Minkowski sums of point sets defined by inequalities

A. Pasko, O. Okunev, V. Savchenko

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1479-1487
Number of pages9
JournalComputers and Mathematics with Applications
Issue number10-11
Publication statusPublished - May 2003
Externally publishedYes


  • Function representation
  • Minkowski sum
  • Projection
  • R-function
  • Shape modeling


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