In this case the latent plane is the plane for which at least one of x, y, z are infinite, that is, the plane 0.x + 0.y + 0.z + a = 0; and the latent conic is the conic in which the cone x² + y² + z² = 0 intersects the latent plane.
It follows from theorems 1 and 2 that there is not one unique congruence-group, but an indefinite number of them. There is one congruence-group corresponding to each closed real quadric, one to each imaginary quadric with a real equation, and one to each imaginary conic in a real plane and with a real equation. The quadric thus associated with each congruence-group is called the absolute for that group, and in the degenerate case of 1 (β) the absolute is the latent plane together with the latent imaginary conic. If the absolute is real, the congruence-group is hyperbolic; if imaginary, it is elliptic; if the absolute is a plane and imaginary conic, the group is parabolic. Metrical geometry is simply the theory of the properties of some particular congruence-group selected for study.
The definition of distance is connected with the corresponding congruence-group by two considerations in respect to a range of five points (A1, A2, P1, P2, P3), of which A1 and A2 are on the absolute.
Let {A1P1A2P2} stand for the cross ratio (as defined above) of the range (A1P1A2P2), with a similar notation for the other ranges. Then
(1)
log {A1P1A2P2} + log {A1P2A2P3} = log {A1P1A2P3},
and
(2), if the points A1, A2, P1, P2 are transformed into A′1, A′2, P′1, P′2 by any transformation of the congruence-group, (α) {A1P1A2P2 = {A′1P′1A′2P′2}, since the transformation is projective, and (β) A′1, A′2 are on the absolute since A1 and A2 are on it. Thus if we define the distance P1P2 to be ½k log {A1P1A2P2}, where A1 and A2 are the points in which the line P1P2 cuts the absolute, and k is some constant, the two characteristic properties of distance, namely, (1) the addition of consecutive lengths on a straight line, and (2) the invariability of distances during a transformation of the congruence-group, are satisfied. This is the well-known Cayley-Klein projective definition[47] of distance, which was elaborated in view of the addition property alone, previously to Lie’s discovery of the theory of congruence-groups. For a hyperbolic group when P1 and P2 are in the region enclosed by the absolute, log {A1P1A2P2} is real, and therefore k must be real. For an elliptic group A1 and A2 are conjugate imaginaries, and log {A1P1A2P2} is a pure imaginary, and k is chosen to be κ/ι, where κ is real and ι = √ −.
Similarly the angle between two planes, p1 and p2, is defined to be (1/2ι) log (t1p1t2p2), where t1 and t2 are tangent planes to the absolute through the line p1p2. The planes t1 and t2 are imaginary for an elliptic group, and also for an hyperbolic group when the planes p1 and p2 intersect at points within the region enclosed by the absolute. The development of the consequences of these metrical definitions is the subject of non-Euclidean geometry.
The definitions for the parabolic case can be arrived at as limits of those obtained in either of the other two cases by making k ultimately to vanish. It is also obvious that, if P1 and P2 be the points (x1, y1, z1) and (x2, y2, z2), it follows from equations (B) above that {(x1 − x2)² + (y1 − y2)² + (z1 − z2)²}1/2 is unaltered by a congruence transformation and also satisfies the addition property for collinear distances. Also the previous definition of an angle can be adapted to this case, by making t1 and t2 to be the tangent planes through the line p1p2 to the imaginary conic. Similarly if p1 and p2 are intersecting lines, the same definition of an angle holds, where t1 and t2 are now the lines from the point p1p2 to the two points where the plane p1p2 cuts the imaginary conic. These points are in fact the “circular points at infinity” on the plane. The development of the consequences of these definitions for the parabolic case gives the ordinary Euclidean metrical geometry.