Agebraic Geomerty I Lectures 6 and 7 Summary: Agebraic Geomerty I Lectures 6 and 7 Chris Portwood October 20, 2008 1 Projective Space In Euclidean geometry, two distinct lines in the plane intersect in exactly in one point except if they are parallel; this exception is a bit of a nuisance. So we add one extra point to the Euclidean plane for each direction (these are called points at infinity) and declare that parallel lines meet at the point at infinity corresponding to their common direction. The corresponding geometry is called projective geometry. Just like Euclidean geometry has an axiomatic definition, so does projective geometry. The former has an analytic description in terms of the "usual" coordinate geometry (Descarte). An analytic description of projective geomerty was introduced by Pl¨ucker, Mobius, and others. 1.1 General Idea Put the xy-plane in 3-dimensions and shift it up by 1. Each point on the xy-plane is in correspondence with a line in through the originwhich is not in the xy-plane. Each line in the xy-plane corresponds to a direction. So we can think of the projective plane as the set of lines through the origin; Collections: Mathematics