On the boundary complexity of the union of fat triangles

A triangle is said to be δ-fat if its smallest angle is at least δ>0. A connected component of the complement of the union of a family of triangles is called hole. It is shown that any family of δ-far triangles in the plane determines at most O (n/δ log 2/δ) holes. This improves on some earlier bounds of (Efrat et al., 1993; Matousek et al., 1994). Solving a problem of (Agarwal and Bern, 1999) we also give a general upper bound for the number of holes determined by n triangles in the plane with given angles. As a corollary, we obtain improved upper bounds for the boundary complexity of the union of fat polygons in the plane, which, in turn, leads to better upper bounds for the running times of some known algorithms for motion planning, for finding a separator line for a set of segments, etc