Unit hyperbola

In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation

When the conjugate of the unit hyperbola is in use, the alternative radial length is

A prominent instance is the depiction of spacetime as a pseudo-Euclidean space.

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.

The curve is first interpreted in the projective plane using homogeneous coordinates.

Then the asymptotes are lines that are tangent to the projective curve at a point at infinity, thus circumventing any need for a distance concept and convergence.

In a common framework (x, y, z) are homogeneous coordinates with the line at infinity determined by the equation z = 0.

For instance, C. G. Gibson wrote:[2] The Minkowski diagram is drawn in a spacetime plane where the spatial aspect has been restricted to a single dimension.

The units of distance and time on such a plane are Each of these scales of coordinates results in photon connections of events along diagonal lines of slope plus or minus one.

The conjugate diameter represents the spatial hyperplane of simultaneity corresponding to rapidity a.

In this context the unit hyperbola is a calibration hyperbola[3][4] Commonly in relativity study the hyperbola with vertical axis is taken as primary: The vertical time axis convention stems from Minkowski in 1908, and is also illustrated on page 48 of Eddington's The Nature of the Physical World (1928).

One finds an early expression of the parametrized unit hyperbola in Elements of Dynamic (1878) by W. K. Clifford.

The following description was given by Russian analysts: Whereas the unit circle is associated with complex numbers, the unit hyperbola is key to the split-complex number plane consisting of z = x + yj, where j 2 = +1.

Then jz = y + xj, so the action of j on the plane is to swap the coordinates.

In fact, this branch is the image of the exponential map acting on the j-axis.

Similar to the ordinary complex plane, a point not on the diagonals has a polar decomposition using the parametrization of the unit hyperbola and the alternative radial length.