Geometry-Free Polygon Splitting

A polygon splitting algorithm is a combinatorial recipe. The description and the implementation of polygon splitting should not depend on the embedding geometry. Whether a polygon is being split in Euclidean, in spherical, in oriented projective, or in hyperbolic geometry should not be part of the description of the algorithm. The algorithm should be purely combinatorial, or geometry free. The geometry ultimately needs to be specified, and the geometric predicates can only be implemented after specifying the coordinate type and the number type. But the geometry, along with the coordinates and number type in that geometry, remain a late “plug-in”, to be added only to the finished algorithm. We describe a kernel for hyperbolic geometry. Once classes and predicates in that geometry are developed, hyperbolic geometry can be used as a plug-in to polygon splitting alongside other geometries. We also describe an algorithm for the splitting of a polygon represented using its bounding lines. The use of this dual representation ensures that all predicates are computed directly from input data. This remains the case even if the same polygon is split multiple times, as occurs in BSP tree construction.