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.
Consider the rectangular hyperbola , and (by convention) pay particular attention to the branch.
First define:
Note that, because of the role played by the natural logarithm:
Finally, extend the definition of hyperbolic angle to that subtended by any interval on the hyperbola. Suppose are positive real numbers such that and , so that and are points on the hyperbola and determine an interval on it. Then the squeeze mapping maps the angle to the standard position angle . By the result of Gregoire de Saint-Vincent, the hyperbolic sectors determined by these angles have the same area, which is taken to be the magnitude of the angle. This magnitude is .
A unit circle has a circular sector with an area half of the circular angle in radians. Analogously, a unit hyperbola has a hyperbolic sector with an area half of the hyperbolic angle.
There is also a projective resolution between circular and hyperbolic cases: both curves are conic sections, and hence are treated as projective ranges in projective geometry. Given an origin point on one of these ranges, other points correspond to angles. The idea of addition of angles, basic to science, corresponds to addition of points on one of these ranges as follows:
Circular angles can be characterised geometrically by the property that if two chords P0P1 and P0P2 subtend angles L1 and L2 at the centre of a circle, their sum L1 + L2 is the angle subtended by a chord P0Q, where P0Q is required to be parallel to P1P2.
The same construction can also be applied to the hyperbola. If P0 is taken to be the point (1, 1), P1 the point (x1, 1/x1), and P2 the point (x2, 1/x2), then the parallel condition requires that Q be the point (x1x2, 1/x11/x2). It thus makes sense to define the hyperbolic angle from P0 to an arbitrary point on the curve as a logarithmic function of the point's value of x. [1] [2]
Whereas in Euclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a pseudo-Euclidean plane steadily moving orthogonally to a ray from the origin traces out a hyperbola. In Euclidean space, the multiple of a given angle traces equal distances around a circle while it traces exponential distances upon the hyperbolic line. [3]
Both circular and hyperbolic angle provide instances of an invariant measure. Arcs with an angular magnitude on a circle generate a measure on certain measurable sets on the circle whose magnitude does not vary as the circle turns or rotates. For the hyperbola the turning is by squeeze mapping, and the hyperbolic angle magnitudes stay the same when the plane is squeezed by a mapping
There is also a curious relation to a hyperbolic angle and the metric defined on Minkowski space. Just as two dimensional Euclidean geometry defines its line element as
the line element on Minkowski space is [4]
Consider a curve embedded in two dimensional Euclidean space,
Where the parameter is a real number that runs between and (). The arclength of this curve in Euclidean space is computed as:
If defines a unit circle, a single parameterized solution set to this equation is and . Letting , computing the arclength gives . Now doing the same procedure, except replacing the Euclidean element with the Minkowski line element,
and defining a unit hyperbola as with its corresponding parameterized solution set and , and by letting (the hyperbolic angle), we arrive at the result of . Just as the circular angle is the length of a circular arc using the Euclidean metric, the hyperbolic angle is the length of a hyperbolic arc using the Minkowski metric.
The quadrature of the hyperbola is the evaluation of the area of a hyperbolic sector. It can be shown to be equal to the corresponding area against an asymptote. The quadrature was first accomplished by Gregoire de Saint-Vincent in 1647 in Opus geometricum quadrature circuli et sectionum coni. As expressed by a historian,
A. A. de Sarasa interpreted the quadrature as a logarithm and thus the geometrically defined natural logarithm (or "hyperbolic logarithm") is understood as the area under y = 1/x to the right of x = 1. As an example of a transcendental function, the logarithm is more familiar than its motivator, the hyperbolic angle. Nevertheless, the hyperbolic angle plays a role when the theorem of Saint-Vincent is advanced with squeeze mapping.
Circular trigonometry was extended to the hyperbola by Augustus De Morgan in his textbook Trigonometry and Double Algebra. [6] In 1878 W.K. Clifford used the hyperbolic angle to parametrize a unit hyperbola, describing it as "quasi-harmonic motion".
In 1894 Alexander Macfarlane circulated his essay "The Imaginary of Algebra", which used hyperbolic angles to generate hyperbolic versors, in his book Papers on Space Analysis. [7] The following year Bulletin of the American Mathematical Society published Mellen W. Haskell's outline of the hyperbolic functions. [8]
When Ludwik Silberstein penned his popular 1914 textbook on the new theory of relativity, he used the rapidity concept based on hyperbolic angle a, where tanh a = v/c, the ratio of velocity v to the speed of light. He wrote:
Silberstein also uses Lobachevsky's concept of angle of parallelism Π(a) to obtain cos Π(a) = v/c. [9]
The hyperbolic angle is often presented as if it were an imaginary number, and so that the hyperbolic functions cosh and sinh can be presented through the circular functions. But in the Euclidean plane we might alternately consider circular angle measures to be imaginary and hyperbolic angle measures to be real scalars, and
These relationships can be understood in terms of the exponential function, which for a complex argument can be broken into even and odd parts and respectively. Then
or if the argument is separated into real and imaginary parts the exponential can be split into the product of scaling and rotation
As infinite series,
The infinite series for cosine is derived from cosh by turning it into an alternating series, and the series for sine comes from making sinh into an alternating series.
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 set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric 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, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively.
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle.
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices.
In hyperbolic geometry, angle of parallelism is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism.
In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar-.
In special relativity, the classical concept of velocity is converted to rapidity to accommodate the limit determined by speed of light. Velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velocity are almost exactly proportional but, for higher velocities, rapidity takes a larger value, with the rapidity of light being infinite.
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted as and .
The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus:
In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. It can also be related to the relativistic velocity addition formula.
In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length
In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rational function of by setting . This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line. The general transformation formula is:
In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent via trigonometric identities,
In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used.