Discrete & Computational Geometry

, Volume 35, Issue 2, pp 223–240

Minkowski Sums of Monotone and General Simple Polygons

Article

DOI: 10.1007/s00454-005-1206-y

Cite this article as:
Oks, E. & Sharir, M. Discrete Comput Geom (2006) 35: 223. doi:10.1007/s00454-005-1206-y

Abstract

Let P be a simple polygon with m edges, which is the disjoint union of k simple polygons, all monotone in a common direction u, and let Q be another simple polygon with n edges, which is the disjoint union of ℓ simple polygons, all monotone in a common direction v. We show that the combinatorial complexity of the Minkowski sum P ⊕ Q is O(kℓmnα(min{m,n})), where α(·) is the inverse Ackermann function. Some structural properties of the case k = ℓ = 1 have been (implicitly) studied in [17]. We rederive these properties using a different proof, apply them to obtain the above complexity bound for k = ℓ = 1, obtain several additional properties of the sum for this special case, and then use them to derive the general bound.

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.School of Computer Science, Tel Aviv University, Tel Aviv 69978Israel
  2. 2.Courant Institute of Mathematical Sciences, New York University, New York, NY 10012USA

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