Klein develops noneuclidean geometries on the projective plane, rather than the euclidean plane. This introduces points at infinity, like the convergence of parallel railroad tracks in the distance. Moreover, geometrical figures constantly change shape as we change our perspectivity. Klein observes that noneuclidean motions can be regarded as rotations. Just as the distance between two points is invariant under translation, the angle between two lines, or two planes, is invariant under rotations. The distinguishing feature of the geometries lies in whether the angle is real or imaginary.
Klein shows that the measure of distance between any two points is a constant multiple of the logarithm of the cross-ratio of four points, two of which are fundamental. The angle between two lines in projective geometry as i/2 times the logarithm of the cross-ratio of the two lines to the lines from their common point to the imaginary points at infinity of the circle. [Klein "On the so-called noneuclidean geometry"]
This fits in with our previous work on the velocity space of constant angular velocity. The conservation of energy is
E= v^2/2- m r,
where m is the gravitational parameter. Hamilton's equation is
dv/dt= m,
to which we introduce the constraint of constant angular velocity, w=r d theta/dt. We then obtain
dv/d theta= mr/w= (v^2-2E)/2w/
In the second inequality we introduced the constant energy equation.
Curvature coincide with Gaussian curvature, K=-2E. For E>0, we get hyperbolic geometry, E<0, elliptic geometry, and E=0, euclidean geometry where straight lines are the geodesics.
The metric is expressed in terms of angle increments
d theta^2 = 4w^2 dv^2/(v^2+K)^2. (A)
The general formula for the distance is
theta= w/(-K)^1/2 ln{(v-(-K)^1/2)/(v+(-K)^1/2}=-[1/(-K)^1/2] coth^(-1)[v/(-K)^1/2].
For K<0, the limit circles are obtained for increasing theta, while the kappa curve for K>0, as shown in the figure.
This is rather counterintuitive since we would expect the hyperbolic tangent to be aperiodic and the tangent to be periodic. However, as the kappa curve, v=(-K)^1/2 cot theta, testifies, it is aperiodic. It was used by Newton to show how straight lines are bent. More specifically, Newton show that kappa curves are the number of lines that are bent evenly. Its periodic counterpart, obtained by making the angle imaginary, seems to have been overlooked. There is an asymptotic winding onto an invariant circle from infinity or from the origin, as shown in the figure below.
For small velocities, we can expand coth and cot in series; to lowest order we get an inverse relationship
theta v= const.,
in both cases. It is noteworthy that this is independent of the curvature so it holds for all three geometries. It affirms that the larger the velocity the smaller the angle, and hence the smaller the distance.
In particular, the cross-ratio of a quadrupole is -1. One half of the logarithm of this is pure imaginary. As Klein explains, we must multiply it by a pure imaginary coefficient in order to get a distance between the two points. However, there are no real points at infinity. The line must curve into a circle. As a consequence, the distance between any two points is only determined up to a multiple of a real period that represents the whole length of the line. Consequently, the measure of the line is that of the "ordinary" measure of a circle.
Thus, the angular measure of distance is built-into velocity space processes of constant angular velocity, w. Moreover, it accounts for the three noneuclidean geometries of constant curvature, K. The angular measure of distance between any two points is implicit in the noneuclidean definition of distance as one-half the logarithm of the cross-ratio, which is also given by the angle
2i/(-K)^1/2 arccos{v/(v^2+K)^1/2},
i.e., +/- i cos^(-1)x=cosh^(-1)x.