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The word "hyperbola" derives from the Greek ὑπερβολή, meaning "over-thrown" or "excessive", from which the English term hyperbole also derives. Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. [2]
Hyperbole (/ h aɪ ˈ p ɜːr b əl i / ⓘ; adj. hyperbolic / ˌ h aɪ p ər ˈ b ɒ l ɪ k / ⓘ) is the use of exaggeration as a rhetorical device or figure of speech.In rhetoric, it is also sometimes known as auxesis (literally 'growth').
Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of circular sector area u and hyperbolic functions depending on hyperbolic sector area u. The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle.
(See Ellipse § Definition as locus of points and Hyperbola § As locus of points.) The foci F 1 , F 2 {\displaystyle F_{1},\,F_{2}} thus determine two pencils of confocal ellipses and hyperbolas. By the principal axis theorem , the plane admits a Cartesian coordinate system with its origin at the midpoint between foci and its axes aligned with ...
The eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a sharp "v" shape. At = the asymptotes are at right angles. With > the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion ...
and defining a unit hyperbola as = with its corresponding parameterized solution set = and = , and by letting < (the hyperbolic angle), we arrive at the result of =. Just as the circular angle is the length of a circular arc using the Euclidean metric, the hyperbolic angle is the length of a hyperbolic arc using the Minkowski metric.
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Hyperbolic may refer to: . of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics . Hyperbolic geometry, a non-Euclidean geometry; Hyperbolic functions, analogues of ordinary trigonometric functions, defined using the hyperbola