Computational geometry in a curved world

We extend the results of straight-edged computational geometry into the curved world by defining a pair of new geometric objects, the splinegon and the splinehedron, as curved generalizations of the polygon and polyhedron. We identify three distinct techniques for extending polygon algorithms to splinegons: the carrier polygon approach, the bounding polygon approach, and the direct approach. By these methods, large groups of algorithms for polygons can be extended as a class to encompass these new objects. In general, if the original polygon algorithm has time complexity O(f(n)), the comparable splinegon algorithm has time complexity at worst O(Kf(n)) where K represents a constant number of calls to members of a set of primitive procedures on individual curved edges. These techniques also apply to splinehedra. In addition to presenting the general methods, we state and prove a series of specific theorems. Problem areas include convex hull computation, diameter computation, intersection detection and computation, kernel computation, monotonicity testing, and monotone decomposition, among others.