DOI: 10.12732/ijam.v37i2.4
ON THE GEOMETRIC PROPERTIES
OF THE MINKOWSKI OPERATOR
Mashrabjon Mamatov 1 , Jalolxon Nuritdinov 2, §
1 Department of Geometry and Topology
of National University of Uzbekistan
Tashkent - 100174, UZBEKISTAN
2 Department of Digital Technologies and
Mathematics of Kokand University
Kokand - 150700, UZBEKISTAN
Abstract. This article is about Minkowski difference of sets, which is one of the Minkowski operators. The necessary and sufficient conditions for the existence of the Minkowski difference of given regular polygons in the plane are derived. The method of finding the Minkowski difference of given regular tetrahedrons in the Euclidean space R3 is explained. Results for finding the Minkowski difference of given n-dimensional cubes in space Rn are also presented.
At the end of the article, the obtained results are summarized and a geometric method for finding the Minkowski difference of the convex set M and compact set N given in Rn is shown. The theory of foliations is applied to find the Minkowski difference of sets. New geometric concepts such as “dense embedding” and “completely dense embedding” are introduced. An important
geometric property of the Minkowski operator is introduced and proved as a theorem.
How
to cite this paper?
DOI: 10.12732/ijam.v37i2.4
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2024
Volume: 37
Issue: 2
References
[14] J.T. Nuritdinov, Minkowski difference of cubes, In: Proceedings of International
Conference on Mathematics and Mathematics Education, Pamukkale University, Denizli, Turkey (2022), 88-90.
[15] K. Sugihara, Invertible Minkowski sum of polygons. In: A. Braquelaire, J.O. Lachaud, A. Vialard (eds), Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 2301, Springer, Berlin-Heidelberg (2002); https://doi.org/10.1007/3-540-45986-3 31.
[16] Y. Feng, Y. Tan, On Minkowski difference-based contact detection in discrete/
discontinuous modelling of convex polygons/polyhedra, Engineering Computations, 37, No 1 (2020), 54-72; https://doi.org/10.1108/EC-03-2019-0124.
[17] Y. Martinez-Maure, Geometric study of Minkowski differences of plane convex bodies, Canad. J. Math., 58, No 3 (2006), 600-624.
[18] I. Tamura, Topology of Foliations, Moscow, Mir (1979).
[19] D. Husemoller, Fibre Bundles, Springer-Verlag, Berlin and Heidelberg GmbH and Co. (1975).
[20] R. Hermann, On the differential geometry of foliations, Annals of Math., 72 (1960), 445-457.
[21] H. Suzuki, Holoriomy groupoids of generalized foliations. Hokkaido Math. J., 19, No 2 (1990), 215-227.
(c) 2010-2024, Academic Publications, Ltd.; https://www.diogenes.bg/ijam/