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A hyperbola and its conjugate hyperbola. In geometry, a conjugate hyperbola to a given hyperbola shares the same asymptotes but lies in the opposite two sectors of the plane compared to the original hyperbola. A hyperbola and its conjugate may be constructed as conic sections obtained from an intersecting plane that meets tangent double cones ...
The transverse axis of a hyperbola coincides with the major axis. [4] In a hyperbola, a conjugate axis or minor axis of length , corresponding to the minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the two vertices (turning points) of the hyperbola, with the two axes intersecting ...
Since both the transverse axis and the conjugate axis are axes of symmetry, the symmetry group 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 sets.
The unit hyperbola is blue, its conjugate is green, and the asymptotes are red. In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation = In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length
The Rytz’s axis construction is a basic method of descriptive geometry to find the axes, the semi-major axis and semi-minor axis and the vertices of an ellipse, starting from two conjugated half-diameters. If the center and the semi axis of an ellipse are determined the ellipse can be drawn using an ellipsograph or by hand (see ellipse).
A ray through the unit hyperbola x 2 − y 2 = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).
Apollonius of Perga gave the following construction of conjugate diameters, given the conjugate hyperbola: "If Q be any point on a hyperbola and CE be drawn from the centre parallel to the tangent at Q to meet the conjugate hyperbola in E, then (1) the tangent at E will be parallel to CQ and (2) CQ and CE will be conjugate diameters." [2]
The hyperbola is the isogonal conjugate of , the line joining the circumcenter and the incenter. [3] This fact leads to a few interesting properties. Specifically all the points lying on the line O I {\displaystyle OI} have their isogonal conjugates lying on the hyperbola.