möbius transformations - cedar crest college · i möbius transformations are deﬁned over the...

OutlineIntroduction

Cardinal FormsCircle Preserving Property

Relationship to SphereConclusion

Möbius Transformations

James M. Hammer, III

May 1, 2008

James M. Hammer, III Möbius Transformations

OutlineIntroduction

IntroductionInformation on Complex Numbers

Cardinal FormsDefinition of Möbius TransformationsComposition of Möbius Transformations

Circle Preserving PropertyDefinition

Relationship to SphereStereographic ProjectionRigid Motion of the SphereClassification of Möbius TransformationsRelation to the Sphere

ConclusionWrap-up

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Information on Complex Numbers

Complex Numbers

I Möbius transformations are defined over the Complexplane (denoted C.)

I Algebraically, the equation, x 2 = �1 should have asolution.

I Imaginary Unit: i =p�1.

I A complex number, z , can be expressed in the formz = x + iy for real numbers x and y .

I x is called the real part of z (Re z = x .)I y is called the imaginary part of z (Im z = y .)

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Complex Numbers

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Complex Numbers

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Complex Numbers

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Complex Numbers

I x is called the real part of z (Re z = x .)

I y is called the imaginary part of z (Im z = y .)

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Complex Numbers

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Complex Algebra

Let z = x + iy and w = s + it , where x ; y ; s ; and t are realnumbers. Then, the following operations are defined as follows:

I Addition: z + w = (x + s) + i (y + t).I Subtraction: z � w = (x � s) + i (y � t).I Multiplication: zw = (xs � yt) + i (xt + ys).

I Division:zw

=(xs + yt) + i (ys � xt)

s2 + t2 ;w 6= 0.

I Modulus (distance from the origin): jz j = px 2 + y2.

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Complex Algebra

I Addition: z + w = (x + s) + i (y + t).

I Subtraction: z � w = (x � s) + i (y � t).I Multiplication: zw = (xs � yt) + i (xt + ys).

I Division:zw

=(xs + yt) + i (ys � xt)

s2 + t2 ;w 6= 0.

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Complex Algebra

I Addition: z + w = (x + s) + i (y + t).I Subtraction: z � w = (x � s) + i (y � t).

I Multiplication: zw = (xs � yt) + i (xt + ys).

I Division:zw

=(xs + yt) + i (ys � xt)

s2 + t2 ;w 6= 0.

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Complex Algebra

I Division:zw

=(xs + yt) + i (ys � xt)

s2 + t2 ;w 6= 0.

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Complex Algebra

I Division:zw

=(xs + yt) + i (ys � xt)

s2 + t2 ;w 6= 0.

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Complex Algebra

I Division:zw

=(xs + yt) + i (ys � xt)

s2 + t2 ;w 6= 0.

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Theorem 1.1

Theoremz z = jz j2

Proof.

z �z = x 2 + y2 + i (xy � xy) Multiplication= x 2 + y2 Additive Inverse

x 2 + y2�2

Square Root= jz j2 Modulus.

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Theorem 1.1

Theoremz z = jz j2

Proof.

z �z = x 2 + y2 + i (xy � xy) Multiplication

= x 2 + y2 Additive Inverse

x 2 + y2�2

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Theorem 1.1

Theoremz z = jz j2

Proof.

x 2 + y2�2

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Theorem 1.1

Theoremz z = jz j2

Proof.

x 2 + y2�2

Square Root

= jz j2 Modulus.

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Theorem 1.1

Theoremz z = jz j2

Proof.

x 2 + y2�2

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Definition of Möbius TransformationsComposition of Möbius Transformations

Definition of Möbius Transformations

I A Möbius transformation is a rational function of the formH (z ) =

az + bcz + d

for a ; b; c; and d in C, where ad � bc 6= 0.

I The restriction on the determinant (ad � bc) is for the sakeof the derivative (slope of the tangent line), which is

H0 =ad � bc(cz + d)2

I If the determinant would be allowed to be 0, then theoriginal transformation, H, would be a constant function.

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az + bcz + d

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az + bcz + d

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Basic Transformations

There are four basic transformations to be aware of:

I Translation: Tb = z + b; b 2 CI Rotation: R� = e i�z ; 0 � � � 2�.I Dilation: D� = �z ; � 2 R

I Reciprocal: R =1z

Each of which can be expressed as the product of two inversions(For proof, see Theorems 2.2-2.4)

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I Translation: Tb = z + b; b 2 C

I Rotation: R� = e i�z ; 0 � � � 2�.I Dilation: D� = �z ; � 2 R

I Reciprocal: R =1z

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I Translation: Tb = z + b; b 2 CI Rotation: R� = e i�z ; 0 � � � 2�.

I Dilation: D� = �z ; � 2 R

I Reciprocal: R =1z

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I Reciprocal: R =1z

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I Reciprocal: R =1z

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I Reciprocal: R =1z

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Decomposing Möbius Transformations

TheoremEvery linear fractional transformation (i.e. Möbiustransformation) is a composition of translations, dilations,and inversions.

Proof.Simple algebra shows that

H =az + bcz + d

=bc � ad

c2 � 1

Which is clearly the composition of dilations, inversions, andtranslations (for details, see Theorem 2.5.)

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H =az + bcz + d

=bc � ad

c2 � 1

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H =az + bcz + d

=bc � ad

c2 � 1

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Definition

Definition of Circle Preserving

A transformation is called circle preserving if it carriesstraight lines and circles into straight lines and circles.

TheoremMöbius transformations are circle preserving.

Proof.This is quite computation intensive; so, only an outline will bepresented. Approach this as a proof by cases.

I Circles in the pre-image map into either circles or lines inthe plane.

I Lines in the pre-image map into either lines or circles inthe plane.

For details, see Theorem 3.1, Hammer 8.

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Definition

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Definition

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Definition

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Definition

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Definition

For details, see Theorem 3.1, Hammer 8.James M. Hammer, III Möbius Transformations

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Stereographic ProjectionRigid Motion of the SphereClassification of Möbius TransformationsRelation to the Sphere

It is the goal of the rest of this presentation to show thatMöbius transformations can be expressed by:

I Map the complex plane onto the unit sphere�S2� via a

stereographic projection.I A rigid transformation of the sphere.I Map the unit sphere back onto the complex plane via a

stereographic projection.

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I A rigid transformation of the sphere.I Map the unit sphere back onto the complex plane via a

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stereographic projection.I A rigid transformation of the sphere.

I Map the unit sphere back onto the complex plane via astereographic projection.

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stereographic projection.I A rigid transformation of the sphere.I Map the unit sphere back onto the complex plane via a

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The Extended Complex Plane

The extended complex plane corresponds to the complexplane with the concept of a point at infinity added on. Thispoint at infinity will become important for completing thestereographic projection.

The extended complex plane will be denoted as C.

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The Extended Complex Plane

The extended complex plane corresponds to the complexplane with the concept of a point at infinity added on. Thispoint at infinity will become important for completing thestereographic projection.

The extended complex plane will be denoted as C.

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Description

To produce a stereographic projection from the extendedcomplex plane to the sphere, take the following steps.

I Allow C to cut through�S2�.

I Draw a line from a point, z , in C to N , the north pole of�S2�.

I The intersection of the line drawn with the outside of thesphere is the point on

�S2� of z .

I Cover N with the point at infinity.

To use a stereographic projection from the sphere to thecomplex plane, draw the line from N to the point on thesphere. The intersection of the line and the complex plane willbe the image of the point on the sphere on the complex plane.

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Description

�S2� of z .

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Description

�S2� of z .

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Description

�S2� of z .

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Description

�S2� of z .

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Equation of the Stereographic Projection

For those who are algebraically inclined, let (�; �; �) define aCartesian coordinate system.

I From the plane to the sphere: z =� + i�1 + �

I From a point in space, P = (�; �; �): � + i� =2z

1 + z z, and

� =1� z z1 + z z

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1 + z z, and

� =1� z z1 + z z

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1 + z z, and

� =1� z z1 + z z

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Circle Preserving Property

It would be extremely nice if the stereographic projectionrespected circles and lines as much as the Möbiustransformations of the complex plane do.

TheoremThe stereographic projection carries circles and lines of theplane into circles and “lines” on the sphere and conversely.

Proof.Substitute the equations for the stereographic projection

into the general equation of a circle, which isC (z ; z ) = Azz +Bz +Cz +D ; where A and D are realnumbers and C and D are complex conjugates.

For explicit details, please see Theorem 4.1, Hammer 13.

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Rigid Motion of the Sphere

What kind of transformations respect circles and lines on thesphere as much as Möbius transformations of the complexplane?

The answer is the rigid transformations of the sphere.

I A rigid motion is a mapping where any pair of imagepoints have the same distance as the corresponding pair ofinverse image points.

I Rotations in any direction clearly preserves distancebetween two points.

I Translations will also preserve distance in this way.I Reflections of the unit sphere are also rigid motions.

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What kind of transformations respect circles and lines on thesphere as much as Möbius transformations of the complexplane? The answer is the rigid transformations of the sphere.

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I Translations will also preserve distance in this way.

I Reflections of the unit sphere are also rigid motions.

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Fixed Points

How many different fixed points are needed to determine aunique Möbius transformation?

TheoremThree different points in the complex plane are all that isneeded to ensure that the Möbius transformation, H, isuniquely determined.

Proof.It is easier to prove an equivalent statement: A Möbiustransformation with three distinct fixed points is necessarily theidentity. For details, See Theorem 4.2, Hammer 15.

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Fixed Points

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Fixed Points

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Other Circle Preserving Transformations

Are there other circle preserving transformations in theextended complex plane that preserve orientations other thanMöbius transformations?

TheoremEvery injective function mapping the complex plane intoitself that is circle preserving is either a Möbiustransformation or an anti-homography

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Are there other circle preserving transformations in theextended complex plane that preserve orientations other thanMöbius transformations? No!

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Proof.Define g : C ! C as H � f or H � f , such that Im (g (i)) � 0.

ShowI g is additive (i.e. g (z1 + z2) = g (z1) + g (z2)

I g (z ) = z , the identity on RI g (z ) = z , the identity on iRI g (z ) = z

This means that either f (z ) = H�1 (z ), in which case, f is aMöbius transformation or f (z ) = H�1 (z ), meaning that f is ananti-homography (i.e. either a reflection followed by a Möbiustransformation or vice versa.)

For more detail, See Theorem 4.3.

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Proof.Define g : C ! C as H � f or H � f , such that Im (g (i)) � 0.Show

I g is additive (i.e. g (z1 + z2) = g (z1) + g (z2)

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I g (z ) = z , the identity on R

I g (z ) = z , the identity on iRI g (z ) = z

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I g (z ) = z , the identity on RI g (z ) = z , the identity on iR

I g (z ) = zThis means that either f (z ) = H�1 (z ), in which case, f is aMöbius transformation or f (z ) = H�1 (z ), meaning that f is ananti-homography (i.e. either a reflection followed by a Möbiustransformation or vice versa.)

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This means that either f (z ) = H�1 (z ), in which case, f is aMöbius transformation

or f (z ) = H�1 (z ), meaning that f is ananti-homography (i.e. either a reflection followed by a Möbiustransformation or vice versa.)

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CorollaryEvery circle preserving transformation of the completedplane, which preserves the sense of rotation (orientation) atone point, is necessarily a Möbius transformation

Proof.See Corollary 4.4

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CorollaryEvery circle preserving transformation of the completedplane, which preserves the sense of rotation (orientation) atone point, is necessarily a Möbius transformation

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Möbius Transformations and The Sphere

CorollaryA rigid motion of the sphere that does not changeorientation corresponds to a Möbius transformation in theextended complex plane.

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Möbius Transformations and The Sphere

CorollaryA rigid motion of the sphere that does not changeorientation corresponds to a Möbius transformation in theextended complex plane.

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Changing Orientation

CorollaryA rigid transformation of the sphere that changesorientation corresponds to an anti-homography (i.e. areflection followed by a Möbius transformation) in theextended complex plane.

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Changing Orientation

CorollaryA rigid transformation of the sphere that changesorientation corresponds to an anti-homography (i.e. areflection followed by a Möbius transformation) in theextended complex plane.

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Wrap-up

In Summary, The following topics have been fleshed out:I General information on the Complex Plane.

I Definition of Möbius transformations.I What a Möbius transformation is composed of.I The circle preserving property of Möbius transformations

and the rigid motions of the sphere.I The relationship between the Möbius transformations of

the complex plane and the unit sphere.

Thank you for your time and generous attention.Any Questions?

OutlineIntroduction

Wrap-up

In Summary, The following topics have been fleshed out:I General information on the Complex Plane.I Definition of Möbius transformations.

I What a Möbius transformation is composed of.I The circle preserving property of Möbius transformations

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Wrap-up

In Summary, The following topics have been fleshed out:I General information on the Complex Plane.I Definition of Möbius transformations.I What a Möbius transformation is composed of.

I The circle preserving property of Möbius transformationsand the rigid motions of the sphere.

I The relationship between the Möbius transformations ofthe complex plane and the unit sphere.

OutlineIntroduction

Wrap-up

In Summary, The following topics have been fleshed out:I General information on the Complex Plane.I Definition of Möbius transformations.I What a Möbius transformation is composed of.I The circle preserving property of Möbius transformations

and the rigid motions of the sphere.

I The relationship between the Möbius transformations ofthe complex plane and the unit sphere.

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Wrap-up

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Wrap-up

Thank you for your time and generous attention.

Any Questions?

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Wrap-up

möbius transformations - cedar crest college · i möbius transformations are deﬁned over the...

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