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...

respectively. Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations...

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...

mathematics, the Gudermannian function relates a hyperbolic angle measure ψ {\textstyle \psi } to a circular angle measure ϕ {\textstyle \phi } called the gudermannian...

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...

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...

not equivalent in hyperbolic geometry; new concepts need to be introduced. Further, because of the angle of parallelism, hyperbolic geometry has an absolute...

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...

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...

). For a given value of a hyperbolic function, the inverse hyperbolic function provides the corresponding hyperbolic angle measure, for example arsinh...

measure of a hyperbolic angle associated with the sector. The hyperbolic angle concept is quite independent of the ordinary circular angle, but shares...

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...

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...

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...

In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than...

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...

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...

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...

lengths, angles, and other geometric figures, constructions can also be made in hyperbolic geometry. There are a couple of models for hyperbolic geometry...

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...

accounted in terms of the hyperbolic tangent function tanh which takes hyperbolic angle (rapidity) as an argument. In fact, the hyperbolic tangent of rapidity...

circular sector which that angle subtends. The analogous hyperbolic angle is likewise defined as twice the area of a hyperbolic sector. Let a {\displaystyle...

more specifically used for lines and planes that intersect to form a right angle, whereas orthogonal is used in generalizations, such as orthogonal vectors...

light being infinite. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each...

geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference...

are conformal since hyperbolic rotations preserve hyperbolic angle, (called rapidity) and the other rotations preserve circular angle. The introduction...

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...

functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side...

by rapidity, a hyperbolic angle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities...

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...