Unit hyperbola

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

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

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