44 CONCRETE REPRESENTATIONS OF 



Project the quadric stereographically, i.e. with the centre 

 of projection on the surface. The two generators through 

 give two fixed points X, F, and any plane section is projected 

 into a conic passing through X, Y. The points at infinity on 

 the quadric project into a fixed conic, also passing through 

 X, F, and the pole of XY with respect to the fixed conic is a 

 point Z, which is the projection of the centre C of the quadric. 

 The tangents at infinity, i.e. the asymptotes, of a diametral 

 section, pass through (7, and their projections therefore pass 

 through Z. Hence the projection consists of the net of conies 

 which we considered in the last section. 



The angle and distance functions can therefore be deduced. 

 At a point P a pencil is determined by the tangents to the 

 diametral sections and the two generators, which correspond 

 in the projection to the two lines passing through X 9 Y. 

 The angle between the lines represented by the diametral 

 sections is then proportional to the logarithm of the cross-ratio 

 of this pencil. In a diametral section a range is determined 

 by two points and the two points at infinity, which correspond 

 in the projection to the intersections with the fixed conic. 

 The distance between the two points is then proportional to 

 the logarithm of the cross-ratio of this configuration on the 

 diametral section. A circle corresponds in the projection 

 to any conic passing through X, Y, i.e. it is represented by 

 any plane section. 



If the quadric is ruled the points X, Y are real and the 

 measure of angle is hyperbolic ; or parabolic if the quadric 

 degenerates to a cone. 



The geometry is Hyperbolic, Parabolic, or Elliptic accord- 

 ing as the quadric is a hyperboloid of two sheets, an elliptic 

 paraboloid, or an ellipsoid. 



37. If the quadric is projected from the centre, diametral 

 sections become straight lines ; the points at infinity give again 

 a fixed conic, the section of the asymptotic cone ; and any 

 plane section projects into a conic having double contact 



