Proceedings 32nd Annual Symposium of Foundations of Computer Science (1991)
San Juan, Puerto Rico
Oct. 1, 1991 to Oct. 4, 1991
J. Matousek , Dept. of Appl. Math., Charles Univ., Praha, Czechoslovakia
It is shown that for every fixed delta >0 the following holds: if F is a union of n triangles, all of whose angles are at least delta , then the complement of F has O(n) connected components, and the boundary of F consists of O(n log log n) segments. This latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges. A randomized algorithm that computes F in expected time O(n2/sup alpha (n)/ log n) is given. Several applications of these results are presented.
randomized algorithm, fat triangles, linearly many holes, connected components, complexity
M. Sharir, E. Welzl, N. Miller, S. Sifrony, J. Matousek and J. Pach, "Fat triangles determine linearly many holes (computational geometry),"  Proceedings 32nd Annual Symposium of Foundations of Computer Science(FOCS), San Juan, Puerto Rico, 1991, pp. 49-58.