  Kun Zhou,et al.  Mesh quilting for geometric texture synthesis , 2006, ACM Trans. Graph..

[11]  N. Mitra,et al.  Partial and approximate symmetry detection for 3D geometry , 2006, ACM Transactions on Graphics.

[12]  Kun Zhou,et al.  Mesh quilting for geometric texture synthesis , 2006, SIGGRAPH 2006.

[13]  Leonidas J. Guibas,et al.  Partial and approximate symmetry detection for 3D geometry , 2006, ACM Trans. Graph..

[14]  Hugues Hoppe,et al.  Design of tangent vector fields , 2007, SIGGRAPH 2007.

[15]  M. Kilian,et al.  Geometric modeling in shape space , 2007, SIGGRAPH 2007.

[16]  Yanxi Liu,et al.  Performance evaluation of state-of-the-art discrete symmetry detection algorithms , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[17]  Leonidas J. Guibas,et al.  Global Intrinsic Symmetries of Shapes , 2008, Comput. Graph. Forum.

[18]  C. Beetle.  Approximate Killing Fields as an Eigenvalue Problem , 2008, 0808.1745.

[19]  Leonidas J. Guibas,et al.  Discovering structural regularity in 3D geometry , 2008, ACM Trans. Graph..

[20]  Ligang Liu,et al.  Partial intrinsic reflectional symmetry of 3D shapes , 2009, ACM Trans. Graph..

[21]  Hans-Peter Seidel,et al.  Symmetry Detection Using Feature Lines , 2009, Comput. Graph. Forum.

[22]  Thomas Müller,et al.  Eurographics/ Ieee-vgtc Symposium on Visualization 2009 the Gödel Engine -an Interactive Approach to Visualization in General Relativity , 2022 .

[23]  Kai Xu,et al.  Partial intrinsic reflectional symmetry of 3D shapes , 2009, SIGGRAPH 2009.

[24]  Hans-Peter Seidel,et al.  A probabilistic framework for partial intrinsic symmetries in geometric data , 2009, 2009 IEEE 12th International Conference on Computer Vision.

[25]  Hans-Peter Seidel,et al.  Symmetry Detection Using Line Features , 2009 .

[26]  E. Akleman,et al.  Cyclic plain-weaving on polygonal mesh surfaces with graph rotation systems , 2009, SIGGRAPH '09.

[27]  Ergun Akleman,et al.  Cyclic Plain-Weaving on Polygonal Mesh Surfaces with Extended Graph Rotation Systems , 2009 .

[28]  Craig S. Kaplan,et al.  Semiregular patterns on surfaces , 2009, NPAR '09.

[29]  Alexander M. Bronstein,et al.  Full and Partial Symmetries of Non-rigid Shapes , 2010, International Journal of Computer Vision.

[30]  Eugene Zhang,et al.  Geometry Synthesis on Surfaces Using Field-Guided Shape Grammars , 2011, IEEE Transactions on Visualization and Computer Graphics.