The curve represents xy = 1. A hyperbolic angle has magnitude equal to the area of the corresponding hyperbolic sector, which is in standard position if a = 1
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation.
The hyperbola xy = 1 is rectangular with a semi-major axis of , analogous to the magnitude of a circular angle corresponding to the area of a circular sector in a circle with radius .
Hyperbolic angle is used as the independent variable for the hyperbolic functions sinh, cosh, and tanh, because these functions may be premised on hyperbolic analogies to the corresponding circular trigonometric functions by regarding a hyperbolic angle as defining a hyperbolic triangle.
The parameter thus becomes one of the most useful in the calculus of real variables.
In geometry, hyperbolicangle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane...
cosh(t) and +sinh(t) respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the...
ar-. For a given value of a hyperbolic function, the inverse hyperbolic function provides the corresponding hyperbolicangle measure, for example arsinh...
about them is exaggerated. Hyperbolicangle, an unbounded variable referring to a hyperbola instead of a circle Hyperbolic coordinates, location by geometric...
not equivalent in hyperbolic geometry; new concepts need to be introduced. Further, because of the angle of parallelism, hyperbolic geometry has an absolute...
t|g_{ij}V^{i}V^{j}\right|}}}.} A hyperbolicangle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function...
the same area). When in standard position, a hyperbolic sector corresponds to a positive hyperbolicangle at the origin, with the measure of the latter...
measure of a hyperbolicangle associated with the sector. The hyperbolicangle concept is quite independent of the ordinary circular angle, but shares...
and v = x y {\displaystyle v={\sqrt {xy}}} . The parameter u is the hyperbolicangle to (x, y) and v is the geometric mean of x and y. The inverse mapping...
sides or edges and three points called angles or vertices. Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always...
A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center, unlike the constant angles of logarithmic spirals...
mathematics, the Gudermannian function relates a hyperbolicangle measure ψ {\textstyle \psi } to a circular angle measure ϕ {\textstyle \phi } called the gudermannian...
could be seen as simply a hyperbolic rotation of the spacetime coordinates, i.e., a rotation through an imaginary angle. This angle therefore represents (in...
hyperbola (/haɪˈpɜːrbələ/ ; pl. hyperbolas or hyperbolae /-liː/ ; adj. hyperbolic /ˌhaɪpərˈbɒlɪk/ ) is a type of smooth curve lying in a plane, defined...
In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than...
right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The sum of the angles of a hyperbolic triangle...
The argument of a circular function is an angle. The argument of a hyperbolic function is a hyperbolicangle. A mathematical function has one or more arguments...
In hyperbolic geometry, angle of parallelism Π ( a ) {\displaystyle \Pi (a)} is the angle at the non-right angle vertex of a right hyperbolic triangle...
Ancient Greek ὀρθός (orthós), meaning "upright", and γωνία (gōnía), meaning "angle". The Ancient Greek ὀρθογώνιον (orthogṓnion) and Classical Latin orthogonium...
Natural logarithm e (mathematical constant) Exponential function HyperbolicangleHyperbolic function Stirling's approximation Bernoulli numbers See also...
radial coordinate or radius, and the angle is called the angular coordinate, or polar angle. From the hyperbolic law of cosines, we get that the distance...
angular invariant measure, on a par with circular angle (invariant under rotation) and hyperbolicangle, with invariance group of squeeze mappings. The...
by rapidity, a hyperbolicangle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities...
geometric transformation. For examples, circular angle is invariant under rotation, hyperbolicangle is invariant under squeeze mapping, and a difference...
Sophus Lie. An example of a one-parameter group is the hyperbolic versor with the hyperbolicangle parameter. This parameter is part of the polar decomposition...
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