Local polyhedra and geometric graphs

We introduce a new realistic input model for geometric graphs and nonconvex polyhedra. A geometric graph G is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor and (2) the lengths of the longest and shortest edges differ by at most a polynomial factor. A polyhedron is local if all its faces are simplices and its edges form a local geometric graph. We show that any boolean combination of any two local polyhedra in IRd, each with n vertices, can be computed in O(n log n) time, using a standard hierarchy of axis-aligned bounding boxes. Using results of de Berg, we also show that any local polyhedron in IRd has a binary space partition tree of size O(n logd-1 n). Finally, we describe efficient algorithms for computing Minkowski sums of local polyhedra in two and three dimensions.

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