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In geometry, the unit hyperbola is the set of points (''x,y'') in the Cartesian plane that satisfies In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative radial length'' : Whereas the unit circle surrounds its center, the unit hyperbola requires the ''conjugate hyperbola'' to complement it in the plane. This pair of hyperbolas share the asymptotes ''y'' = ''x'' and ''y'' = −''x''. When the conjugate of the unit hyperbola is in use, the alternative radial length is The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals 〔(Wolfram Mathworld )〕 The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space. There the asymptotes of the unit hyperbola form a light cone. Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function and the modern parametrization of the hyperbola by sector areas. When the notions of conjugate hyperbolas and hyperbolic angles are understood, then the classical complex numbers, which are built around the unit circle, can be replaced with numbers built around the unit hyperbola. ==Asymptotes== (詳細はalgebraic geometry and the theory of algebraic curves there is a different approach to asymptotes. The curve is first interpreted in the projective plane using homogeneous coordinates. Then the asymptotes are lines that are tangent to the projective curve at a point at infinity, thus circumventing any need for a distance concept and convergence. In a common framework (''x, y, z'') are homogeneous coordinates with the line at infinity determined by the equation ''z'' = 0. For instance, C. G. Gibson wrote:〔C.G. Gibson (1998) ''Elementary Geometry of Algebraic Curves'', p 159, Cambridge University Press ISBN 0-521-64140-3〕 :For the standard rectangular hyperbola in R2 the corresponding projective curve is which meets ''z'' = 0 at the points ''P'' = (1 : 1 : 0) and ''Q'' = (1 : −1 : 0). Both ''P, Q'' are simple on ''F'', with tangents ''x'' + ''y'' = 0, ''x'' − ''y'' = 0; thus we recover the familiar 'asymptotes' of elementary geometry. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「unit hyperbola」の詳細全文を読む スポンサード リンク
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