Rectangular Hyperbola

A hyperbola for which the asymptotes are perpendicular, also called an equilateral hyperbola or right hyperbola. This occurs when the semimajor and semiminor axes are equal. This corresponds to taking , giving eccentricity . Plugging into the general equation of a hyperbola with semimajor axis parallel to the x-axis and semiminor axis parallel to the y-axis (i.e., vertical conic section directrix),

(1)

therefore gives

(2)

The rectangular hyperbola opening to the left and right has polar equation

(3)

and the rectangular hyperbola opening in the first and third quadrants has the Cartesian equation

(4)

The parametric equations for the right branch of a rectangular hyperbola are given by

			(5)
			(6)

where is the hyperbolic cosine and is the hyperbolic sine. The curvature, arc length, and tangential angle for the above parametrization with are

			(7)
			(8)
			(9)
			(10)
			(11)

where is an elliptic integral of the second kind, is the gamma function, is a hypergeometric function, is an incomplete beta function, and is a hyperbolic tangent.

A parametrization which gives both branches is given by

			(12)
			(13)

with and discontinuities at .

The inverse curve of a rectangular hyperbola with inversion center at the center of the hyperbola is a lemniscate (Wells 1991).

If the three vertices of a triangle lie on a rectangular hyperbola, then so does the orthocenter (Wells 1991). Equivalently, if four points form an orthocentric system, then there is a family of rectangular hyperbolas through the points. Moreover, the locus of centers of these hyperbolas is the nine-point circle of the triangle (Wells 1991).

If four points do not form an orthocentric system, then there is a unique rectangular hyperbola passing through them, and its center is given by the intersection of the nine-point circles of the points taken three at a time (Wells 1991).