Projective Metric

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.

REFERENCES

Efimov, N. V. Vysshaia geometriia, 5th ed. Moscow, 1971.
Klein, F. Neevklidova geometriia. Moscow-Leningrad, 1936. (Translated from German.)
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The Galilean space G3 is a Cayley-Klein space equipped with the projective metric of signature (0,0,+,+), as in E.