On Counting and Analyzing Empty Pseudo-triangles in a Point Set

We address the problems of counting and analyzing empty pseudo-triangles defined by a set of points in the plane. First, we assume that the three convex vertices are fixed, and provide algorithms, which solve the counting problem in quadratic time using linear space, for the cases when the pseudo-triangles can be arbitrary or must be star-shaped. In addition, we demonstrate that our approach leads to a fairly simple method for retrieving empty pseudo-triangles optimal in a certain sense in cubic time and linear space. The relaxation of our assumption increases the time complexity by factor n(3) in each case.