BHPolygons

BHPolygons are a class of polygonal shapes used in computational geometry to create multi-resolution representations of planar figures. The concept, based on a binary halving (BH) operation, generates a hierarchy of polygons from an initial polygon while approximately preserving area and overall geometry. They are designed to support efficient rendering, collision detection, and shape analysis.

Construction of BHPolygons begins with a simple polygon P with vertices v1, v2, ..., vn. The BH operation pairs successive edges and replaces each pair with a single edge whose direction is the average of the two edge directions. The replacement vertex is placed so that the polygon’s centroid remains near that of P. By applying the operation iteratively, one obtains a sequence of BHPolygons P0, P1, P2, ... with decreasing vertex counts. The number of vertices roughly halves with each iteration, though the exact count depends on local geometry and a user-specified parameter controlling alignment with curvature.

Properties of BHPolygons include their non-uniqueness: the construction depends on the order of processing and the averaging rule chosen. They tend to smooth sharp corners, roughly preserve area, and maintain simple polygon topology when starting from simple polygons. The representation is multi-resolution, enabling level-of-detail rendering and progressive transmission in graphics pipelines.

Applications of the BHPolygon framework appear in vector graphics editors for contour simplification, real-time rendering for level-of-detail control, and collision-detection systems that require fast, approximate bounding polygons. See also polygon simplification, multiresolution polygons, vertex hull, convex hull, and mesh simplification.