Computational Geometry from the Viewpoint of Affine Differential Geometry

In this paper, we consider Voronoi diagrams from the view point of affine differential geometry. A main object of affine differential geometry is to study hypersurfaces in an affine space that are invariant under the action of the group of affine transformations. Since incidence relations (configurations of vertexes, edges, etc.) in computational geometry are invariant under affine transformations, we may say that affine differential geometry gives a new sight in computational geometry. The Euclidean distance function can be generalized by a divergence function in affine differential geometry. For such divergence functions, we show that Voronoi diagrams on statistical manifolds are invariant under (-1)-conformal transformations. We then give some typical figures of Voronoi diagrams on a manifold. These figures may give good intuition for Voronoi diagrams on a manifold because the figures or constructing algorithms on a manifold strongly depend on the realization or on the choice of local coordinate systems. We also consider the upper envelope type theorems on statistical manifolds, and give a constructing algorithm of Voronoi diagrams on (-1)-conformally flat statistical manifolds.