Mathematical modeling of the interaction of non-oriented convex polytopes
An Φ-function for two non-oriented convex polytopes is set up. The Φ-function can be used to construct a mathematical model of packing optimization problem for non-oriented polytopes. An example of an Φ-function for two non-oriented parallelepipeds is given.
Keywordsmathematical modeling Φ-function non-oriented convex polytopes packing optimization problem
Unable to display preview. Download preview PDF.
- 1.Yu. G. Stoyan and S. V. Yakovlev, Mathematical Models and Optimization Methods of Geometric Design [in Russian], Naukova Dumka, Kyiv (1986).Google Scholar
- 2.N. Mayer, E. Fogel, and D. Halperin, “Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space,” in: Proc. 14th ACM Symp. on Solid and Physical Modeling, Haifa, Israel (2010), pp. 1–10.Google Scholar
- 3.O. Takayuki, Approaches to 3D Free-Form Cutting and Packing Problems and their Applications: A Survey, IBM Research Laboratory, Tokyo (1998).Google Scholar
- 6.M. Gan, N. Gopinathan, X. Jia, and R. A. Williams, “Predicting packing characteristics of particles of arbitrary shapes,” KONA, No. 22, 82–93 (2004).Google Scholar
- 7.V. S. Doroshenko, “3D-technologies in foundry,” Vynakhidnyk i Ratsionalizator, No. 2, 12–15 (2009).Google Scholar
- 8.F. Eisenbrand, S. Funke, A. Karrenbauer, J. Reichel, and E. Schömer, “Packing a trunk — now with a twist!,” in: Proc. SPM 2005 ACM Symp. on Solid and Physical Modeling, New York, USA (2005), pp. 197–206.Google Scholar
- 10.J. Bennell and J. Oliveira, “The geometry of nesting problems: A tutorial,” Europ. J. of Operational Research, No. 184, 397–415 (2008).Google Scholar
- 11.Yu. G. Stoyan, “Φ-function and its basic properties,” Dokl. NAN Ukr., Ser. A, No. 8, 112–117 (2001).Google Scholar
- 15.Yu. Stoyan, A. Pankratov, and G. Sheithauer, “Translational packing non-convex polytopes into a parallelepiped,” J. of Mechanical Engineering, 12, No. 3, 67–76 (2009).Google Scholar
- 16.K. Kuratowski, Topology, Vol. 1, Academic Press, New York (1966).Google Scholar