Computational Geometry from the Viewpoint of Affine Differential Geometry

Incidence relations (configurations of vertexes, edges, etc.) are important in computational geometry. Incidence relations are invariant under the group of affine transformations. On the other hand, affine differential geometry is to study hypersurfaces in an affine space that are invariant under the group of affine transformation. Therefore affine differential geometry gives a new sight in computational geometry. From the viewpoint of affine differential geometry, algorithms of geometric transformation and dual transformation are discussed. The Euclidean distance function is generalized by a divergence function in affine differential geometry. A divergence function is an asymmetric distance-like function on a manifold, and it is an important object in information geometry. For divergence functions, the upper envelope type theorems on statistical manifolds are given. Voronoi diagrams determined from divergence functions are also discussed.