Minkowski sums of point sets defined by inequalities.

Pasko, A., Okounev, O. and Savchenko, V. V., 2003. Minkowski sums of point sets defined by inequalities. Computers and Mathematics with Applications, 45 (10/11), pp. 1479-1487.

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Official URL: http://hyperfun.org/Mink.pdf

DOI: 10.1016/S0898-1221(03)00131-7

Abstract

The existing approaches support Minkowski sums for the boundary, set-theoretic, and ray representations of solids. In this paper, we consider the Minkowski sum operation in the context of geometric modeling using real functions. The problem is to find a real function f3(X) for the Minkowski sum of two objects defined by the inequalities f1(X) ≥ 0 and f2(X) ≥ 0. We represent the Minkowski sum as a composition of other operations: the Cartesian product, resulting in a higher-dimensional object, and a mapping to the original space. The Cartesian product is realized as an intersection in the higher-dimensional space, using an R-function. The mapping projects the resulting object along n coordinate axes, where n is the dimension of the original space. We discuss the properties of the resulting function and the problems of analytic and numeric implementation, especially for the projection operation. Finally, we apply Minkowski sums to implement offsetting and metamorphosis between set-theoretic solids with curvilinear boundaries.

Item Type:Article
Subjects:Science > Mathematics
Generalities > Computer Science and Informatics
Technology > Manufacturing and Design > Design
Group:Media School
ID Code:9629
Deposited By:Professor Alexander Pasko
Deposited On:10 Mar 2009 18:58
Last Modified:07 Mar 2013 15:07

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