Efficient Angle Summation Algorithm for Point Inclusion Test and Its Robustness

A winding angle summation approach to the point-in-polygon problem is considered. The winding angle summation is backward stable and resistant to possible gaps between subsequent edges. A precomputed bounding volume hierarchy can be used to accelerate point queries for winding angle summation. The main theorem states that a BVH-accelerated point query takes no longer than O ( K log n K ) time if all the bounding boxes are tight. Here, n is the number of edges in the polygon, and K is the scaled absolute winding angle (called the condition number). It is proved that the absolute winding angle never exceeds the absolute turning angle for a closed piecewise smooth regular curve. Hence, there is a non-trivial bound on query time complexity, independent of the point being tested.