 A graph of the unit hyperbola (where , in blue) and its conjugate (, in green) as well as the asymptotes in red.

A hyperbola is a type of conic section. A hyperbola is a set of points where the distances between the line called the directrix and point called the focus remain in a constant ratio (this ratio is the hyperbola's eccentricity). A hyperbola is visually similar to a parabola, but with two mirrored sides. The formula for a hyperbola (assuming they are orientated facing left and right rather than up and down) is with being the distance between the vertices and the origin and being equal to the slope of the asymptotes.

Similarly to how the unit circle defines trigonometric functions, the unit hyperbola, , defines hyperbolic functions.

## Rectangular Hyperbola

The reciprocal function which shows one variable is inversely proportional to another also describes a hyperbola but oriented differently. It can be shown that this equation is in fact a hyperbola rotated 45 degrees about the origin. Start with the Cartesian form Rewrite in a parametric/vector form. Pre-multiply by rotation matrix representing a 45 degree anti-clockwise rotation about the origin from the x-axis (we are making a new equation).  Take out the factor of half square root 2 