_{1}

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In this paper, I have provided a brief introduction on M?bius transformation and explored some basic properties of this kind of transformation. For instance, M?bius transformation is classified according to the invariant points. Moreover, we can see that M?bius transformation is hyperbolic isometries that form a group action PSL (2, R) on the upper half plane model.

Möbius transformations have applications to problems in physics, engineering and mathematics. Furthermore, the conformal mapping is represented as bilinear translation, linear fractional transformation and Mobius transformation.

Möbius transformations are also called homographic transformations, linear fractional transformations, or fractional linear transformations and it is a bijective holomorphic function (conformal map) [

The purpose of this paper is studied the properties of Möbius transformations in detail, and some definitions and theorems are given. The basic properties of these transformations are introduced and classified according to the invariant points. Möbius transformations are formed a group action PSL (2,Â) on the upper half plane model.

A Möbius transformation of the plane is a map f:

which sending each point to a corresponding point, where z is the complex variable and the coefficients a, b, c, d are complex numbers [

Definition (1-1).

The upper half plane model is defined by the set

and the boundary of is defined by

The lines (geodesics) are vertical rays and semicircles orthogonal to ¶H. The angles are Euclidean angles.

Definition (1-2).

A Möbius transformations form a group which is denoted by

Remark (1-3).

Since Möbius transformation takes the form

If the point

1)

2) If c = 0

3) If

Lemma (1-4).

A Möbius transformation consists of four composition functions.

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The four functions are:

1) translation by

2) inversion and reflection with respect to real axis

3) dilation and rotation

4) translation by

Remark (1-5).

We can write Möbius transformations as follows

The inverse Möbius transformation is evaluated from the inverse of the metric

then

Theorem (1-6).

Möbius transformations also preserve cross ratio.

Proof.

Given four distinct points z_{1}, z_{2}, z_{3}, z_{4}, their cross ratio is defined by

The cross ratio is invariant of the group of all Möbius transformation so if we transform the four points z_{i} into

Since translation, rotation and dilation preserve cross ratio and Möbius transformation consists of them so Möbius transformation preserves cross ratio.

Corollary (1-7).

If

and therefore

If any one of z_{i} = 0 for example z_{3} = 0, then

Since the trace of matrix A is tr(A) = a + b and this trace is invariant under conjugation, this is mean,

Every Möbius transformation can be represented by normalized matrix A such that its determinant equal one which mean ad − bc = 1.

Lemma (1-8).

Two Möbius transformations A, B with

Poof.

Let

Since matrix A and B are Möbius transformations, then

Since

If and only if

A Möbius transformation is

Since fixed points (i.e. invariant points) is defined by f(z) = z, then

This mean

For non parabolic transformation, there are two fixed points 0, ¥ but for parabolic transformation, there is only fixed points ¥ because the fixed points are coincide.

There are Parabolic, elliptic, hyperbolic and loxodromic which are distinguished by looking at the trace tr(A) = a + b.

tr^{2}(A) = 4, the parabolic Möbius transformations forms subgroup isomorphic to the group of matrices

which describes a translation

which describes a rotation

which describes a rotation

which describes a dilation (homothety)

The difference between orientation preserving (invariant) and orientation reversing:

1) Rotation and translation are orientation-preserving.

2) Reflection and glide-reflection are orientation-reversing.

3) A composition of orientation-preserving functions is orientation-preserving.

4) A composition of two orientation-reversing functions is orientation-preserving.

5) A composition of one orientation-preserving function and one orientation-reversing function is orientation- reversing.

6) The determinant of the matrix A = 1 (which mentioned above) then the orientation-preserving but if the determinant of the matrix A = ‒1 then the orientation reversing

7)

8) In Orientation preserving all non collinear points A, B, C, the proper angle measures of the angles ABC and A'B'C' have the same sign but in orientation reversing all non collinear points A, B, C, the proper angle measures of the angles ABC and A'B'C' have opposite signs.

9) Orientation preserving isometries takes counterclockwise angles to counterclockwise angles, and it takes clockwise angles to clockwise angles. An orientation reversing isometries takes counterclockwise angles to clockwise angles, and it takes clockwise angles to counterclockwise angles.

Definition (4-1).

The group

by Möbius transformations and also the matrices of this group conjugate to the matrix

that

Definition (4-2).

Let

with metric

Remark (4-3).

From this definition the geodesic between two points (x_{0}, y_{1}) and (x_{0}, y_{2}) on the vertical line with y_{2} > y_{1} has length ln(y_{2}/y_{1}) but if two points do not lie on a vertical line so the geodesics is circular arc with center on the x-axis as seen in

Remark (4-4).

From the definition (1-1) we can define the isometry of hyperbolic plane

Let a mapping f:

Theorem (4-4).

Möbius transformations act isometries in

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Möbius transformations preserve distance. A bijective map that preserves distance is called an isometry because an isometry is a transformation which preserves distance. Thus Möbius transformations are isometries of H.

A second proof.

Since the form of Möbius transformations are

Since

Then

From this equation we remark that Möbius transformations preserve the hyperbolic metric so that Möbius transformations are hyperbolic isometries.

A third proof.

From the definition of hyperbolic distance, we want to show that

Since

The plane as boundary of half space model of hyperbolic space

Let

and from

Since the left hand side is

Since

Then

We get the left hand side equal the right hand side, and then the proof is complete.

Lemma (4-5).

Let Mobius transformations

Proof.

The right hand side

We get the left hand side equal the right hand side, and then the proof is complete.

Remark (4-6).

The group

This action is faithful and

Theorem (4-7).

All orientation-preserving isometries of

Proof.

The isometry group of hyperbolic plane is denoted by

and then, we get:

Let f(z) is an isometry of

Let z_{1}, z_{2} be two points lie in positive imaginary axis. Let the point z not lie in positive imaginary axis and draw two hyperbolic circles with center z_{1} and z_{2} and passing through z, we find these circles intersect in z,

The first case:

If

serving isometries is given by the map

The second case:

If

serving isometries is given by the rotation z → kz and inversion z → –1/z. This means all orientation-preserving isometry of

Theorem (4-8).

Möbius transformations preserve circles and lines (

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Let the transformation w = 1/z is an inversion and every Möbius transformation (

Since w = u + iv and z = x + iy, then

Circle-preserving maps from the plane to itself

Möbius transformation is composition of multiple inversions

From the equation of the circle

But if A = 0, it is a line, if

We can write again the Equation (1-33) w.r.t u, v as follows

If D = 0, it is a line, if

So Möbius transformations preserve circles and lines.

Remark (4-9).

From the last theorem (1-5), we find that the circle goes through the origin may be mapped to the circle or the line.

Theorem (4-10).

Möbius transformations preserve distance.

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From theorem (1-2) Möbius transformations act isometries in

The properties of Möbius transformations are introduced in detail, and some definitions and theorems are given to show that Möbius transformations are one-to-one, onto and conformal mapping. Also, Möbius transformations map circles to circles and also, map the real line to the real line such that the coefficients a, b, c and d are real. Every orientation-preserving isometrics of the hyperbolic plane is Möbius transformations. Every orientation-reversing isometrics of the hyperbolic plane is a composition of Möbius transformations and reflection.

I wish to express my gratitude towards to Professor Dr. William M. Goldman, University of Maryland and Distinguished Scholar-Teacher Professor, Department of Mathematics, for his valuable, guidance, patience and support. I consider myself very fortunate for being able to work with a very considerate and encouraging professor like him.