2D Computational Spherical Geometry

Studying spherical geometry before classical (Chapter 11) and oriented (Chapter 14) projective geometries provides considerable insight to these geometries. Assigning coordinates to spherical objects also makes the assignment of homogeneous coordinates to projective objects more intuitive.

But spherical geometry is also interesting in its own right as many geometric systems are implemented on the sphere. Such systems will most naturally be implemented not using two spherical coordinates (latitude and longitude), but by operating on vectors in Euclidean 3D space (Chapter 3). Systems using spherical geometry could indeed be implemented by operating on vectors without abstracting the structure of the sphere, but such systems are inherently flawed. The notions that need to be abstracted are repeatedly defined within the system; the operations risk being multiply defined; and debugging the system must proceed in an ad-hoc manner. These flaws are removed by encapsulating objects and predicates in a software layer for spherical geometry, the topic of this chapter.

At the time of this writing, the CGAL (see Chapter 18) kernel exclusively contains Euclidean geometry objects, but an extension package for spherical geometry that includes a number of algorithms not discussed here has been designed and implemented [92] for LEDA [67].