conjugate hyperbola

The equation of the conjugate hyperbola of is .

Given a hyperbola with asymptote A, its reflection in A produces the conjugate hyperbola.

The second diagram showed the conjugate hyperbola to calibrate space, where a similar stretching leaves the impression of FitzGerald contraction.

Whereas the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola to complement it in the plane.

When abstracted to a line drawing, then any figure showing conjugate hyperbolas, with a selection of conjugate diameters, falls into this category.

The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements:

For example, the angle θ of the conjugate hyperbola equals 90 minus the angle of the original hyperbola.

Hence, the conjugate hyperbola does not in general correspond to a 90 rotation of the original hyperbola; the two hyperbolas are generally different in shape.

Every hyperbola has a conjugate hyperbola, in which the transverse and conjugate axes are exchanged without changing the asymptotes.

This terminology stems from the use of two conjugate hyperbolas in the pseudo-Euclidean plane: conjugate diameters of these hyperbolas are hyperbolic-orthogonal.

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

In 1900 Alexander published "Hyperbolic Quaternions" with the Royal Society in Edinburgh, and included a sheet of nine figures, two of which display conjugate hyperbolas.

The product of the perpendicular distances from a point P on a hyperbola or on its conjugate hyperbola to the asymptotes is a constant independent of the location of P.

Osgood and Graustein used the rectangular hyperbola, its conjugate hyperbola, and conjugate diameters to rationalize tie rods at 15 degree radial spacing, to a square of girders, from its center.

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.

Five elements constitute the diagram Hermann Minkowski used to describe the relativity transformations: the unit hyperbola, its conjugate hyperbola, the axes of the hyperbola, a diameter of the unit hyperbola, and the conjugate diameter.

If the graph of the conjugate hyperbola is rotated 90 to restore the east-west opening orientation (so that x becomes y and vice versa), the equation of the resulting rotated conjugate hyperbola is the same as the equation of the original hyperbola except with a and b exchanged.