Conjugate hyperbola

In geometry, a conjugate hyperbola to a given hyperbola shares the same asymptotes but lies in the opposite two sectors of the plane compared to the original hyperbola.

A hyperbola and its conjugate hyperbola

A hyperbola and its conjugate may be constructed as conic sections derived from parallel intersecting planes and cutting tangent double cones sharing the same apex.

Using analytic geometry, the hyperbolas satisfy the symmetric equations

${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1$ with vertices (a,0) and (–a,0), and

${\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}=1$ with vertices (0,b) and (0,–b).

In case a = b they are rectangular hyperbolas, and a reflection of the plane in an asymptote exchanges the conjugates.

History

Light cone and conjugate hyperbolas in Minkowski (1908)

The conjugate hyperbola arises in the study of conjugate diameters of conic sections.

Elements of Dynamic (1878) by W. K. Clifford identifies the conjugate hyperbola.

In 1894 Alexander Macfarlane used an illustration of conjugate right hyperbolas in his study "Principles of elliptic and hyperbolic analysis".

In 1895 W. H. Besant noted conjugate hyperbolas in his book on conic sections. George Salmon illustrated a conjugate hyperbola as a dotted curve in this Treatise on Conic Sections (1900).

In 1908 conjugate hyperbolas were used by Herman Minkowski to demarcate units of duration and distance in a spacetime diagram illustrating a plane in his Minkowski space.

The principle of relativity may be stated as "Any pair of conjugate diameters of conjugate hyperbolas can be taken for the axes of space and time".

In 1957 Barry Spain illustrated conjugate rectangular hyperbolas.