The only points to which the metrical geometry applies are those within the region enclosed by the quadric; the other points are “improper ideal points.” The angle (θ12) between two planes, l1x + m1y + n1z + r1w = 0 and l2x + m2y + n2z + r2w = 0, is given by
cos θ12 = (l1l2 + m1m2 + n1n2 − r1r2) / {(l1² + m1² + n1² − r1²) (l2² + m2² + n2² − r2²)}1/2
(2)
These planes only have a real angle of inclination if they possess a line of intersection within the actual space, i.e. if they intersect. Planes which do not intersect possess a shortest distance along a line which is perpendicular to both of them. If this shortest distance is δ12, we have
cosh (δ12/γ) = (l1l2 + m1m2 + n1n2 − r1r2) / {(l1² + m1² + n1² − r1²) (l2² + m2² + n2² − r2²)}1/2
(3)
| Fig. 67. |
Thus in the case of the two planes one and only one of the two, θ12 and δ12, is real. The same considerations hold for coplanar straight lines (see VII. Axioms of Geometry). Let O (fig. 67) be the point (0, 0, 0, 1), OX the line y = 0, z = 0, OY the line z = 0, x = 0, and OZ the line x = 0, y = 0. These are the coordinate axes and are at right angles to each other. Let P be any point, and let ρ be the distance OP, θ the angle POZ, and φ the angle between the planes ZOX and ZOP. Then the coordinates of P can be taken to be
sinh (ρ/γ) sin θ cos φ, sinh (ρ/γ) sin θ sin φ, sinh (ρ/γ) cos θ, cosh (ρ/γ).
If ABC is a triangle, and the sides and angles are named according to the usual convention, we have