Spline Orbifolds

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, E and H. 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, the geometry of the unit sphere S of euclidean E, and the geometry of the hyperbolic plane H. The geometries of E and S 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. Curves and Surfaces with Applications in CAGD 445 A. Le Méhauté, C. Rabut, and L. L. Schumaker (eds.), pp. 445–464. Copyright oc 1997 by Vanderbilt University Press, Nashville, TN. ISBN 0-8265-1293-3. All rights of reproduction in any form reserved. 446 J. Wallner, H. Pottmann Fig. 1. Projective model of H: (a) points and non-intersecting lines, (b) hyperbolic reflection κ. 1.1 The Projective Model of Hyperbolic Geometry Consider the real projective plane P 2 equipped with a homogeneous coordinate system, where a point with homogeneous coordinates (x0 : x1 : x2) has affine coordinates (x1/x0, x2/x0). We will not distinguish between the point and its homogeneous coordinate vectors. Every time when a coordinate vector of a point appears in a formula, it is tacitly understood that any scalar multiple of this coordinate vector could be there as well. We define an orthogonality relation between points: Let β be a symmetric bilinear form defined in IR, and let β have two negative squares, for instance β(x, y) = x0y0 − x1y1 − x2y2. An equivalent formulation is β(x, x) = x Jx, J being the diagonal matrix with entries 1, −1 and −1. We call x and y orthogonal, if β(x, y) = 0. Points with β(x, x) = 0 are called ideal points. The set of all ideal points is a conic and will be called the ideal circle. If we choose β as above, the ideal circle is nothing but the euclidean unit circle. Now a point x ∈ P 2 shall belong to the hyperbolic plane H if it is contained in the interior of the ideal circle, x ∈ H ⇐⇒ β(x, x) > 0. The lines of the hyperbolic plane are the intersections of projective lines with H. We define two lines to be parallel if they have no point in common. It is now obvious that for all lines l and all non-incident points p, there are a lot of lines parallel to l and containing p. A picture of the projective model can be found in Figure 1. So far we have dealt with the incidence structure of the hyperbolic plane. We now come to metric properties. We define the hyperbolic distance d(x, y) between points x and y of H by coshd(x, y) = |β(x, y)| √ β(x, x)β(y, y) . Spline Orbifolds 447 We leave the verification of the fact that always β(x, x)β(y, y) ≤ β(x, y) to the reader. This metric satisfies the triangle inequality and is compatible with the definition of lines, in the sense that they are precisely the geodesic curves with respect to this metric. Hyperbolic congruence transformations will be those projective transformations, which map H onto H and preserve hyperbolic distances. For this reason and also because it is shorter, we will call them isometries or motions. We express the isometric property in matrix form: for each projective transformation κ there is a matrix such that in homogeneous coordinates