To illustrate, consider three points A,B,C on the Euclidean plane, and let a,b,c denote points on the opposite edges of the triangle formed by those points. We arrive at similar conditions on the coordinates of P from the fact that it lies on the lines Bb and Cc. The duality is formally ensured by choosing the axioms of projective geometry in dual pairs, so that every theory automatically implies a dual theorem. This duality is also expressed by the facts that every two points fall on a single line, and every two lines meet in a single point. Also, if each of the variables is regarded as a distance, the equation as written does not have homogeneous units.
Similar comments apply to the v plane and space.
Since the dot product of two vectors vanishes if and only if the vectors are perpendicular, this implies that for a given point in the u space the locus of points in the v space that satisfy this equation consists of all the vectors in the plane perpendicular to u. P under this projection. The relevance of projective geometry to relativistic spacetime goes beyond the simple velocity composition law. Recalling the discussion of null coordinates in Section 1.7, it's clear that lightline axes can be regarded as projective coordinates, since the scale factor along these rays is indeterminate. In addition, we saw in Section 2.6 how the entire set of null direction at any point in spacetime can be identified with the points of a Riemann sphere, and sterographically projected onto the complex plane, and how the effect of an arbitrary Lorentz transformations corresponds to the application of a simple linear fractional mapping.