We s h o w that for every xed > 0 the following holds: If F is a union of n triangles, all of whose angles are at least , then the complement o f F has O(n) connected components, and the boundary of F consists of O(n log log n) straight segments (where the constants of proportionality d e p e n d o n ). This latter complexity becomes linear if all triangles are of roughly the same size or if they are all in nite wedges.
Matousek et al. REF show that the union of n δ-fat triangles has complexity O(n log log n) while for n general triangles this is Ω(n 2 ).
Fat Triangles Determine Linearly Many Holes
FOCS
Mathematics Computer Science