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 ...
and
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, a Möbius transformation of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
is a
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
of the form
of one
complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' − ''bc'' ≠ 0.
Geometrically, a Möbius transformation can be obtained by first performing
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
from the plane to the
unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane.
These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle.
The Möbius transformations are the
projective transformations of the
complex projective line. They form a
group called the Möbius group, which is the
projective linear group PGL(2,C). Together with its
subgroups, it has numerous applications in mathematics and physics.
Möbius transformations are named in honor of
August Ferdinand Möbius; they are also variously named
homographies, homographic transformations, linear fractional transformations, bilinear transformations, fractional linear transformations, and spin transformations (in relativity theory).
Overview
Möbius transformations are defined on the
extended complex plane
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
(i.e., the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
augmented by the
point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. ...
).
Stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
identifies
with a sphere, which is then called the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
; alternatively,
can be thought of as the complex
projective line . The Möbius transformations are exactly the
bijective conformal maps from the Riemann sphere to itself, i.e., the
automorphisms of the Riemann sphere as a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
; alternatively, they are the automorphisms of
as an algebraic variety. Therefore, the set of all Möbius transformations forms a
group under
composition. This group is called the Möbius group, and is sometimes denoted
.
The Möbius group is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the group of orientation-preserving
isometries of
hyperbolic 3-space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
and therefore plays an important role when studying
hyperbolic 3-manifolds.
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the
identity component of the
Lorentz group acts on the
celestial sphere
In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphe ...
in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of
twistor theory
In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic ar ...
.
Certain
subgroups of the Möbius group form the automorphism groups of the other
simply-connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
Riemann surfaces (the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
and the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ' ...
). As such, Möbius transformations play an important role in the theory of
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s. The
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
of every Riemann surface is a
discrete subgroup of the Möbius group (see
Fuchsian group and
Kleinian group). A particularly important discrete subgroup of the Möbius group is the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
; it is central to the theory of many
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
s,
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory o ...
s,
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
s and
Pellian equations.
Möbius transformations can be more generally defined in spaces of dimension ''n'' > 2 as the bijective conformal orientation-preserving maps from the
''n''-sphere to the ''n''-sphere. Such a transformation is the most general form of conformal mapping of a domain. According to
Liouville's theorem a Möbius transformation can be expressed as a composition of translations,
similarities, orthogonal transformations and inversions.
Definition
The general form of a Möbius transformation is given by
where ''a'', ''b'', ''c'', ''d'' are any
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s satisfying . If , the rational function defined above is a constant since
and is thus not considered a Möbius transformation.
In case , this definition is extended to the whole
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
by defining
If , we define
Thus a Möbius transformation is always a bijective
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
from the Riemann sphere to the Riemann sphere.
The set of all Möbius transformations forms a
group under
composition. This group can be given the structure of a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
in such a way that composition and inversion are
holomorphic maps. The Möbius group is then a
complex Lie group
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mat ...
. The Möbius group is usually denoted
as it is the
automorphism group of the Riemann sphere.
Fixed points
Every non-identity Möbius transformation has two
fixed points on the Riemann sphere. Note that the fixed points are counted here with
multiplicity; the parabolic transformations are those where the fixed points coincide. Either or both of these fixed points may be the point at infinity.
Determining the fixed points
The fixed points of the transformation
are obtained by solving the fixed point equation ''f''(''γ'') = ''γ''. For ''c'' ≠ 0, this has two roots obtained by expanding this equation to
and applying the
quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
. The roots are
with discriminant
Parabolic transforms have coincidental fixed points due to zero discriminant. For ''c'' nonzero and nonzero discriminant the transform is elliptic or hyperbolic.
When ''c'' = 0, the quadratic equation degenerates into a linear equation and the transform is linear. This corresponds to the situation that one of the fixed points is the point at infinity. When ''a'' ≠ ''d'' the second fixed point is finite and is given by
In this case the transformation will be a simple transformation composed of
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
s,
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s, and
dilation
Dilation (or dilatation) may refer to:
Physiology or medicine
* Cervical dilation, the widening of the cervix in childbirth, miscarriage etc.
* Coronary dilation, or coronary reflex
* Dilation and curettage, the opening of the cervix and surgi ...
s:
If ''c'' = 0 and ''a'' = ''d'', then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation:
Topological proof
Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of the sphere being 2:
Firstly, the
projective linear group PGL(2,''K'') is
sharply 3-transitive – for any two ordered triples of distinct points, there is a unique map that takes one triple to the other, just as for Möbius transforms, and by the same algebraic proof (essentially
dimension counting
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
, as the group is 3-dimensional). Thus any map that fixes at least 3 points is the identity.
Next, one can see by identifying the Möbius group with
that any Möbius function is homotopic to the identity. Indeed, any member of the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
can be reduced to the identity map by Gauss-Jordan elimination, this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. The
Lefschetz–Hopf theorem states that the sum of the indices (in this context, multiplicity) of the fixed points of a map with finitely many fixed points equals the
Lefschetz number of the map, which in this case is the trace of the identity map on homology groups, which is simply the Euler characteristic.
By contrast, the projective linear group of the real projective line, PGL(2,R) need not fix any points – for example
has no (real) fixed points: as a complex transformation it fixes ±''i''
[Geometrically this map is the ]stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
of a rotation by 90° around ±''i'' with period 4, which takes – while the map 2''x'' fixes the two points of 0 and ∞. This corresponds to the fact that the Euler characteristic of the circle (real projective line) is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more.
Normal form
Möbius transformations are also sometimes written in terms of their fixed points in so-called normal form. We first treat the non-parabolic case, for which there are two distinct fixed points.
''Non-parabolic case'':
Every non-parabolic transformation is
conjugate to a dilation/rotation, i.e., a transformation of the form
(''k'' ∈ C) with fixed points at 0 and ∞. To see this define a map
which sends the points (''γ''
1, ''γ''
2) to (0, ∞). Here we assume that γ
1 and γ
2 are distinct and finite. If one of them is already at infinity then ''g'' can be modified so as to fix infinity and send the other point to 0.
If ''f'' has distinct fixed points (''γ''
1, ''γ''
2) then the transformation
has fixed points at 0 and ∞ and is therefore a dilation:
. The fixed point equation for the transformation ''f'' can then be written
Solving for ''f'' gives (in matrix form):
or, if one of the fixed points is at infinity:
From the above expressions one can calculate the derivatives of ''f'' at the fixed points:
and
Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers (''k'') of ''f'' as the characteristic constant of ''f''. Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant:
For loxodromic transformations, whenever , ''k'', > 1, one says that γ
1 is the repulsive fixed point, and γ
2 is the attractive fixed point. For , ''k'', < 1, the roles are reversed.
''Parabolic case'':
In the parabolic case there is only one fixed point ''γ''. The transformation sending that point to ∞ is
or the identity if ''γ'' is already at infinity. The transformation
fixes infinity and is therefore a translation:
Here, β is called the translation length. The fixed point formula for a parabolic transformation is then
Solving for ''f'' (in matrix form) gives
or, if ''γ'' = ∞:
Note that ''β'' is ''not'' the characteristic constant of ''f'', which is always 1 for a parabolic transformation. From the above expressions one can calculate:
Poles of the transformation
The point
is called the
pole of
; it is that point which is transformed to the point at infinity under
.
The inverse pole
is that point to which the point at infinity is transformed. The point midway between the two poles is always the same as the point midway between the two fixed points:
These four points are the vertices of a
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
which is sometimes called the characteristic parallelogram of the transformation.
A transform
can be specified with two fixed points ''γ''
1, ''γ''
2 and the pole
.
This allows us to derive a formula for conversion between ''k'' and
given
:
which reduces down to
The last expression coincides with one of the (mutually reciprocal)
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
ratios
of the matrix
representing the transform (compare the discussion in the preceding section about the characteristic constant of a transformation). Its
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The c ...
is equal to
which has roots
Simple Möbius transformations and composition
A Möbius transformation can be
composed as a sequence of simple transformations.
The following simple transformations are also Möbius transformations:
*
is a
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
.
*
is a combination of a (
homothety
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by th ...
and a
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
). If
then it is a rotation, if
then it is a homothety.
*
(
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
and
reflection with respect to the real axis)
Composition of simple transformations
If
, let:
*
(
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
by ''d''/''c'')
*
(
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
and
reflection with respect to the real axis)
*
(
homothety
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by th ...
and
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
)
*
(translation by ''a''/''c'')
Then these functions can be
composed, giving
That is,
with
This decomposition makes many properties of the Möbius transformation obvious.
Elementary properties
A Möbius transformation is equivalent to a sequence of simpler transformations. The composition makes many properties of the Möbius transformation obvious.
Formula for the inverse transformation
The existence of the inverse Möbius transformation and its explicit formula are easily derived by the composition of the inverse functions of the simpler transformations. That is, define functions ''g''
1, ''g''
2, ''g''
3, ''g''
4 such that each ''g
i'' is the inverse of ''f
i''. Then the composition
gives a formula for the inverse.
Preservation of angles and generalized circles
From this decomposition, we see that Möbius transformations carry over all non-trivial properties of
circle inversion. For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilations and
isometries (translation, reflection, rotation), which trivially preserve angles.
Furthermore, Möbius transformations map
generalized circles to generalized circles since circle inversion has this property. A generalized circle is either a circle or a line, the latter being considered as a circle through the point at infinity. Note that a Möbius transformation does not necessarily map circles to circles and lines to lines: it can mix the two. Even if it maps a circle to another circle, it does not necessarily map the first circle's center to the second circle's center.
Cross-ratio preservation
Cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, t ...
s are invariant under Möbius transformations. That is, if a Möbius transformation maps four distinct points
to four distinct points
respectively, then
If one of the points
is the point at infinity, then the cross-ratio has to be defined by taking the appropriate limit; e.g. the cross-ratio of
is
The cross ratio of four different points is real if and only if there is a line or a circle passing through them. This is another way to show that Möbius transformations preserve generalized circles.
Conjugation
Two points ''z''
1 and ''z''
2 are conjugate with respect to a generalized circle ''C'', if, given a generalized circle ''D'' passing through ''z''
1 and ''z''
2 and cutting ''C'' in two points ''a'' and ''b'', are in
harmonic cross-ratio (i.e. their cross ratio is −1). This property does not depend on the choice of the circle ''D''. This property is also sometimes referred to as being symmetric with respect to a line or circle.
Two points ''z'', ''z''
∗ are conjugate with respect to a line, if they are
symmetric with respect to the line. Two points are conjugate with respect to a circle if they are exchanged by the
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
with respect to this circle.
The point ''z''
∗ conjugate to ''z'' when ''L'' is the line determined by the vector based ''e
iθ'' at the point ''z''
0 can be explicitly given as
The point ''z''
∗ conjugate to ''z'' when ''C'' is the circle of radius ''r'' centered ''z''
0 can be explicitly given as
Since Möbius transformations preserve generalized circles and cross-ratios, they preserve also the conjugation.
Projective matrix representations
The natural
action of PGL(2,C) on the
complex projective line CP
1 is exactly the natural action of the Möbius group on the Riemann sphere, where the projective line CP
1 and the Riemann sphere are identified as follows:
Here
1:''z''2">'z''1:''z''2are
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
on CP
1; the point
:0corresponds to the point ∞ of the Riemann sphere. By using homogeneous coordinates, many concrete calculations involving Möbius transformations can be simplified, since no case distinctions dealing with ∞ are required.
With every
invertible complex 2×2 matrix
we can associate the Möbius transformation
The condition ''ad'' − ''bc'' ≠ 0 is equivalent to the condition that the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of above matrix be nonzero, i.e. that the matrix be invertible.
It is straightforward to check that then the
product of two matrices will be associated with the composition of the two corresponding Möbius transformations. In other words, the map
from the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL(2,C) to the Möbius group, which sends the matrix
to the transformation ''f'', is a
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
.
Note that any matrix obtained by multiplying
by a complex scalar λ determines the same transformation, so a Möbius transformation determines its matrix only
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
scalar multiples. In other words: the
kernel of consists of all scalar multiples of the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
''I'', and the
first isomorphism theorem of group theory states that the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
GL(2,C) / ((C\)''I'') is isomorphic to the Möbius group. This quotient group is known as the
projective linear group and is usually denoted PGL(2,C).
The same identification of PGL(2,''K'') with the group of fractional linear transformations and with the group of projective linear automorphisms of the projective line holds over any field ''K'', a fact of algebraic interest, particularly for finite fields, though the case of the complex numbers has the greatest geometric interest.
If one restricts
to matrices of determinant one, the map restricts to a surjective map from the
special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...
SL(2,C) to the Möbius group; in the restricted setting the kernel is formed by plus and minus the identity, and the quotient group SL(2,C) / , denoted by PSL(2,C), is therefore also isomorphic to the Möbius group:
From this we see that the Möbius group is a 3-dimensional complex Lie group (or a 6-dimensional real Lie group). It is a
semisimple
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
non-
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
Lie group.
Note that there are precisely two matrices with unit determinant which can be used to represent any given Möbius transformation. That is, SL(2,C) is a
double cover of PSL(2,C). Since SL(2,C) is
simply-connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
it is the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
of the Möbius group. Therefore, the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
of the Möbius group is Z
2.
Specifying a transformation by three points
Given a set of three distinct points
on the Riemann sphere and a second set of distinct points
, there exists precisely one Möbius transformation
with
for
. (In other words: the
action of the Möbius group on the Riemann sphere is ''sharply 3-transitive''.) There are several ways to determine
from the given sets of points.
Mapping first to 0, 1,
It is easy to check that the Möbius transformation
with matrix
maps
to
respectively. If one of the ''
'' is
, then the proper formula for
is obtained from the above one by first dividing all entries by ''
'' and then taking the limit ''
''.
If
is similarly defined to map
to
then the matrix
which maps
to
becomes
.
The stabilizer of
(as an unordered set) is a subgroup known as the
anharmonic group.
Explicit determinant formula
The equation
is equivalent to the equation of a standard
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) 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, ca ...
in the
-plane. The problem of constructing a Möbius transformation
mapping a triple
to another triple
is thus equivalent to finding the coefficients
of the hyperbola passing through the points
. An explicit equation can be found by evaluating the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
by means of a
Laplace expansion along the first row. This results in the determinant formulae
for the coefficients
of the representing matrix
. The constructed matrix
has determinant equal to
which does not vanish if the
resp. ''
'' are pairwise different thus the Möbius transformation is well-defined. If one of the points ''
'' or ''
'' is
, then we first divide all four determinants by this variable and then take the limit as the variable approaches
.
Subgroups of the Möbius group
If we require the coefficients
of a Möbius transformation to be real numbers with
, we obtain a subgroup of the Möbius group denoted as
PSL(2,R). This is the group of those Möbius transformations that map the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
to itself, and is equal to the group of all
biholomorphic (or equivalently:
bijective,
conformal and orientation-preserving) maps . If a proper
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
is introduced, the upper half-plane becomes a model of the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ' ...
''H'', the
Poincaré half-plane model
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.
Equivalently the Poincar� ...
, and PSL(2,R) is the group of all orientation-preserving isometries of ''H'' in this model.
The subgroup of all Möbius transformations that map the open disk to itself consists of all transformations of the form
with
∈ R, ''b'' ∈ C and , ''b'', < 1. This is equal to the group of all biholomorphic (or equivalently: bijective, angle-preserving and orientation-preserving) maps . By introducing a suitable metric, the open disk turns into another model of the hyperbolic plane, the
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk ...
, and this group is the group of all orientation-preserving isometries of ''H'' in this model.
Since both of the above subgroups serve as isometry groups of ''H'', they are isomorphic. A concrete isomorphism is given by
conjugation
Conjugation or conjugate may refer to:
Linguistics
*Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
*Complex conjugation, the change ...
with the transformation
which bijectively maps the open unit disk to the upper half plane.
Alternatively, consider an open disk with radius ''r'', centered at ''r'i''. The Poincaré disk model in this disk becomes identical to the upper-half-plane model as ''r'' approaches ∞.
A
maximal compact subgroup of the Möbius group
is given by
and corresponds under the isomorphism
to the
projective special unitary group PSU(2,C) which is isomorphic to the
special orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
SO(3) of rotations in three dimensions, and can be interpreted as rotations of the Riemann sphere. Every finite subgroup is conjugate into this maximal compact group, and thus these correspond exactly to the polyhedral groups, the
point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometrie ...
.
Icosahedral groups of Möbius transformations were used by
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
to give an analytic solution to the
quintic equation
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
in ; a modern exposition is given in .
If we require the coefficients ''a'', ''b'', ''c'', ''d'' of a Möbius transformation to be
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s with ''ad'' − ''bc'' = 1, we obtain the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
PSL(2,Z), a discrete subgroup of PSL(2,R) important in the study of
lattices in the complex plane,
elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those ...
s and
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
s. The discrete subgroups of PSL(2,R) are known as
Fuchsian groups; they are important in the study of
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s.
Classification
In the following discussion we will always assume that the representing matrix
is normalized such that
.
Non-identity Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and loxodromic, with the hyperbolic ones being a subclass of the loxodromic ones. The classification has both algebraic and geometric significance. Geometrically, the different types result in different transformations of the complex plane, as the figures below illustrate.
The four types can be distinguished by looking at the
trace . Note that the trace is invariant under
conjugation
Conjugation or conjugate may refer to:
Linguistics
*Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
*Complex conjugation, the change ...
, that is,
and so every member of a conjugacy class will have the same trace. Every Möbius transformation can be written such that its representing matrix
has determinant one (by multiplying the entries with a suitable scalar). Two Möbius transformations
(both not equal to the identity transform) with
are conjugate if and only if
Parabolic transforms
A non-identity Möbius transformation defined by a matrix
of determinant one is said to be ''parabolic'' if
(so the trace is plus or minus 2; either can occur for a given transformation since
is determined only up to sign). In fact one of the choices for
has the same
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The c ...
''X''
2−2''X''+1 as the identity matrix, and is therefore
unipotent
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''.
In particular, a square matrix ''M'' is a unipoten ...
. A Möbius transform is parabolic if and only if it has exactly one fixed point in the
extended complex plane
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
, which happens if and only if it can be defined by a matrix
conjugate to
which describes a translation in the complex plane.
The set of all parabolic Möbius transformations with a ''given'' fixed point in
, together with the identity, forms a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
isomorphic to the group of matrices
this is an example of the
unipotent radical
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''.
In particular, a square matrix ''M'' is a unipote ...
of a
Borel subgroup (of the Möbius group, or of SL(2,C) for the matrix group; the notion is defined for any
reductive Lie group).
Characteristic constant
All non-parabolic transformations have two fixed points and are defined by a matrix conjugate to
with the complex number ''λ'' not equal to 0, 1 or −1, corresponding to a dilation/rotation through multiplication by the complex number ''k'' = λ
2, called the characteristic constant or multiplier of the transformation.
Elliptic transforms
The transformation is said to be ''elliptic'' if it can be represented by a matrix
whose trace is
real with
A transform is elliptic if and only if , ''λ'', = 1 and ''λ'' ≠ ±1. Writing
, an elliptic transform is conjugate to
with ''α'' real.
Note that for ''any''
with characteristic constant ''k'', the characteristic constant of
is ''k
n''. Thus, all Möbius transformations of finite
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
are elliptic transformations, namely exactly those where λ is a
root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
, or, equivalently, where α is a
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
multiple of
. The simplest possibility of a fractional multiple means ''α'' = /2, which is also the unique case of
, is also denoted as a ; this corresponds geometrically to rotation by 180° about two fixed points. This class is represented in matrix form as:
There are 3 representatives fixing , which are the three transpositions in the symmetry group of these 3 points:
which fixes 1 and swaps 0 with ''∞'' (rotation by 180° about the points 1 and −1),
, which fixes ''∞'' and swaps 0 with 1 (rotation by 180° about the points 1/2 and ''∞''), and
which fixes 0 and swaps 1 with ''∞'' (rotation by 180° about the points 0 and 2).
Hyperbolic transforms
The transform is said to be ''hyperbolic'' if it can be represented by a matrix
whose trace is
real with
A transform is hyperbolic if and only if ''λ'' is real and ''λ'' ≠ ±1.
Loxodromic transforms
The transform is said to be ''loxodromic'' if
is not in
,4 A transformation is loxodromic if and only if
.
Historically,
navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation ...
by
loxodrome
In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north.
Introduction
The effect of following a rhumb li ...
or
rhumb line
In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north.
Introduction
The effect of following a rhumb l ...
refers to a path of constant
bearing; the resulting path is a
logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes. See the geometric figures below.
General classification
The real case and a note on terminology
Over the real numbers (if the coefficients must be real), there are no non-hyperbolic loxodromic transformations, and the classification is into elliptic, parabolic, and hyperbolic, as for real
conics. The terminology is due to considering half the absolute value of the trace, , tr, /2, as the
eccentricity of the transformation – division by 2 corrects for the dimension, so the identity has eccentricity 1 (tr/''n'' is sometimes used as an alternative for the trace for this reason), and absolute value corrects for the trace only being defined up to a factor of ±1 due to working in PSL. Alternatively one may use half the trace ''squared'' as a proxy for the eccentricity squared, as was done above; these classifications (but not the exact eccentricity values, since squaring and absolute values are different) agree for real traces but not complex traces. The same terminology is used for the
classification of elements of SL(2, R) (the 2-fold cover), and
analogous classifications are used elsewhere. Loxodromic transformations are an essentially complex phenomenon, and correspond to complex eccentricities.
Geometric interpretation of the characteristic constant
The following picture depicts (after stereographic transformation from the sphere to the plane) the two fixed points of a Möbius transformation in the non-parabolic case:
The characteristic constant can be expressed in terms of its
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
:
When expressed in this way, the real number ρ becomes an expansion factor. It indicates how repulsive the fixed point ''γ''
1 is, and how attractive ''γ''
2 is. The real number ''α'' is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about ''γ''
1 and clockwise about ''γ''
2.
Elliptic transformations
If ''ρ'' = 0, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be ''elliptic''. These transformations tend to move all points in circles around the two fixed points. If one of the fixed points is at infinity, this is equivalent to doing an affine rotation around a point.
If we take the
one-parameter subgroup
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
:\varphi : \mathbb \rightarrow G
from the real line \mathbb (as an additive group) to some other topological group G.
If \varphi i ...
generated by any elliptic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the ''same'' two points. All other points flow along a family of circles which is nested between the two fixed points on the Riemann sphere. In general, the two fixed points can be any two distinct points.
This has an important physical interpretation.
Imagine that some observer rotates with constant angular velocity about some axis. Then we can take the two fixed points to be the North and South poles of the celestial sphere. The appearance of the night sky is now transformed continuously in exactly the manner described by the one-parameter subgroup of elliptic transformations sharing the fixed points 0, ∞, and with the number α corresponding to the constant angular velocity of our observer.
Here are some figures illustrating the effect of an elliptic Möbius transformation on the Riemann sphere (after stereographic projection to the plane):
These pictures illustrate the effect of a single Möbius transformation. The one-parameter subgroup which it generates ''continuously'' moves points along the family of circular arcs suggested by the pictures.
Hyperbolic transformations
If ''α'' is zero (or a multiple of 2), then the transformation is said to be ''hyperbolic''. These transformations tend to move points along circular paths from one fixed point toward the other.
If we take the
one-parameter subgroup
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
:\varphi : \mathbb \rightarrow G
from the real line \mathbb (as an additive group) to some other topological group G.
If \varphi i ...
generated by any hyperbolic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the ''same'' two points. All other points flow along a certain family of circular arcs ''away'' from the first fixed point and ''toward'' the second fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere.
This too has an important physical interpretation. Imagine that an observer accelerates (with constant magnitude of acceleration) in the direction of the North pole on his celestial sphere. Then the appearance of the night sky is transformed in exactly the manner described by the one-parameter subgroup of hyperbolic transformations sharing the fixed points 0, ∞, with the real number ρ corresponding to the magnitude of his acceleration vector. The stars seem to move along longitudes, away from the South pole toward the North pole. (The longitudes appear as circular arcs under stereographic projection from the sphere to the plane.)
Here are some figures illustrating the effect of a hyperbolic Möbius transformation on the Riemann sphere (after stereographic projection to the plane):
These pictures resemble the field lines of a positive and a negative electrical charge located at the fixed points, because the circular flow lines subtend a constant angle between the two fixed points.
Loxodromic transformations
If both ''ρ'' and ''α'' are nonzero, then the transformation is said to be ''loxodromic''. These transformations tend to move all points in S-shaped paths from one fixed point to the other.
The word "
loxodrome
In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north.
Introduction
The effect of following a rhumb li ...
" is from the Greek: "λοξος (loxos), ''slanting'' + δρόμος (dromos), ''course''". When
sailing
Sailing employs the wind—acting on sails, wingsails or kites—to propel a craft on the surface of the ''water'' (sailing ship, sailboat, raft, windsurfer, or kitesurfer), on ''ice'' (iceboat) or on ''land'' ( land yacht) over a chose ...
on a constant
bearing – if you maintain a heading of (say) north-east, you will eventually wind up sailing around the
north pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Ma ...
in a
logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
. On the
mercator projection
The Mercator projection () is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and s ...
such a course is a straight line, as the north and south poles project to infinity. The angle that the loxodrome subtends relative to the lines of longitude (i.e. its slope, the "tightness" of the spiral) is the argument of ''k''. Of course, Möbius transformations may have their two fixed points anywhere, not just at the north and south poles. But any loxodromic transformation will be conjugate to a transform that moves all points along such loxodromes.
If we take the
one-parameter subgroup
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
:\varphi : \mathbb \rightarrow G
from the real line \mathbb (as an additive group) to some other topological group G.
If \varphi i ...
generated by any loxodromic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the ''same'' two points. All other points flow along a certain family of curves, ''away'' from the first fixed point and ''toward'' the second fixed point. Unlike the hyperbolic case, these curves are not circular arcs, but certain curves which under stereographic projection from the sphere to the plane appear as spiral curves which twist counterclockwise infinitely often around one fixed point and twist clockwise infinitely often around the other fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere.
You can probably guess the physical interpretation in the case when the two fixed points are 0, ∞: an observer who is both rotating (with constant angular velocity) about some axis and moving along the ''same'' axis, will see the appearance of the night sky transform according to the one-parameter subgroup of loxodromic transformations with fixed points 0, ∞, and with ρ, α determined respectively by the magnitude of the actual linear and angular velocities.
Stereographic projection
These images show Möbius transformations
stereographically projected onto the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
. Note in particular that when projected onto a sphere, the special case of a fixed point at infinity looks no different from having the fixed points in an arbitrary location.
Iterating a transformation
If a transformation
has fixed points γ
1, γ
2, and characteristic constant ''k'', then
will have
.
This can be used to
iterate a transformation, or to animate one by breaking it up into steps.
These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants.
And these images demonstrate what happens when you transform a circle under Hyperbolic, Elliptical, and Loxodromic transforms. Note that in the elliptical and loxodromic images, the α value is 1/10 .
Higher dimensions
In higher dimensions, a Möbius transformation is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
of
, the
one-point compactification In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Al ...
of
, which is a finite composition of
inversions in spheres and
reflections in
hyperplanes.
Liouville's theorem in conformal geometry states that in dimension at least three, all
conformal transformations are Möbius transformations. Every Möbius transformation can be put in the form
where
,
,
is an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity m ...
, and
is 0 or 2. The group of Möbius transformations is also called the Möbius group.
The orientation-preserving Möbius transformations form the connected component of the identity in the Möbius group. In dimension , the orientation-preserving Möbius transformations are exactly the maps of the Riemann sphere covered here. The orientation-reversing ones are obtained from these by complex conjugation.
The domain of Möbius transformations, i.e.
, is homeomorphic to the ''n''-dimensional sphere
. The canonical isomorphism between these two spaces is the
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform i ...
, which is itself a Möbius transformation of
. This identification means that Möbius transformations can also be thought of as conformal isomorphisms of
. The ''n''-sphere, together with action of the Möbius group, is a geometric structure (in the sense of Klein's
Erlangen program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
) called
Möbius geometry.
Applications
Lorentz transformation
An isomorphism of the Möbius group with the
Lorentz group was noted by several authors: Based on previous work of
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
(1893, 1897) on
automorphic functions related to hyperbolic geometry and Möbius geometry,
Gustav Herglotz
Gustav Herglotz (2 February 1881 – 22 March 1953) was a German Bohemian physicist best known for his works on the theory of relativity and seismology.
Biography
Gustav Ferdinand Joseph Wenzel Herglotz was born in Volary num. 28 to a public ...
(1909) showed that
hyperbolic motion
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geom ...
s (i.e.
isometric automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s of a
hyperbolic space) transforming the
unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
into itself correspond to Lorentz transformations, by which Herglotz was able to classify the one-parameter Lorentz transformations into loxodromic, elliptic, hyperbolic, and parabolic groups. Other authors include
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing l ...
(1957),
H. S. M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
(1965), and
Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus f ...
,
Wolfgang Rindler (1984),
Tristan Needham (1997) and W. M. Olivia (2002).
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
consists of the four-dimensional real coordinate space R
4 consisting of the space of ordered quadruples (''x''
0,''x''
1,''x''
2,''x''
3) of real numbers, together with a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
Borrowing terminology from
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
, points with are considered ''timelike''; in addition, if , then the point is called ''future-pointing''. Points with are called ''spacelike''. The
null cone ''S'' consists of those points where ; the ''future null cone'' ''N''
+ are those points on the null cone with . The
celestial sphere
In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphe ...
is then identified with the collection of rays in ''N''
+ whose initial point is the origin of R
4. The collection of
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s on R
4 with positive
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
preserving the quadratic form ''Q'' and preserving the time direction form the
restricted Lorentz group SO
+(1,3).
In connection with the geometry of the celestial sphere, the group of transformations SO
+(1,3) is identified with the group PSL(2,C) of Möbius transformations of the sphere. To each , associate the
hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...
The
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the matrix ''X'' is equal to . The
special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...
acts on the space of such matrices via
for each ''A'' ∈ SL(2,C), and this action of SL(2,C) preserves the determinant of ''X'' because . Since the determinant of ''X'' is identified with the quadratic form ''Q'', SL(2,C) acts by Lorentz transformations. On dimensional grounds, SL(2,C) covers a neighborhood of the identity of SO(1,3). Since SL(2,C) is connected, it covers the entire restricted Lorentz group SO
+(1,3). Furthermore, since the
kernel of the action () is the subgroup , then passing to the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
gives the
group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two gr ...
Focusing now attention on the case when (''x''
0,''x''
1,''x''
2,''x''
3) is null, the matrix ''X'' has zero determinant, and therefore splits as the
outer product
In linear algebra, the outer product of two coordinate vectors is a matrix. If the two vectors have dimensions ''n'' and ''m'', then their outer product is an ''n'' × ''m'' matrix. More generally, given two tensors (multidimensional arrays of nu ...
of a complex two-vector ξ with its complex conjugate:
The two-component vector ξ is acted upon by SL(2,C) in a manner compatible with (). It is now clear that the kernel of the representation of SL(2,C) on hermitian matrices is .
The action of PSL(2,C) on the celestial sphere may also be described geometrically using
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
. Consider first the hyperplane in R
4 given by ''x''
0 = 1. The celestial sphere may be identified with the sphere ''S''
+ of intersection of the hyperplane with the future null cone ''N''
+. The stereographic projection from the north pole (1,0,0,1) of this sphere onto the plane ''x''
3 = 0 takes a point with coordinates (1,''x''
1,''x''
2,''x''
3) with
to the point
Introducing the
complex coordinate
the inverse stereographic projection gives the following formula for a point (''x''
1, ''x''
2, ''x''
3) on ''S''
+:
The action of SO
+(1,3) on the points of ''N''
+ does not preserve the hyperplane ''S''
+, but acting on points in ''S''
+ and then rescaling so that the result is again in ''S''
+ gives an action of SO
+(1,3) on the sphere which goes over to an action on the complex variable ζ. In fact, this action is by fractional linear transformations, although this is not easily seen from this representation of the celestial sphere. Conversely, for any fractional linear transformation of ζ variable goes over to a unique Lorentz transformation on ''N''
+, possibly after a suitable (uniquely determined) rescaling.
A more invariant description of the stereographic projection which allows the action to be more clearly seen is to consider the variable ζ = ''z'':''w'' as a ratio of a pair of homogeneous coordinates for the complex projective line CP
1. The stereographic projection goes over to a transformation from C
2 − to ''N''
+ which is homogeneous of degree two with respect to real scalings
which agrees with () upon restriction to scales in which
The components of () are precisely those obtained from the outer product
In summary, the action of the restricted Lorentz group SO
+(1,3) agrees with that of the Möbius group PSL(2,C). This motivates the following definition. In dimension ''n'' ≥ 2, the Möbius group Möb(''n'') is the group of all orientation-preserving
conformal isometries of the round sphere ''S''
''n'' to itself. By realizing the conformal sphere as the space of future-pointing rays of the null cone in the Minkowski space R
1,n+1, there is an isomorphism of Möb(''n'') with the restricted Lorentz group SO
+(1,''n''+1) of Lorentz transformations with positive determinant, preserving the direction of time.
Coxeter began instead with the equivalent quadratic form
He identified the Lorentz group with transformations for which is
stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
. Then he interpreted the x's as
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
and , the
null cone, as the
Cayley absolute Cayley may refer to:
__NOTOC__ People
* Cayley (surname)
* Cayley Illingworth (1759–1823), Anglican Archdeacon of Stow
* Cayley Mercer (born 1994), Canadian women's ice hockey player
Places
* Cayley, Alberta, Canada, a hamlet
* Mount Cayley, ...
for a hyperbolic space of points . Next, Coxeter introduced the variables
so that the Lorentz-invariant quadric corresponds to the sphere
Coxeter notes that
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
also wrote of this correspondence, applying stereographic projection from (0, 0, 1) to the complex plane
Coxeter used the fact that circles of the inversive plane represent planes of hyperbolic space, and the general homography is the product of inversions in two or four circles, corresponding to the general hyperbolic displacement which is the product of inversions in two or four planes.
Hyperbolic space
As seen above, the Möbius group PSL(2,C) acts on Minkowski space as the group of those isometries that preserve the origin, the orientation of space and the direction of time. Restricting to the points where ''Q''=1 in the positive light cone, which form a model of
hyperbolic 3-space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
''H'', we see that the Möbius group acts on ''H'' as a group of orientation-preserving isometries. In fact, the Möbius group is equal to the group of orientation-preserving isometries of hyperbolic 3-space.
If we use the
Poincaré ball model, identifying the unit ball in R
3 with ''H'', then we can think of the Riemann sphere as the "conformal boundary" of ''H''. Every orientation-preserving isometry of ''H'' gives rise to a Möbius transformation on the Riemann sphere and vice versa; this is the very first observation leading to the
AdS/CFT correspondence
In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter ...
conjectures in physics.
See also
*
Bilinear transform
The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.
The bilinear t ...
*
Conformal geometry
*
Fuchsian group
*
Generalised circle
In geometry, a generalized circle, also referred to as a "cline" or "circline", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and ...
*
Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
*
Infinite compositions of analytic functions
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the ...
*
Inversion transformation
*
Kleinian group
*
Lie sphere geometry
*
Linear fractional transformation
In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form
:z \mapsto \frac ,
which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfo ...
*
Liouville's theorem (conformal mappings)
*
Lorentz group
*
Modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
*
Poincaré half-plane model
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.
Equivalently the Poincar� ...
*
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, ...
*
Projective line over a ring
*
Representation theory of the Lorentz group
Notes
References
Specific
General
*
*
* ''(See Chapter 6 for the classification, up to conjugacy, of the Lie subalgebras of the Lie algebra of the Lorentz group.)''
* ''See Chapter 2''.
* .
* ''(See Chapters 3–5 of this classic book for a beautiful introduction to the Riemann sphere, stereographic projection, and Möbius transformations.)''
* ''(Aimed at non-mathematicians, provides an excellent exposition of theory and results, richly illustrated with diagrams.)''
* ''(See Chapter 3 for a beautifully illustrated introduction to Möbius transformations, including their classification up to conjugacy.)''
*
* ''(See Chapter 2 for an introduction to Möbius transformations.)''
*
Further reading
*
External links
*
Conformal maps gallery*
{{DEFAULTSORT:Mobius transformation
Projective geometry
Conformal geometry
Lie groups
Riemann surfaces
Functions and mappings
Kleinian groups
Conformal mappings