Explore: *hyperbolas

of a hyperbola is the Klein four-group. The rectangular hyperbolas xy = constant admit group actions by squeeze mappings which have the hyperbolas as invariant

ellipses and hyperbolas have two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture

unit hyperbola requires the conjugate hyperbola y 2 − x 2 = 1 {\displaystyle y^{2}-x^{2}=1} to complement it in the plane. This pair of hyperbolas share

the complex coordinate plane C2, ellipses and hyperbolas are not distinct: one may consider a hyperbola as an ellipse with an imaginary axis length. For

sometimes called the numerical eccentricity. In the case of ellipses and hyperbolas the linear eccentricity is sometimes called the half-focal separation

Retrieved 2007-09-06. "The Geometry of Orbits: Ellipses, Parabolas, and Hyperbolas". www.bogan.ca. "7.1 Alternative Characterization". Archived from the

diameters of both hyperbolas." "The two hyperbolas so constructed are called conjugate hyperbolas, and [the] last drawn is the hyperbola conjugate to the

{\displaystyle x^{2}+y^{2}=a^{2}+b^{2}} (see below), The orthoptic of a hyperbola x 2 a 2 − y 2 b 2 = 1 ,   a > b {\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac

reflections of each other over the asymptote of a given hyperbola. Two particular hyperbolas are frequently used in the plane: xy = 1 with y = 0 as asymptote

describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term bipolar coordinates is reserved