In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, hyperbolic angle is a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
determined by the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of the corresponding
hyperbolic sector of ''xy'' = 1 in Quadrant I of the
Cartesian plane. The hyperbolic angle parametrises the
unit hyperbola
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 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative ra ...
, which has
hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s as coordinates. In mathematics, hyperbolic angle is an
invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping ...
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
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
corresponding to the area of a
circular sector in a circle with radius
.
Hyperbolic angle is used as the
independent variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or dema ...
for the
hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s 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
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
of
real variables.
Definition
Consider the rectangular hyperbola
, and (by convention) pay particular attention to the ''branch''
.
First define:
* The hyperbolic angle in ''standard position'' is the
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
at
between the ray to
and the ray to
, where
.
* The magnitude of this angle is the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of the corresponding
hyperbolic sector, which turns out to be
.
Note that, because of the role played by the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
:
* Unlike the circular angle, the hyperbolic angle is ''unbounded'' (because
is unbounded); this is related to the fact that the
harmonic series is unbounded.
* The formula for the magnitude of the angle suggests that, for
, the hyperbolic angle should be negative. This reflects the fact that, as defined, the angle is ''directed''.
Finally, extend the definition of ''hyperbolic angle'' to that subtended by any interval on the hyperbola. Suppose
are
positive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used f ...
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
.
Comparison with circular angle
A
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
has a
circular sector with an area half of the circular angle in radians. Analogously, a
unit hyperbola
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 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative ra ...
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 section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
s, and hence are treated as
projective range
In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For inst ...
s in
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
. 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 ''P''
0''P''
1 and ''P''
0''P''
2 subtend angles ''L''
1 and ''L''
2 at the centre of a circle, their sum is the angle subtended by a chord ''PQ'', where ''PQ'' is required to be parallel to ''P''
1''P''
2.
The same construction can also be applied to the hyperbola. If ''P''
0 is taken to be the point , ''P''
1 the point , and ''P''
2 the point , then the parallel condition requires that ''Q'' be the point . It thus makes sense to define the hyperbolic angle from ''P''
0 to an arbitrary point on the curve as a logarithmic function of the point's value of ''x''.
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.
Both circular and hyperbolic angle provide instances of an
invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping ...
. 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
:(''x'', ''y'') ↦ (''rx'', ''y'' / ''r''), with ''r'' > 0 .
Relation To The Minkowski Line Element
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
:
Consider a curve imbedded in two dimensional Euclidean space,
:
Where the parameter
is a real number that runs between
and
(