Point-in-Polygon Algorithm Based on Monolithic Calculation for Included Angle of Half Plane Continuous Chains

摘要

The point-in-polygon test which query about whether a point lies within a polygon or not is a fundamental problem in geometry, and of importance in various applications in GIS (Geographic Information System) and other areas. In taking advantage of the basic idea of the sum of included angle algorithm, a novel improvement for the point-in-polygon test is proposed in this paper. A new concept, the half plane continuous chain is presented, the continuous segments whose endpoints lies similar side by the line through the tested point will be organized as a half plane continuous chain. The monolithic calculation method of included angle for half plane continuous chain is founded, which accumulate the included angle value of each contained edge by directly calculating the included angle between the two endpoints of half plane continuous chains, all intermediate edges' included angle value calculation in each half plane continuous chain are omitted. As a result, the computation time is cut down. The improved algorithm for inclusion test consisting of three phases: (1) organizing edges of a polygon into a minimal number of half plane continuous chains and (2) calculating each chain's included angle value, and accumulating them to a sum and (3) comparing the sum with the constant: ±2π (or ±360°) means included and 0 means not. In the first phase, the computer for splitting polygonal chains into half plane continuous chain will just process Boolean compares. In the second phase, the included angle computation and accumulating times ranges from 0 to 2m, depending on the geometry of the polygon and the test direction, here, m is the fewer number of half plane continuous chains and is always smaller, often much smaller, than the number n of edges. In the case of afield polygon and convex polygon, the number of included angle calculating and accumulating times could be reduced from n to 0 or 5. Analysis shows except in the case of saw-shaped polygon, the improved algorithm is faster than the original in most cases, especially for polygons with large amounts of edges.