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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

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

with $a$ being the distance between the vertices and the origin and $\frac{a}{b}$ being equal to the slope of the asymptotes.

Similarly to how the unit circle defines trigonometric functions, the unit hyperbola, $x^2 - y^2 = 1$, 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. $y=\frac{k}{x}$

Start with the Cartesian form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

Rewrite in a parametric/vector form. $\left (\begin{matrix}x \\ y \end{matrix} \right ) = \left (\begin{matrix}a \tan(t) \\ b \sec(t) \end{matrix} \right )$

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).

$\left (\begin{matrix}x \\ y \end{matrix} \right ) = \begin{pmatrix} \cos{\pi/4} & -\sin{\pi/4} \\ \sin{\pi/4} & \cos{\pi/4} \end{pmatrix} \left (\begin{matrix}a \tan(t) \\ b \sec(t) \end{matrix} \right )$
$\left (\begin{matrix}x \\ y \end{matrix} \right ) = \left (\begin{matrix}a\tan{t}\sqrt{2}/2-b\sec{t}\sqrt{2}/2 \\ a\tan{t}\sqrt{2}/2+b\sec{t}\sqrt{2}/2 \end{matrix} \right )$

Take out the factor of half square root 2 $\left (\begin{matrix}x \\ y \end{matrix} \right ) = \frac{\sqrt{2}}{2}\left (\begin{matrix}a\tan{t}-b\sec{t} \\ a\tan{t}+b\sec{t} \end{matrix} \right )$