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...
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respectively. Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations...
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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...
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Gudermannian function (redirect from Hyperbolic amplitude)
mathematics, the Gudermannian function relates a hyperbolic angle measure ψ {\textstyle \psi } to a circular angle measure ϕ {\textstyle \phi } called the gudermannian...
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t|g_{ij}V^{i}V^{j}\right|}}}.} A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function...
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and v = x y {\displaystyle v={\sqrt {xy}}} . The parameter u is the hyperbolic angle to (x, y) and v is the geometric mean of x and y. The inverse mapping...
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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...
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not equivalent in hyperbolic geometry; new concepts need to be introduced. Further, because of the angle of parallelism, hyperbolic geometry has an absolute...
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unit hyperbola. A hyperbolic sector in standard position has a = 1 and b > 1. The argument of hyperbolic functions is the hyperbolic angle, which is defined...
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). For a given value of a hyperbolic function, the inverse hyperbolic function provides the corresponding hyperbolic angle measure, for example arsinh...
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Squeeze mapping (redirect from Hyperbolic rotation)
measure of a hyperbolic angle associated with the sector. The hyperbolic angle concept is quite independent of the ordinary circular angle, but shares...
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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...
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t,\ \sinh t).} This parameter t is the hyperbolic angle, which is the argument of the hyperbolic functions. One finds an early expression of the...
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Sophus Lie. An example of a one-parameter group is the hyperbolic versor with the hyperbolic angle parameter. This parameter is part of the polar decomposition...
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In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than...
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The argument of a circular function is an angle. The argument of a hyperbolic function is a hyperbolic angle. A mathematical function has one or more arguments...
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Split-complex number (redirect from Hyperbolic number)
algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying j 2 = 1 {\displaystyle...
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Non-Euclidean geometry (section Hyperbolic geometry)
quadrilateral with three right angles. The fourth angle of a Lambert quadrilateral is acute if the geometry is hyperbolic, a right angle if the geometry is Euclidean...
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Velocity-addition formula (section Hyperbolic geometry)
accounted in terms of the hyperbolic tangent function tanh which takes hyperbolic angle (rapidity) as an argument. In fact, the hyperbolic tangent of rapidity...
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lengths, angles, and other geometric figures, constructions can also be made in hyperbolic geometry. There are a couple of models for hyperbolic geometry...
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Orthogonality (section Hyperbolic orthogonality)
more specifically used for lines and planes that intersect to form a right angle, whereas orthogonal is used in generalizations, such as orthogonal vectors...
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the lengths of two sides and their angle, or the length of one side and the two adjacent angles (see hyperbolic trigonometry). Once again, the euclidean...
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Rapidity (category Hyperbolic functions)
light being infinite. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each...
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geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference...
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Hyperbola (redirect from Hyperbolic arc)
circular sector which that angle subtends. The analogous hyperbolic angle is likewise defined as twice the area of a hyperbolic sector. Let a {\displaystyle...
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Triangle (redirect from Angle proofs)
above, the internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less...
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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...
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Natural logarithm e (mathematical constant) Exponential function Hyperbolic angle Hyperbolic function Stirling's approximation Bernoulli numbers See also...
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Conformal map (redirect from Angle-preserving)
are conformal since hyperbolic rotations preserve hyperbolic angle, (called rapidity) and the other rotations preserve circular angle. The introduction...
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Conformal group (section Angle analysis)
by rapidity, a hyperbolic angle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities...
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