Projective Metric

a method of measuring lengths and angles by means of projective geometry. In a projective metric, some figure is taken as an absolute determining the given metric geometry, and the transformations that map the absolute into itself and thus generate a corresponding group of motions are singled out from the group of all projective transformations. For example, the metric of the Lobachevskian plane is obtained if a nondegenerate real quadratic curve is taken as the absolute. The length of the line segment A B is then equal to λ In (ABPQ), where P and Q are the points at which the line A B intersects the absolute, (ABPQ) is the cross ratio, and λ is a constant identical for all segments. If a quadratic curve without real points is used to measure lengths and angles, elliptic geometry is obtained. Degenerate quadratic curves are used to construct Euclidean and Minkowskian geometry.