Optimal shortest path queries in a simple polygon
SCG '87 Proceedings of the third annual symposium on Computational geometry
On translating a set of rectangles
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Structured visibility profiles with applications to problems in simple polygons (extended abstract)
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Assembly sequencing with toleranced parts
SMA '95 Proceedings of the third ACM symposium on Solid modeling and applications
Monotonicity of rectilinear geodesics in d-space (extended abstract)
Proceedings of the twelfth annual symposium on Computational geometry
A general framework for assembly planning: the motion space approach
Proceedings of the fourteenth annual symposium on Computational geometry
Geometric algorithms for static leaf sequencing problems in radiation therapy
Proceedings of the nineteenth annual symposium on Computational geometry
On the qualitative structure of a mechanical assembly
AAAI'92 Proceedings of the tenth national conference on Artificial intelligence
GD'10 Proceedings of the 18th international conference on Graph drawing
Monotone drawings of graphs with fixed embedding
GD'11 Proceedings of the 19th international conference on Graph Drawing
Computing shortest heterochromatic monotone routes
Operations Research Letters
Optimal uniformly monotone partitioning of polygons with holes
Computer-Aided Design
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We study the class of problems associated with the detection and computation of monotone paths among a set of disjoint obstacles. We give an &Ogr;(nE) algorithm for finding a monotone path (if one exists) between two points in the plane in the presence of polygonal obstacles. (Here, E is the size of the visibility graph defined by the n vertices of the obstacles.) If all of the obstacles are convex, we prove that there always exists a monotone path between any two points s and t. We give an &Ogr;(n log n) algorithm for finding such a path for any s and t, after an initial &Ogr;(E + n log n) preprocesing. We introduce the notions of “monotone path map”, and “shortest monotone path map” and give algorithms to compute them. We apply our results to a class of separation and assembly problems, yielding polynomial-time algorithms for planning an assembly sequence (based on separations by single translations) of arbitrary polygonal parts in two dimensions.