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 the longest and shortest edges differ in length 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 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−2 n) and depth O(log n); these bounds are tight in the worst case when d ≤ 3. Finally, we describe efficient algorithms for computing Minkowski sums of local polyhedra in two and three dimensions. ∗ A preliminary version of this paper was presen...

• Type: Conference
• Year: 2003
• Where: COMPGEOM
• Authors: Jeff Erickson
• Keywords: COMPGEOM 2003, Geometric Graph, Local Geometric Graph, Local Polyhedra