In geometry, given a pair of conjugate hyperbolas, two conjugate diameters are hyperbolically orthogonal. This relationship of diameters was described by Apollonius of Perga and has been modernized using analytic geometry. Hyperbolically orthogonal lines appear in special relativity as temporal and spatial directions that show the relativity of simultaneity.

Keeping time and space axes hyperbolically orthogonal, as in Minkowski space, gives a constant result when measurements are taken of the speed of light.

Two lines are hyperbolic orthogonal when they are reflections of each other over the asymptote of a given hyperbola. Two particular hyperbolas are frequently used in the plane:

When reflected in the x-axis, a line y = mx becomes y = −mx.

The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class. Thus, for a given hyperbola and asymptote A, a pair of lines (a, b) are hyperbolic orthogonal if there is a pair (c, d) such that ${\displaystyle a\rVert c,\ b\rVert d}$, and c is the reflection of d across A.

Similar to the perpendularity of a circle radius to the tangent, a radius to a hyperbola is hyperbolic orthogonal to a tangent to the hyperbola.

A bilinear form is used to describe orthogonality in analytic geometry, with two elements orthogonal when their bilinear form vanishes. In the plane of complex numbers ${\displaystyle z_{1}=u+iv,\quad z_{2}=x+iy}$, the bilinear form is ${\displaystyle xu+yv}$, while in the plane of hyperbolic numbers ${\displaystyle w_{1}=u+jv,\quad w_{2}=x+jy,}$ the bilinear form is ${\displaystyle xu-yv.}$

The bilinear form may be computed as the real part of the complex product of one number with the conjugate of the other. Then