Deconstructing Approximate Offsets

Eric Berberich, Dan Halperin, Michael Kerber, Roza Pogalnikova

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance ε in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(nlogn)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using the cgal library, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter δ its running time additionally depends on δ. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.

Original languageEnglish
Pages (from-to)964-989
Number of pages26
JournalDiscrete and Computational Geometry
Volume48
Issue number4
DOIs
StatePublished - Dec 2012

Keywords

  • Deconstruction
  • Minkowski sums
  • Offsets
  • Polygonal smoothing

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