# 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.)