Editing Point in polygon (section)

=== Limited precision ===
If implemented on a computer with [[finite precision arithmetics]], the results may be incorrect if the point lies very close to that boundary, because of rounding errors. For some applications, like video games or other entertainment products, this is not a large concern since they often favor speed over precision. However, for a [[Formal verification|formally correct]] [[computer program]], one would have to introduce a [[Numerical analysis|numerical]] [[Tolerance (engineering)|tolerance]] ε and test in line whether ''P'' (the point) lies within ε of ''L'' (the Line), in which case the algorithm should stop and report "''P'' lies very close to the boundary."

Most implementations of the ray casting algorithm consecutively check intersections of a ray with all sides of the polygon in turn. In this case the following problem must be addressed. If the ray passes exactly through a [[vertex (geometry)|vertex]] of a polygon, then it will intersect 2 segments at their endpoints. While it is OK for the case of the topmost vertex in the example or the vertex between crossing 4 and 5, the case of the rightmost vertex (in the example) requires that we count one intersection for the algorithm to work correctly. A similar problem arises with horizontal segments that happen to fall on the ray. The issue is solved as follows: If the intersection point is a vertex of a tested polygon side, then the intersection counts only if the other vertex of the side lies below the ray. This is effectively equivalent to considering vertices ''on'' the ray as lying slightly ''above'' the ray.

Once again, the case of the ray passing through a vertex may pose numerical problems in [[finite precision arithmetics]]: for two sides adjacent to the same vertex the straightforward computation of the intersection with a ray may not give the vertex in both cases. If the polygon is specified by its vertices, then this problem is eliminated by checking the y-coordinates of the ray and the ends of the tested polygon side before actual computation of the intersection. In other cases, when polygon sides are computed from other types of data, other tricks must be applied for the [[numerical robustness]] of the algorithm.