Hyperbola in Cartesian coordinates

An east-west opening hyperbola centered at (h,k) has the equation

$\frac{\left( x-h \right)^2}{a^2} - \frac{\left( y-k \right)^2}{b^2} = 1.$

The major axis runs through the center of the hyperbola and intersects both arms of the hyperbola at the vertices (bend points) of the arms. The foci lie on the extension of the major axis of the hyperbola.

The minor axis runs through the center of the hyperbola and is perpendicular to the major axis.

In both formulas a is the semi-major axis (half the distance between the two arms of the hyperbola measured along the major axis), and b is the semi-minor axis (half the distance between the asymptotes along a line tangent to the hyperbola at a vertex).

If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, the sides tangent to the hyperbola are 2b in length while the sides that run parallel to the line between the foci (the major axis) are 2a in length. Note that b may be larger than a despite the names minor and major.

If one calculates the distance from any point on the hyperbola to each focus, the absolute value of the difference of those two distances is always 2a.

The eccentricity is given by

$\varepsilon = \sqrt{1+\frac{b^2}{a^2}} = \sec\left(\arctan\left(\frac{b}{a}\right)\right) = \cosh\left(\operatorname{arsinh}\left(\frac{b}{a}\right)\right)$