On fat partitioning, fat covering and the union size of polygons

Abstract

The complexity of the contour of the union of simple polygons with n vertices in total can be O( n 2) in general. A notion of fatness for simple polygons is introduced that extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity O( n log log n), which is a generalization of a similar result for fat triangles (Matoušek et al., 1994). Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, and more. The result is based on a new method to partition a fat simple polygon P with n vertices into O( n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(l) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any exact cover of a fat simple polygon by a linear number of fat triangles.