Linear Algebra

Topic: Projective Geometry 337Note though that projective points on the equator don’t project up to the plane.Instead, these project ‘out to infinity’. We can thus think of projective spaceas consisting of the Euclidean plane with some extra points adjoined — the Euclideanplane is embedded in the projective plane. These extra points, theequatorial points, are the ideal points or points at infinity and the equator isthe ideal line or line at infinity (note that it is not a Euclidean line, it is aprojective line).The advantage of the extension to the projective plane is that some of theawkwardness of Euclidean geometry disappears. For instance, the projectivelines shown above in (∗) cross at antipodal spots, a single projective point, onthe sphere’s equator. If we put those lines into (∗∗) then they correspond toEuclidean lines that are parallel. That is, in moving from the Euclidean plane tothe projective plane, we move from having two cases, that lines either intersector are parallel, to having only one case, that lines intersect (possibly at a pointat infinity).The projective case is nicer in many ways than the Euclidean case but hasthe problem that we don’t have the same experience or intuitions with it. That’sone advantage of doing analytic geometry, where the equations can lead us tothe right conclusions. Analytic projective geometry uses linear algebra. Forinstance, for three points of the projective plane t, u, and v, setting up theequations for those points by fixing vectors representing each, shows that thethree are collinear — incident in a single line — if and only if the resulting threeequationsystem has infinitely many row vector solutions representing that line.That, in turn, holds if and only if this determinant is zero.∣ t 1 u 1 v 1∣∣∣∣∣t 2 u 2 v 2∣t 3 u 3 v 3Thus, three points in the projective plane are collinear if and only if any threerepresentative column vectors are linearly dependent. Similarly (and illustratingthe Duality Principle), three lines in the projective plane are incident on asingle point if and only if any three row vectors representing them are linearlydependent.The following result is more evidence of the ‘niceness’ of the geometry of theprojective plane, compared to the Euclidean case. These two triangles are saidto be in perspective from P because their corresponding vertices are collinear.OT 1U 1V 1T 2U 2V 2Consider the pairs of corresponding sides: the sides T 1 U 1 and T 2 U 2 , the sidesT 1 V 1 and T 2 V 2 , and the sides U 1 V 1 and U 2 V 2 . Desargue’s Theorem is that