Fat Triangles Determine Linearly Many Holes
Matousek Jiri, János Pach, Micha Sharir, Shmuel Sifrony, Emo Welzl · SIAM Journal on Computing · 1994
The authors show 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 $O(n\log \log n)$ straight segments (where the constants of proportionality depend on $\delta $). This latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges.