NationMaster - Encyclopedia: Möbius transformation

In mathematics, a Möbius transformation, also called a homographic function, is a conformal mapping that is a bijection on the extended complex plane (that is, the complex plane augmented by the point at infinity, written ∞.) It is named in honor of August Ferdinand Möbius. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ... August Ferdinand Möbius (November 17, 1790, Schulpforta, Saxony, Germany - September 26, 1868, Leipzig) was a German mathematician and theoretical astronomer. ...

Möbius transformations are closely related to the group of isometries of the hyperbolic plane and the hyperbolic 3-manifold. Both the Fuchsian groups and the Kleinian groups are certain discrete subgroups of the group of Möbius transformations; these play an important role in the theory of Riemann surfaces and hyperbolic 3-manifolds. A particularly important subgroup is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannnian metric of constant sectional curvature -1. ... In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. ... In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... The Mandelbrot set, named after its discoverer, is a famous example of a fractal. ... A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ... In mathematics, an elliptic curve is a non-singular projective algebraic curve of genus 1 over a field K, together with a distinguished point defined over K. A more accessible (though less accurate) definition is that an elliptic curve is a plane curve defined by an equation of the form... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

Contents

1 Geometry

8 Geometric interpretation of the characteristic constant

9 Iterating a transformation

10 Poles of the transformation

11 Specifying a transformation by three points

Geometry

With two exceptions given below, the general formula for a Möbius transformation is given by

where a, b, c, d are any complex numbers satisfying ad - bc ≠ 0. Since multiplying the numerator and the denominator by the same constant gives the same transformation, one usually works with a representative transformation satisfying ad-bc=1. There are two special cases not covered by the formula above:

the point is mapped to
the point is mapped to

We can have Möbius transformations over the real numbers, as well as for the complex numbers. In both cases, we need to augment the domain with a point at infinity. The text or formatting below is generated by a template which has been proposed for deletion. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ... The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ...

It can be shown that the inverse and composition of two Möbius transformations are similarly defined, and so the Möbius transformations form a group under composition - called the Möbius group. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

The geometric interpretation of the Möbius group is that it is the group of automorphisms of the Riemann sphere. The bilinear transform is a special case of a Möbius transformation. In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... In digital signal processing, the bilinear transform is a conformal mapping, often used to convert a transfer function of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function of a linear, shift-invariant filter in the discrete-time domain...

Any Möbius map can be composed from the elementary transformations - dilations, translations and inversions. If we define a line to be a circle passing through infinity, then it can be shown that a Möbius transformation maps circles to circles, by looking at each elementary transformation. Dilation in physiological context may mean: pupil dilation (mydriasis) dilation of blood vessels (vasodilation) cervical dilation (or dilation of the cervix) in childbirth Dilation and curettage (surgical dilation) In mathematics: Dilation This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Translation is an activity comprising the interpretation of the meaning of a text in one language—the called the source text—and the production of a new, equivalent text in another language—called the target text, or the translation. ... Inversion has different meanings in different fields of knowledge: Something that is inverted or the process by which an inverse is obtained. ...

The Möbius transformation cross-ratio preservation theorem states that the cross-ratio In mathematics, the cross-ratio cr( w, x, y, z ) of an ordered quadruple of complex numbers (which may be real numbers) is Cross-ratios are preserved by linear fractional transformations, i. ...

is invariant under a Möbius transformation that maps from z to w.

Equations

The transformation

can be usefully expressed as a matrix

In this form, the matrix may be multiplied by any scalar λ and still represent the same transformation. This means that a Möbius transformation on C therefore has six real degrees of freedom.The matrix view of a Möbius transformation corresponds to a projectivity on the projective line over C. The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ... In mathematics, the projective line is a fundamental example of an algebraic curve. ...

Composition

Let be two Möbius transformations:

If these transformations are carried out in succession, first then to obtain , the result can be readily seen to be another Möbius transformation which appears as the product of the two matrices

Thus, Möbius transformations form a group. The term group can refer to several concepts: Look up Group in Wiktionary, the free dictionary In music, a group is another term for band or other musical ensemble. ...

Inversion

The inverse of a Möbius transformation can be derived as

and so

Classification

Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and 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, In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ...

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 In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ...

All other non-identity transformations have two fixed points. All non-parabolic (non-identity) transforms are conjugate to

with $λ$ not equal to 0,1 or -1. The square $k = λ 2$ 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 $| λ | = 1$ . Writing $λ = e i α$ , an elliptic transform is conjugate to

with $α$ real. Note that for any , the characteristic constant of is $k n$ . 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 number. In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

Hyperbolic transforms The transform is said to be hyperbolic if

A transform is hyperbolic if and only if λ is real. The word real has many different meanings: Real is something that exists, in the physical sense. ...

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. There are several traditions of navigation. ... Line crossing all meridians at the same angle. ... Line crossing all meridians at the same angle. ... In navigation, a bearing is the angle between the direction to an object and a reference direction. ...

Fixed points

Any Möbius transformation which is not the identity mapping will have two fixed points $γ 1,γ 2$ , invariant under transformation by . 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: this will happen when $c = 0$ . If this is the case, then the transformation will be an affine transformation (some combination of rotation, dilation, and translation). If both fixed points are at infinity, then the transformation is a pure translation with parameters $a = 1$ , $b = Δ$ , $c = 0$ , $d = 1$ i.e. the map . An affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation. ...

Determination of fixed points

The fixed points are obtained by solving the fixed point equation

For , this has two roots

(proof)

and, for $c = 0$ , one of the fixed points is at This mathematics article is devoted entirely to providing proofs and backup support for claims and statements made in the article Möbius transformation. ...

and the other fixed point is the point at infinity. If $a = 1$ , then both fixed points are at infinity, and the mobius transform corrsponds to a pure translation. The above formulas presume the normalization ad-bc=1.

Construction of Möbius transformations with prescribed fixed points

Non-parabolic case:

Let us first discuss the case where the transformation has two different fixed points which are finite.

A Möbius transformation is uniquely defined by its set of fixed points ${γ 1,γ 2}$ together with the characteristic constant $k$ :

The representation is unique once the two fixed points have been suitably labeled; the two matrices and define the same Möbius transformation. The above transform can be written in a normalized form by multiplying each entry with the (non-zero) scalar . The resulting matrix representing the same Möbius transformation has matrix trace equal to k+1 and determinant equal to k. This implies that the characteristic polynomial of the matrix has roots equal to , . Thus the characteristic constant k coincides with one of the two different ratios of eigenvalues of the matrix representing the transformation (note that each ratio is invariant under multiplication of matrices with non-zero scalars). In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ...

Note that every non-parabolic transformation (every transformation having two different fixed points) is conjugate or similar to a linear transformation having one fixed point at infinity and another at 0, i.e the map given by In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ... Several equivalence relations in mathematics are called similarity. ...

with $k = λ 2$ .

Parabolic case:

If there is only one (finite) fixed point $γ$ then the Möbius transformation is of the form

where β is called the translation length.

Normal form

The fixed point expressions are also sometimes written in the so-called normal form. For the non-parabolic case, this form is

where

It is for this reason that the characteristic constant is sometimes called the multiplier. Note that for the derivatives, one has

w'(γ 1) = k

and

w'(γ 2) = 1 / k

Thus, once te multiplier is fixed, the two fixed points can be distinguished. In general, 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.

In the parabolic case, the normal form can be written as

Here, β is called the translation length. When the fixed point γ is at infinity, one then has $- γ 2 β$ being finite and non-zero, corresponding to the translation .

Geometric interpretation of the characteristic constant

The following picture depicts the two fixed points of a Möbius transformation in the non-parabolic case:

Mobius Transformations File links The following pages link to this file: Möbius transformation ...

The characteristic constant can be expressed in terms of its logarithm: The natural logarithm is the logarithm to the base e, where e is approximately equal to 2. ...

When expressed in this way, $ρ$ becomes an expansion factor. It indicates how repulsive the fixed point $γ 1$ is, and how attractive $γ 2$ is. If $ρ = 0$ , 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, the this is equivalent to doing an affine rotation around a point.

Mobius Transformations File links The following pages link to this file: Möbius transformation ... Mobius Transformations File links The following pages link to this file: Möbius transformation ...

$α$ is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about $γ 1$ and clockwise about $γ 2$ . If $α$ is zero (or a multiple of $2π$ ), then the transformation is said to be hyperbolic. These transformations tend to move points along circular paths from one fixed point toward the other.

Mobius Transformations File links The following pages link to this file: Möbius transformation ... Mobius Transformations File links The following pages link to this file: Möbius transformation ...

I'ts not surprising that these look very much like the field lines of bar magnets. Circular arcs are those that subtend a constant angle between two points.

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.

Mobius Transformations File links The following pages link to this file: Möbius transformation ... Mobius Transformations File links The following pages link to this file: Möbius transformation ...

Iterating a transformation

If a transformation has fixed points $γ 1,γ 2$ , and expansion and rotation factors $ρ$ and $α$ , then will have $γ 1' = γ 1,γ 2' = γ 2,ρ' = n ρ,α' = n α$ .

This can be used to continuously iterate a transformation. Iteration is the repetition of a process, typically within a computer program. ...

These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants

Continuous iteration of a mobius transformation. ... Continuous iteration of a mobius transformation. ... Continuous iteration of a mobius transformation. ... Continuous iteration of a mobius transformation. ... Continuous iteration of a mobius transformation. ...

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 $γ 1,γ 2$ and the pole .

This allows us to derive a formula for conversion between $k$ and given $γ 1,γ 2$ :

Which reduces down to

The last expression coincides with one of the (mutually reciprocal) eigenvalue ratios of the matrix In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

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 $z 2 - z 1$ and $Z 2 - Z 1$ as previously noted.

The matrix to map $z 1,2,3$ onto $Z 1,2,3$ 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.

Alternate method using cross-ratios of point quadruples

This construction exploits the fact (mentioned in the first section) that the cross-ratio In mathematics, the cross-ratio cr( w, x, y, z ) of an ordered quadruple of complex numbers (which may be real numbers) is Cross-ratios are preserved by linear fractional transformations, i. ...

is invariant under a Möbius transformation mapping a quadruple $(z 1, z 2, z 3, z 4)$ to $(w 1, w 2, w 3, w 4)$ via . If maps a triple $(z 1, z 2, z 3)$ of pairwise different z_i to another triple $(w 1, w 2, w 3)$ , 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.

References

Not to be confused with:

The Möbius transform should not be confused with Möbius transformations. ... The classical Möbius function is an important multiplicative function in number theory and combinatorics. ...

References

Svetlana Katok, Fuchsian Groups (1992), University of Chicago Press, Chicago ISBN 0-226-42583-5 (See chapter 2).
Alan F. Beardon, The Geometry of Discrete Groups (1995), Graduate Texts in Mathematics, vol. 91, Springer Verlag, New York, ISBN 0-3879-0788-2.
Hershel M. Farkas, Irwin Kra, Theta Constants, Riemann Surfaces and the Modular Group, American Mathematical Society, Providence RI, ISBN 0-8218-1392-7 (See chapter 1. This is a very advanced reference, and only covers Möbius transformations in a very condensed form in the first four pages.)

External link

A java applet allowing you to specify a transformation via its fixed points and so on may be found here (http://www.users.bigpond.com/pmurray/Java/MoebApplet.html).

This page contains material from this article (http://planetmath.org/encyclopedia/MobiusTransformation.html) and this article (http://planetmath.org/?method=src&from=objects&id=4222&op=getobj) at PlanetMath, used under the GFDL by permission. PlanetMath is a free, collaborative, online mathematics encyclopedia. ... GFDL redirects here. ...

Categories: Projective geometry | Conformal geometry

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Equations

Composition

Inversion

Classification

Fixed points

Normal form

Geometric interpretation of the characteristic constant

Iterating a transformation

Poles of the transformation

Specifying a transformation by three points

References

See also

References

External link

Geometry