In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation x 2 − y 2 = 1. {\displaystyle x^{2}-y^{2}=1...
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In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution...
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Pythagorean trigonometric identity Riemannian circle Radian Unit disk Unit sphere Unit hyperbola Unit square Turn (angle) z-transform Smith chart For further...
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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. A hyperbolic...
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Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic...
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Split-complex number (redirect from Hyperbolic unit)
through (a, 0) and (−a, 0). The case a = 1 is called the unit hyperbola. The conjugate hyperbola is given by { z : ‖ z ‖ 2 = − a 2 } {\displaystyle \left\{z:\lVert...
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{arsinh} x)=x.} Hyperbolic angle measure is the length of an arc of a unit hyperbola x 2 − y 2 = 1 {\displaystyle x^{2}-y^{2}=1} as measured in the Lorentzian...
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the unit hyperbola, since ( c / a ) 2 − ( b / a ) 2 = 1 , {\displaystyle (c/a)^{2}-(b/a)^{2}=1,} satisfying the equation for the unit hyperbola. The...
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functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t,...
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x^{2}-y^{2}} , when set equal to one, produces the unit hyperbola, which plays the role of the "unit circle" in the plane of split-complex numbers. Similarly...
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split-complex numbers, which has an algebraic structure built upon the unit hyperbola. The elements hj and hk also determine such subalgebras. Furthermore...
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hyperbolic-orthogonal at a point of the unit hyperbola. The parametric representation of the unit hyperbola via radius vector is p(a) = (cosh a, sinh...
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Cubic plane curve Quartic plane curve Fractal Conic sections Unit circle Unit hyperbola Folium of Descartes Cissoid of Diocles Conchoid of de Sluze Right...
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Numerically, the angle and 2 × area measures for the unit circle are identical. Fig. 7-1b shows a unit hyperbola with sinh(a) and cosh(a), where a is likewise...
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the unit hyperbola. Such matrices form the indefinite orthogonal group O(1,1) with one-dimensional Lie algebra spanned by the anti-diagonal unit matrix...
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arbitrariness of which diameter of the unit hyperbola was used to determine a world-line. Using the parametrization of the hyperbola with hyperbolic angle, the theory...
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the unit hyperbola x2 − y2 = 1 and its conjugate x2 − y2 = −1 can be used to form a polar decomposition based on the branch of the unit hyperbola through...
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first diagram used a branch of the unit hyperbola t 2 − x 2 = 1 {\displaystyle t^{2}-x^{2}=1} to show the locus of a unit of proper time depending on velocity...
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as conics or conic sections and include Circle Unit circle Ellipse Parabola Hyperbola Unit hyperbola Cubic plane curves include Cubic parabola Folium...
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measure of the unit hyperbola is generated by the hyperbolic angle of segments on the hyperbola. For instance, a measure of one unit is given by the...
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Geometrically, the hyperbolic tangent function is the hyperbolic angle on the unit hyperbola x 2 − y 2 = 1 {\displaystyle x^{2}-y^{2}=1} , which factors as ( x +...
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Versor (redirect from Unit quaternion)
Lie groups that contain hyperbolic versors include the group on the unit hyperbola and the special unitary group SU(1,1). The word is derived from Latin...
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Semi-major and semi-minor axes (section Hyperbola)
the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and...
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}=(x+y\mathbf {j} )(x-y\mathbf {j} )=x^{2}-y^{2}\!} and {z | z z* = 1} is the unit hyperbola. When ε2 = 0, then z is a dual number. This approach to non-Euclidean...
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identity of the algebra is at unit distance from the zero. With this basis a unit hyperbola may be compared to the unit circle of the ordinary complex...
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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...
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unit square (whose diagonal has endpoints at (0, 0) and (1, 1)), the unit hyperbola, and so on. The two axes divide the plane into four right angles, called...
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point at infinity. For example, one may identify the asymptotes to the unit hyperbola in this manner. Asymptotes are often considered only for real curves...
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one-parameter groups occur in the split-complex number plane as the unit hyperbola { exp ( j t ) = cosh ( t ) + j sinh ( t ) : t ∈ R } , j 2 = +...
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Standard basis Unit interval Unit square, cube, circle, sphere, and hyperbola Vector notation Vector of ones Unit matrix Weisstein, Eric W. "Unit Vector". Wolfram...
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