A surface of equal distance is a sphere whose centre is improper; and both types of surface are included in the family
k² (w² − x² − y² − z²) = (ax + by + cz + dw)²
(9).
But this family also includes a third type of surfaces, which can be looked on either as the limits of spheres whose centres have approached the absolute, or as the limits of surfaces of equal distance whose central planes have approached a position tangential to the absolute. These surfaces are called limit-surfaces. Thus (9) denotes a limit-surface, if d² − a² − b² − c² = 0. Two limit-surfaces only differ in position. Thus the two limit-surfaces which touch the plane YOZ at O, but have their concavities turned in opposite directions, have as their equations
w² − x² − y² − z² = (w ± x)².
The geodesic geometry of a sphere is elliptic, that of a surface of equal distance is hyperbolic, and that of a limit-surface is parabolic (i.e. Euclidean). The equation of the surface (cylinder) of equal distance (δ) from the line OX is
(w² − x²) tanh² (δ/γ) − y² − z² = 0.
This is not a ruled surface. Hence in this geometry it is not possible for two straight lines to be at a constant distance from each other.
Secondly, let the equation of the absolute be x² + y² + z² + w² = 0. The absolute is now imaginary and the geometry is elliptic.
The distance (d12) between the two points (x1, y1, z1, w1) and (x2, y2, z2, w2) is given by