Hyperbolic Analytic Geometry

Thle individual wh1o encounters hyperbolic geometry for the first time in such a book as the one by Wolfe [31 has the stimu-lating experience of developing the analogue of a substantial part of euclidean geometry uising the same essential spirit and methods as those of Euclid. T'his is followed by a development of hyperbolic trigonometry which provides the reader with all the rudimentary data for solving triangle problems in the hyperbolic plane. IIowever, only a few texts, like t'he one of Sommerville [2], make an attempt to dleveloe a hyperbolic analytic geometry in a way which parallels thie usual freshmaan course in analytics, and the student hlas only to attem,r)t as simple a problem as the determination of the eqtuation of the altitude of an arbitrary triangle to realize the inadequacy of his equipment. Other texts, such as that by Coxeter [I] approach hiyperbolic geometry from the viewpoint of pro,jective geometry. To a student familiar with projective geometry it is then a relatively simple matter to think of the hyperbolic plane as embedded in the projective d1arne and develop its analytic geometry in the usual projective coordinates. Fronm the beginner's point of view it seems an unniatural way of getting analytic results for the hyperbolic plane. Tre aim of this paper is to present a brief development of hyperbolic analytic geometry following the usual procedures of analytics quite familiar to students. Only a single use is made of calculus, and thlis could probably have been avoided. From the discussion follows naturally thie idea of introducing a different coordinate system wh-tic'h amounts in fact to embedding in the proj ective plane. Tnus the ,)rocedure of establishing hyperbolic geometry as a subgeometry of projective geometry is obtained naturally from within in an elementary way, With the exception of the last section, which utilizes a little projective geometry and the already noted single use of calculus, nothing is assumee(I beyond analytic geometry and an introduction to nyperbolic g,eometry. rhe question of applicability of the results and methods to binocular sensory space [4] remains to be examined.