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...

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...

Pythagorean trigonometric identity Riemannian circle Radian Unit disk Unit sphere Unit hyperbola Unit square Turn (angle) z-transform Smith chart For further...

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...

Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic...

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...

{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...

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...

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,...

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...

split-complex numbers, which has an algebraic structure built upon the unit hyperbola. The elements hj and hk also determine such subalgebras. Furthermore...

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...

Cubic plane curve Quartic plane curve Fractal Conic sections Unit circle Unit hyperbola Folium of Descartes Cissoid of Diocles Conchoid of de Sluze Right...

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...

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...

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...

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...

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...

as conics or conic sections and include Circle Unit circle Ellipse Parabola Hyperbola Unit hyperbola Cubic plane curves include Cubic parabola Folium...

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...

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 +...

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...

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...

}=(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...

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...

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...

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...

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...

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 = +...

Standard basis Unit interval Unit square, cube, circle, sphere, and hyperbola Vector notation Vector of ones Unit matrix Weisstein, Eric W. "Unit Vector". Wolfram...