Vertex Configurations and Their Relationship on Orthogonal Pseudo-Polyhedra

Authors: Jefri Marzal, Hong Xie, Chun Che Fung

Abstract:

Vertex configuration for a vertex in an orthogonal pseudo-polyhedron is an identity of a vertex that is determined by the number of edges, dihedral angles, and non-manifold properties meeting at the vertex. There are up to sixteen vertex configurations for any orthogonal pseudo-polyhedron (OPP). Understanding the relationship between these vertex configurations will give us insight into the structure of an OPP and help us design better algorithms for many 3-dimensional geometric problems. In this paper, 16 vertex configurations for OPP are described first. This is followed by a number of formulas giving insight into the relationship between different vertex configurations in an OPP. These formulas will be useful as an extension of orthogonal polyhedra usefulness on pattern analysis in 3D-digital images.

Keywords: Orthogonal Pseudo Polyhedra, Vertex configuration

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1083579

References:

[1] D. Ayala and J. Rodriquez, "Connected component labeling based on the EVM Model," in The 18th spring conference on Computer graphics, 2002, pp. 63-71.
[2] B. Yip and R. Klette, "Angle Counts for Isothetic Polygons and Polyhedra," Pattern Recognition Letter, vol. 24, pp. 1275-1278, 2003.
[3] A. Aquilera and D. Ayala, "Solving point and plane vs orthogonal polyhedra using the extreme vertices model (EVM)," presented at the The Sixth International Conference in Central Europe on Computer Graphics and Visualization'98, 1998.
[4] J. Czyzowicz, et al., "Illuminating rectangles and triangles in the plane," Journal of Combinatorial Theory Series B archive, vol. 57, 1993.
[5] M. d. Berg, et al., Computational Geometry, Second ed.: Springer, 2000.
[6] F. P. Preparata and M. I. Shamos, Computational Geometry an Introduction. New York: Springer-Verlag, 1985.
[7] K. Tang and T. C. Woo, "Algorithmic aspects of alternating sum of volumes. Part 1: Data structure and difference operation," Computer- Aided Design, vol. 23, pp. 357-366, June 1991.
[8] J. R. Rossignac and A. A. G. Requicha, "Construcitve Non-Regularized Geometry," Computer - Aided Design, vol. 23, pp. 21-32, 1991.
[9] H. S. M. Coxeter, Regular polytopes. New York: Dover Publications, 1973.
[10] J. D. Foley, et al., Computer Graphics: Principles and Practice in C, 2nd edition ed.: Addison-Wesley, 1996.
[11] T. Biedl and B. Genc, "Reconstructing orthogonal polyhedra from putative vertex sets," technical reports, 2007.
[12] R. Juan-Arinyo, "Domain extension of isothetic polyhedra with minimal CSG representation," Computer Graphics Forum, vol. 5, pp. 281-293, 1995.
[13] K. Voss, Discrete Images, Objects, and Functions in Zn. Berlin: Springer, 1993.
[14] S. L. Senk, Advanced Algebra. Chicago: Scott Foresman/Addison Wesley, 1998.