Spline Orbifolds Curves and Surfaces with Applications in Cagd 445

In order to obtain a global principle for modeling closed surfaces of arbitrary genus, rst 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 geome-tries S 2 , E 2 and H 2. This well known connection can be exploited to deene spline function spaces on abstract closed surfaces and use them e. g. for approximation and interpolation problems. x1. Geometries of Constant Curvature We are going to deene 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 deene 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 Poincar e and the Klein model, are isomorphic. We start from a diierent point of view: We rst deene 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 diierence to euclidean or spherical geometry. All rights of reproduction in any form reserved.