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A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse.
A diameter of one hyperbola is conjugate to its reflection in the asymptote, which is a diameter of the other hyperbola. As perpendicularity is the relation of conjugate diameters of a circle, so hyperbolic orthogonality is the relation of conjugate diameters of rectangular hyperbolas.
The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals 2 . {\displaystyle {\sqrt {2}}.} [ 1 ] The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry.
Feuerbach Hyperbola. In geometry, the Feuerbach hyperbola is a rectangular hyperbola passing through important triangle centers such as the Orthocenter, Gergonne point, Nagel point and Schiffler point. The center of the hyperbola is the Feuerbach point, the point of tangency of the incircle and the nine-point circle. [1]
For a rectangular or equilateral hyperbola, one whose asymptotes are perpendicular, there is an alternative standard form in which the asymptotes are the coordinate axes and the line x = y is the principal axis. The foci then have coordinates (c, c) and (−c, −c). [9] Circle: + =, Ellipse:
Hyperbola#Rectangular hyperbola Hyperbola with perpendicular asymptotes To a section : This is a redirect from a topic that does not have its own page to a section of a page on the subject.
In 1850 the Irish bishop Charles Graves proved and published the following method for the construction of confocal ellipses with help of a string: [2] If one surrounds a given ellipse E by a closed string, which is longer than the given ellipse's circumference, and draws a curve similar to the gardener's construction of an ellipse (see diagram ...
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a) and (b, 1/b) on the rectangular hyperbola xy = 1, or the corresponding region when this hyperbola is re-scaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola.