Möbius transformation
- Möbius transformations should not be confused with the Möbius transform or the Möbius function.
In geometry, a Möbius transformation is a function:
where z, a, b, c, d are complex numbers satisfying ad − bc ≠ 0.
Möbius transformations are named in honor of August Ferdinand Möbius, although they are also called homographic transformations or fractional linear transformations.
Overview
Möbius transformation is a bijective conformal map of the extended complex plane (i.e. the complex plane augmented by the point at infinity):
The set of all Möbius transformations forms a group under composition called the Möbius group.
The Möbius group is the automorphism group of the Riemann sphere, sometimes denoted
Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces (the complex plane and the hyperbolic plane). As such, Möbius transformations play an important role in the theory of Riemann surfaces. The covering group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). Möbius transformations are also closely related to isometries of hyperbolic 3-manifolds.
A particularly important subgroup of the Möbius group is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations.
In physics, the identity component of the Lorentz group acts on the celestial sphere 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.
Definition
The general form of a Möbius transformation is given by
where a, b, c, d are any complex numbers satisfying ad − bc ≠ 0. This definition can be extended to the whole Riemann sphere (the complex plane plus the point at infinity).
The set of all Möbius transformations forms a group under composition. This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps. The Möbius group is then a complex Lie group. The Möbius group is usually denoted as it is the automorphism group of the Riemann sphere.
Decomposition and Elementary properties
A mobius transformation is equivalent to a sequence of simpler transformations. Let:
* (translation) * (inversion and reflection) * (dilation and rotation) * (translation)
then:
This decompostion makes many properties of the mobius transform obvious.
For example, the preservation of angles is reduced to proving the angle preservation property of circle inversion, since all other transformation are isometries, which trivially preserve angles and distances.
The existence of an inverse mobius transformation function and its explicit formula is easily derived by a composition of the inverse function of the simpler transformations. That is, define functions g1, g2, g3, g4 such that g1 is the inverse of f1 and so on. Then, Composition[g1,g2,g3,g4][z] would be the explicit expression for the inverse mobius transformation, which is: (D Z - B)/(-C Z + A)
From this decomposition, we also see that mobius transformation carries over all non-trivial properties of circle inversion. Namely, that circles are mapped to circles, and angles are preserved. Also, because of the circle inversion, is carried over the convenience of defining möbius transformation over a plane with a point at infinity, which makes statements and concepts of möbius transformation's properties simpler.
For another example, look at f3. If A D - B C = 0, then the transformation collapses to the point 0, then f4 moves to A/C. Collapsing to a point is not an interesting transformation, thus we require in the definition of mobius transformation that .
Preservation of Angles and Circles
As seen from the above decomposition, mobius transformation contains this transformation 1/z, called complex inversion. Geometrically, a complex inversion is a circle inversion followed by a reflection around the x-axis.
In circle inversion, Circles are mapped to circles (here, lines are considered as circles with infinite radius), and angles are preserved. See circle inversion for various properties and proofs.
Cross-ratio preservation
The cross-ratio preservation theorem states that the cross-ratio
is invariant under a Möbius transformation that maps from z to w.
The action of the Möbius group on the Riemann sphere is sharply 3-transitive in the sense that there is a unique Möbius transformation which takes any three distinct points on the Riemann sphere to any other set of three distinct points. See the section below on specifying a transformation by three points.
Projective matrix representations
The transformation
can be usefully expressed as a matrix
The condition ad − bc ≠ 0 is equivalent to the condition that the determinant of above matrix be nonzero (i.e. the matrix should be non-singular). Note that multiplying by any complex number λ gives rise to the same transformation. Such matrix representations are called projective representations for reasons explained below. It is often convenient to normalize so that its determinant is equal to 1. The matrix is then unique up to sign.
The usefulness of this representation is that the composition of two Möbius transformations corresponds precisely to matrix multiplication of the corresponding matrices. That is, if we define a map
from the general linear group GL(2,C) to the Möbius group which sends the matrix to the transformation f, then this map is a group homomorphism.
The map is not an isomorphism, since it maps any scalar multiple of to the same transformation. The kernel of this homomorphism is then the set of all scalar matrices kI, which is the center of GL(2,C). The quotient group GL(2,C)/Z(GL(2,C)) is called the projective linear group and is usually denoted PGL(2,C). By the first isomorphism theorem of group theory we conclude that the Möbius group is isomorphic to PGL(2,C). Since Z(GL(2,C)) is the kernel of the group action given by GL(2,C) acting on itself by conjugation, PGL(2, C) is isomorphic to the inner automorphism group of GL(2,C). Moreover, the natural action of PGL(2,C) on the complex projective line CP1 is exactly the natural action of the Möbius group on the Riemann sphere when the sphere and the projective line are identified as follows:
Here [z1:z2] are homogeneous coordinates on CP1.
If one normalizes so that the determinant is equal to one, the map restricts to a surjective map from the special linear group SL(2,C) to the Möbius group. The Möbius group is therefore also isomorphic to PSL(2,C). We then have the following isomorphisms:
From the last identification we see that the Möbius group is a 3-dimensional complex Lie group (or a 6-dimensional real 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 it is the universal cover of the Möbius group. The fundamental group of the Möbius group is then Z2.
Classification
Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and loxodromic (actually hyperbolic is a special case of loxodromic). 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. These types can be distinguished by looking at the trace . Note that the trace is invariant under conjugation, 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 .
In the following discussion we will always assume that the representing matrix is normalized such that .
Parabolic transforms The transform is said to be parabolic if
- .
A transform is parabolic if and only if it has one fixed point in the compactified complex plane . It is parabolic if and only if it is conjugate to
- .
The subgroup consisting of all parabolic transforms of this form:
is an example of a Borel subgroup, which generalizes the idea to higher dimensions.
All other non-identity transformations have two fixed points. All non-parabolic (non-identity) transforms are conjugate to
with not equal to 0,1 or -1. The square is called the characteristic constant or multiplier of the transformation.
Elliptic transforms The transform is said to be elliptic if
- .
A transform is elliptic if and only if . Writing , an elliptic transform is conjugate to
with real. Note that for any , the characteristic constant of is . Thus, the only Möbius transformations of finite order are the elliptic transformations, and these only when λ is a root of unity; equivalently, when α is a rational multiple of pi.
Hyperbolic transforms The transform is said to be hyperbolic if
- .
A transform is hyperbolic if and only if λ is real and positive.
Loxodromic transforms The transform is said to be loxodromic if is not in the closed interval of [0,4]. Hyperbolic transforms are thus a special case of loxodromic transformations. A transformation is loxodromic if and only if . Historically, navigation by loxodrome or rhumb line refers to a path of constant bearing; the resulting path is a logarithmic spiral, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes. See the geometric figures below.
Transformation | Trace squared | Multipliers | Class representative | |
---|---|---|---|---|
Elliptic | ||||
Parabolic | ||||
Hyperbolic | ||||
Loxodromic |
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; for parabolic transformations, the fixed points coincide. Either or both of these fixed points may be the point at infinity.
The fixed points of the transformation
are obtained by solving the fixed point equation . For , this has two roots (proof):
Note that for parabolic transformations, which satisfy , the fixed points coincide.
When one of the fixed points is at infinity, the other is given by
The transformation will be a simple transformation composed of translations, rotations, and dilations: .
If and , then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation: .
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, i.e. a transformation of the form
with fixed points at 0 and ∞. To see this define a map
which sends the points to . Here we assume that both and are 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 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 , one says that is the repulsive fixed point, and is the attractive fixed point. For , the roles are reversed.
Parabolic case:
In the parabolic case there is only one fixed point . The transformation sending point to ∞ is
or the identity if 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 :
Note that is not the characteristic constant of f, which is always 1 for a parabolic transformation. From the above expressions one can calculate:
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:
When expressed in this way, the real number becomes an expansion factor. It indicates how repulsive the fixed point is, and how attractive is. The real number is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about and clockwise about .
Elliptic transformations
If , then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be elliptical. 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 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 , and with the number 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):
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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 is zero (or a multiple of ), 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 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 , with the real number 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):
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It is not surprising that these pictures look very much like the field lines of bar magnets, since 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" is from the Greek: "loxos, slanting + dromos, course". When sailing on a constant bearing - if you maintain a heading of (say) north-east, you will eventually wind up sailing around the north pole in a logarithmic spiral. On the mercator projection 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 (ie, 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 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 : an observer who is both rotating (with constant angular velocity) about some axis and boosting (with constant magnitude acceleration) 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 , and with determined respectively by the magnitude of acceleration and angular velocity.
Here are some pictures illustrating the effect of a loxodromic transformation:
These images show Möbius transformations stereographically projected onto the Riemann sphere. Note in particular that when projected onto a sphere, the special case of a fixed point at infinity looks no different to having the fixed points in an arbitrary location.
Elliptic | Hyperbolic | Loxodromic | |
One fixed point at Infinity | ![]() |
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Fixed points diametrically opposite | ![]() |
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Fixed points in an arbitrary location | ![]() |
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Iterating a transformation
If a transformation has fixed points , 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.
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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 which is sometimes called the characteristic parallelogram of the transformation.
A transform can be specified with two fixed points and the pole .
This allows us to derive a formula for conversion between and given :
Which reduces down to
The last expression coincides with one of the (mutually reciprocal) eigenvalue ratios of the matrix
representing the transform (compare the discussion in the preceding section about the characteristic constant of a transformation). Its characteristic polynomial is equal to
which has roots
Specifying a transformation by three points
Direct approach
Any set of three points
uniquely defines a transformation . To calculate this out, it is handy to make use of a transformation that is able to map three points onto (0,0), (1, 0) and the point at infinity.
One can get rid of the infinities by multiplying out by and as previously noted.
The matrix to map onto then becomes
You can multiply this out, if you want, but if you are writing code then it's easier to use temporary variables for the middle terms.
Explicit determinant formula
The problem of constructing a Möbius transformation mapping a triple to another triple is equivalent to finding the equation of a standard hyperbola
in the (z,w)-plane passing through the points . An explicit equation can be found by evaluating the determinant
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 zi resp. wi are pairwise different thus the Möbius transformation is well-defined.
Remark: A similar determinant (with replaced by ) leads to the equation of a circle through three different (non collinear) points in the plane.
Alternate method using cross-ratios of point quadruples
This construction exploits the fact (mentioned in the first section) that the cross-ratio
is invariant under a Möbius transformation mapping a quadruple to via . If maps a triple of pairwise different zi to another triple , then the Möbius transformation is determined by the equation
or written out in concrete terms:
The last equation can be transformed into
Solving this equation for one obtains the sought transformation.
Relation to the fixed point normal form
Assume that the points are the two (different) fixed points of the Möbius transform i.e. . Write . The last equation
then reads
In the previous section on normal form a Möbius transform with two fixed points was expressed using the characteristic constant k of the transform as
Comparing both expressions one derives the equality
where is different from the fixed points and is the image of z1 under . In particular the cross-ratio does not depend on the choice of the point z (different from the two fixed points) and is equal to the characteristic constant.
See also
- Fuchsian group
- Hyperbolic geometry
- Inversive ring geometry
- Kleinian group
- Lorentz group
- Modular group
- Poincaré half-plane model
- Projective geometry
- Bilinear transform
References
- David Mumford, Caroline Series and David Wright (2002). Indra's Pearls; The Vision of Felix Klein. Cambridge University Press. ISBN 0-521-35253-3. (Aimed at non-mathematicians, provides an excellent exposition of theory and results, richly illustrated with diagrams.)
- Konrad Knopp (1952). Elements of the Theory of Functions. New York: Dover. ISBN 0-486-60154-4. (See Chapters 3-5 of this classic book for a beautiful introduction to the Riemann sphere, stereographic projection, and Möbius transformations.)
- Tristam Needham (1997). Visual Complex Analysis. Oxford: Clarendon Press. ISBN 0-19-853446-9. (See Chapter 3 for a beautifully illustrated introduction to Möbius transformations, including their classification up to conjugacy.)
- Alan F. Beardon (1995). The Geometry of Discrete Groups. New York: Springer-Verlag. ISBN 0-3879-0788-2.
- Svetlana Katok (1992). Fuchsian Groups. Chicago:University of Chicago Press. ISBN 0-226-42583-5. See Chapter 2.
- Gareth A. Jones and David Singerman (1987). Complex Functions: an Algebraic and Geometric Viewpoint. Cambridge: Cambridge Univerisy Press. ISBN 0-521-313666-X. (See Chapter 2 for various isomorphisms, and for the Lorentz group viewed as a Galois group.)
- G. S. Hall (2004). Symmetries and Curvature Structure in General Relativity. Singapore: World Scientific. ISBN 981-02-1051-5. (See Chapter 6 for the classification, up to conjugacy, of the Lie subalgebras of the Lie algebra of the Lorentz group.)
External links
- A java applet allowing you to specify a transformation via its fixed points and so on can be found at [1].
- conformal maps gallery [2]