Convex polyhedra with triangular faces and cone triangulation
DOI:
https://doi.org/10.2298/YJOR1101079SKeywords:
Triangulation of polyhedra, minimal triangulation, graph algorithms, abstract data type of graphAbstract
Considering the problem of the minimal triangulation for a given polyhedra (dividing polyhedra into tetrahedra) it is known that the cone triangulation provides the number of tetrahedra which is the smallest, or the closest to it. It is also shown that when we want to know whether the cone triangulation is the minimal one, it is necessary to find the order of all vertices, as well as the order of “separating circles”. Here, we will give algorithms for testing the necessary condition for the cone triangulation if it is the minimal one. The algorithm for forming the cone triangulation will also be given.References
Baumgart, B.G. (1975) A polyhedron representation for computer vision. u: Proceedings of the 1975 National Computer Conference. AFIPS Conference Proceedings, Reston, Va: AFIPS Press, vol. 44
Below, A., Loera, J.A., Gebert, J.R. (2000) The complexity of finding small triangulations of convex 3-polytopes. arXiv, math 0012177v1
Brinkmann, G., McKay, B.D. (2007) Fast generation of planar graphs. Communications in Mathematical and in Computer Chemistry / MATCH, vol. 58, br. 2, str. 323-357
Chin, F.Y.L., Fung, S.P.Y., Wang, C.A. (2001) Approximation for minimum triangulations of simplicial convex 3-polytopes. Discrete Computational Geometry, 26, br. 4, 499-511
Develin, M. (2004) Mаximal triangulations of a regular prism. J Combin. Theory Ser. A, Ser. A, 106, br. 1, 159-164
Edelsbrunner, H., Preparata, F.P., West, D.B. (1990) Tetrahedrizing point sets in three dimensions. Journal of Symbolic Computation, 10, 4, 335-347
Goodrich, M., Tamassia, R. (2001) Data structures and algorithms in Java. John Wiley & Sons
Mccormell, J. (2001) Analysis of algorithms: An active learning approach. Jones and Bartlett Publishers
Ruppert, J., Seidel, R. (1992) On the difficulty of triangulating three-dimensional nonconvex polyhedra. Discrete and Computational Geometry, 7, 3, 227-253
Sleator, D.D., Tarjan, R.E., Thurston, W.P. (1988) Rotation distance, triangulations, and hyperbolic geometry. J Am Math Soc, 1, 3, 647-681
Stojanović, M. (2008) Triangulations of some cases of polyhedra with a small number of tetrahedra. Kragujevac Journal of Mathematics, br. 31, str. 85-93
Stojanović, M., Vučković, M. (2007) Algorithms for investigating optimality of cone triangulation for a polyhedron. Kragujevac Journal of Mathematics, br. 30, str. 327-342
Stojanović, M.A. (2005) Algorithms for triangulating polyhedra into a small number of tetrahedra. Matematički vesnik, vol. 57, br. 1-2, str. 1-9
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