Abstract
This paper proposes a new formulation of the Minkowski algebra for figures. In the conventional Minkowski algebra, the sum operation was always defined, but its inverse was not necessarily defined. On the other hand, the proposed algebra forms a group, and hence every element has its inverse, and the sum and the inverse operation can be taken freely. In this new algebraic system, some of the elements does not correspond to the figures in an ordinary sense; we call these new elements “hyperfigures”. Physical interpretations and practical usage of the hyperfigures are also discussed.
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References
Barrera, A.H.: Computing the Minkowski sum of monotone polygons. Technical Report of the Special Interest Group on Algorithms of the Information Processing Society of Japan, 96-AL-50-9, 1996.
Boissonnat, J.-D., de Lange, E., and Teillaud, M.: Minkowski operations for satellite antenna layout. Proceedings of the13th Annual ACM Symposium on Computational Geometry, pp. 67–76, 1997.
Bourbaki, N.: Éléments de Mathématique, Algébre 1. Hermann, Paris, 1964.
Farin, G.: Curves and Surfaces for Computer Aided Geometric Design, 4th Edition, Academic Press, Boston, 1999.
Ghosh, P.K.: A mathematical model for shape description using Minkowski operators. Computer Vision, Graphics and Image Processing, vol. 44, pp. 239–269, 1988.
Ghosh, P.K.: A solution of polygon containment, spatial planning, and other related problems using Minkowski operators. Computer Vision, Graphics and Image Processing, vol. 49, pp. 1–35, 1990.
Ghosh, P.K.: An algebra of polygons through the notion of negative shapes. CVGIP: Image Understanding, vol. 54, pp. 119–144, 1991.
Ghosh, P.K.: Vision, geometry, and Minkowski operators. Contemporary Mathematics, vol. 119, pp. 63–83, 1991.
Ghosh, P.K., and Haralick, R. M.: Mathematical morphological operations of boundary-represented geometric objects. Journal of Mathematical Imaging and Vision, vol. 6, pp. 199–222, 1996.
Guibas, L. J., Ramshaw, L., and Stol., J.: A kinetic framework for computational geometry. Proceedings of the 24th Annual IEEE Symposium on Foundation of Computer Sciences, pp. 100–111, 1983.
Guibas, L. J., and Seidel, R.: Computing convolutions by reciprocal search. Discrete and Computational Geometry, vol. 2, pp. 157–193, 1987.
Haralick, R. M., Sternbery, S.R., and Zhuang, X.: Image analysis using mathematical morphology. IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. PAMI-9, pp. 532–550, 1987.
Har-Peled, S., Chan, T.M., Aronov, B., Halperin, D., and Snoeyink, J.: The complexity of a single face of a Minkowski sum. Proc. of the 7th Canadian Conference on Computational Geometry, pp. 91–96, 1995.
Hataguchi, T., and Sugihara, K.: Exact algorithm for Minkowski operators. Proc. of the 2nd Asian Conference on Computer Vision, vol. 3, pp. 392–396, 1995.
Kaul, A., O’Connor, M.A., and Srinivasan, V.: Computing Minkowski sums ofregular polygons. Proc. of the 3rd Canadian Conf. on Computational Geometry, pp. 74–77, 1991.
Kaul, A., and Farouki, R.T.: Computing Minkowski sums of plane curves. International Journal of Computational Geometry and Applications, vol. 5 (1995), pp. 413–432.
Kedem, K., Livne, R., Pach, J., and Sharir, M.: On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete and Computational Geometry, vol. 1, pp. 59–71, 1986.
Leven, D., and Sharir, M.: Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagram. Discrete and computational Geometry, vol. 2, pp. 9–31, 1987.
Lozano-Perez, T., and Wesley, M.A.: An algorithm for planning collision-free paths among polyhedral obstacles. Commun. of the ACM, vol. 22, pp. 560–570, 1979.
Mikusiński, J.: Operational Calculus (in Japanese), Shokabo, Tokyo, 1963.
Mount, D., and Silverman, R.: Combinatorial and computational aspects of Minkowski decompositions. Contemporary Mathematics, vol. 119, pp. 107–124, 1991.
Ramkumar, G.D.: An algorithm to compute the Minkowski sum outer-face of two simple polygons. Proceedings of the12th Annual ACM Symposium on Computational Geometry, pp. 234–241, 1996.
Rossignac, J.R., and Requicha, A.A. G.: Offseting operations in solid modeling. Computer Aided Geometric Modeling, vol. 3, pp. 129–148, 1986.
Schmitt, M.: Support function and Minkowski addition of non-convex sets. P. Maragos, R. W. Schafer and M. A. Butt (eds.): Mathematical Morphology and its Applications to Image and Signal Processing, Kluwer Academic Publishers, 1996, pp. 15–22.
Schwartz, J.T.: Finding the minimum distance between two convex polygons. Information Processing Letters, vol. 13 pp. 168–170, 1981.
Serra, J.: Image Analysis and Mathematical Morphology, 2nd Edition, Academic Press, 1988.
Sugihara, K., Imai, T., and Hataguchi, T.: An algebra for slope-monotone closed curves. International Journal of Shape Modeling, vol. 3, pp. 167–183, 1997.
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Sugihara, K. (2003). Hyperfigures and Their Interpretations. In: Asano, T., Klette, R., Ronse, C. (eds) Geometry, Morphology, and Computational Imaging. Lecture Notes in Computer Science, vol 2616. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36586-9_15
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DOI: https://doi.org/10.1007/3-540-36586-9_15
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