Unit hyperbola

In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation ${\displaystyle x^{2}-y^{2}=1.}$ In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length

${\displaystyle r={\sqrt {x^{2}-y^{2}}}.}$

Whereas the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola ${\displaystyle y^{2}-x^{2}=1}$ to complement it in the plane. This pair of hyperbolas share the asymptotes y = x and y = −x. When the conjugate of the unit hyperbola is in use, the alternative radial length is ${\displaystyle r={\sqrt {y^{2}-x^{2}}}.}$

The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals ${\displaystyle {\sqrt {2}}.}$^[1]

The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space. There the asymptotes of the unit hyperbola form a light cone. Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function and the modern parametrization of the hyperbola by sector areas. When the notions of conjugate hyperbolas and hyperbolic angles are understood, then the classical complex numbers, which are built around the unit circle, can be replaced with numbers built around the unit hyperbola.