Deconstructing approximate offsets

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 shape P; then, P’s offset constitutes an accurate, vertexreduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. An alternative algorithm, based purely on rational arithmetic, answers the same deconstruction problem, up to an uncertainty parameter δ, and its running time depends on the parameter δ (in addition to the other input parameters: n, ε and the radius of the disk). If the input shape is found to be approximable, the rational-arithmetic algorithm also computes an approximate solution shape for the problem. For convex shapes, the complexity of the exact decision ...

• Type: Journal
• Year: 2011
• Where: COMPGEOM
• Authors: Eric Berberich, Dan Halperin, Michael Kerber, Roza Pogalnikova
• Keywords: COMPGEOM 2011, Decision Algorithm, Discrete Geometry, Polygonal Shape, Problem Complexity