On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Finding minimal convex nested polygons
Information and Computation
On the computational geometry of pocket machining
On the computational geometry of pocket machining
An algorithm for generating NC tool paths for arbitrarily shaped pockets with islands
ACM Transactions on Graphics (TOG)
A rational rotation method for robust geometric algorithms
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
On the significance of the theory of convex polyhedra for formal algebra
ACM SIGSAM Bulletin
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Exact and approximate construction of offset polygons
Computer-Aided Design
Approximation of an open polygonal curve with a minimum number of circular arcs and biarcs
Computational Geometry: Theory and Applications
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We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance µ 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, vertex-reduced, 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, and its running time depends on the parameter δ (in addition to the other input parameters: n, δ 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 algorithm drops to O(n), which is also the time required to compute a solution shape P with at most one more vertex than a vertex-minimal one. Our study is motivated by applications from two different domains. However, since the offset operation has numerous uses, we anticipate that the reverse question that we study here will be still more broadly applicable. We present results obtained with our implementation of the rational-arithmetic algorithm.