Spline orbifolds

Abstract. In order to obtain a global principle for modeling closed surfaces of arbitrary genus, first hyperbolic geometry and then discrete groups of motions in planar geometries of constant curvature are studied. The representation of a closed surface as an orbifold leads to a natural parametrization of the surfaces as a subset of one of the classical geometries S 2, E 2 and H 2. This well known connection can be exploited to define spline function spaces on abstract closed surfaces and use them e. g. for approximation and interpolation problems. §1. Geometries of Constant Curvature We are going to define three geometries consisting of a set of points, a set of lines, and a group of congruence transformations: The geometry of the euclidean plane E 2, the geometry of the unit sphere S 2 of euclidean E 3, and the geometry of the hyperbolic plane H 2. The geometries of E 2 and S 2 are well known: the hyperbolic plane will be presented in the next subsections. For more details, see for instance (Alekseevskij et al., 1988). It is possible to define hyperbolic geometry in a completely synthetic way. We could use a system of axioms for euclidean geometry and then negate the parallel postulate or one of its equivalents. Any structure satisfying the axioms would be called a model of hyperbolic geometry. We would have to verify that all models, including the classical ones, the Poincaré and the Klein model, are isomorphic. We start from a different point of view: We first define a set of points, lines and congruence transformations, as linear as possible, and then show some structures isomorphic to it. The reader then will see the difference to euclidean or spherical geometry