Fat Triangles Determine Linearly Many Holes

Abstract

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.