Quadric surfaces in hyperbolic space

Abstract

The first broad line of demarcationi is naturally between those surfaces which have a vanishing discriminant, and those which have a non-vanishing one. We shall occupy ourselves only with surfaces of the latter type. The next great division is into ruled surfaces, non-ruled ones, and surfaces whose equations are definite forms. This classification depends upon the signs of the discriminant and its leading minors, according to certain familiar principles. We then come to the metrical properties of quadries. These arise from the various possible relations to the Absolute, so that the next problem is that of the classification of the mutual relations of two quadrics. This has been successfully solved by a number of writers, in particular, by HESSE t and CLEBSCH.1 Yet from the strict point of view of a dweller in a hyperbolic space this last classification is not wholly satisfactory, for it is at once too inclusive and too ill-defined. It is too inclusive, for we are interested only in real surfaces, and must distinguish between real and imaginary curves of intersection. The Absolute has no real generators, so that we must exclude those cases where the two surfaces cut in one or more generators. It is not sharp enough, for if we search for the shape of our surface, we wish to know whether the curve which it cuts from the Absolute-the absolute curve, let us say-and the corresponding focal developable are real or imaginiary; whether there be real vertices to the common self-conjugate tetrahedron; etc. Furthermore, we care not at all for those surfaces, analyt-