Given a triangulation of a simple polygon P, we present linear-time algorithms for solving a collection of problems concerning shortest paths and visibility within P. These problems include calculation of the collection of all shortest paths inside P from a given source vertex s to all the other vertices of P, calculation of the subpolygon of P consisting of points that are visible from a given segment within P, preprocessing P for fast "ray shooting" queries, and several related problems.
An O(n log n) algorithm for planning a purely translational motion for a simple convex object amidst polygonal barriers in two-dimensional space is given. The algorithm is based on a new generalization of Voronoi diagrams (similar to that proposed by Chew and Drysdale [1] and by Fortune [2]), and adapts and uses a recent technique of Yap for the efficient construction of these diagrams.
We show that the number of critical positions of a convex polygonal object B moving amidst polygonal barriers in two-dimensional space, at which it makes three simultaneous contacts with the obstacles but does not penetrate into any obstacle is O(knAs(kn)) for some s<6, where k is the number of boundary segments of B, n is the number of wall segments, and As(q) is an almost linear function of q yielding the maximal number of "breakpoints" along the lower envelope (i.e., pointwise minimum) of a set of q continueus functions each pair of which intersect in at most s points (here a breakpoint is a point at which two of the functions simultaneously attain the minimum). We also present an example where the number of such critical contacts is ft(k2n2), showing that in the worst case our upper bound is almost optimal.
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