pointwithinpolygon

Point within polygon, or point-in-polygon (PIP), is a classic problem in computational geometry. Given a polygon defined by an ordered list of vertices, the question is whether a query point lies inside the polygon’s interior, on its boundary, or outside. Depending on the application, boundary points may be treated as inside, outside, or as a separate category.

The most common method is the ray casting algorithm. A horizontal or vertical ray is projected from the query point to infinity, and the number of times it intersects polygon edges is counted. If the count is odd, the point is inside; if even, it is outside. Special care is required for intersections at polygon vertices and for points lying precisely on edges. A separate boundary test checks if the point lies on any edge by testing for collinearity and within-segment bounds. Another approach is the winding number method, which sums the signed angles or edge orientations around the point; a nonzero winding number indicates inside, and zero indicates outside. Winding numbers tend to be more robust for complex polygons, including those that wrap or overlap in certain representations.

Performance and data considerations: for a polygon with n vertices, a straightforward PIP test runs in O(n) time per query. For many queries, preprocessing and spatial indexing (or using polygons with holes as multipolygons) can improve efficiency. Floating-point precision and degeneracies (nearly collinear edges, points on vertices) require epsilon-tolerant checks. Applications span geographic information systems, computer graphics, and robotics.