This formula (1) shows a change of a single point (, , ). If the point has nothing to do with other points,

The number of elements is nine. But it is actually eight because we can devide all elements by one of nonzero elements, which becomes 1. Even so, we can not see what the matrix of tells us. Moreover, we do not know how to find the value of

rule

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Let us define the congruence.

If two figures

We say figures

Look at Fig. 1.

They are examples of figures moved by projective transformation. Let us determine elements of transformation matrix.

The elements of an ordinary 3x3 matrix are definitely determined if three points not on a straight line are given before and after transformation. But in homogeneous coordinates four points are to be given before and after transformation because point (, , ) and point (k, k, k) are the identical. (

Suppose points and , i = 1, 2, 3, 4, are given before and after transformation respectively, elements of transformation matrix in (1) will be fixed as;

We previously constructed Klein disk from imaginary sphere. It was based on the fact that

It looks so tight. But projective transformation on condition (2) forms group. Becase whatever elements are, they forms a regular matrix. It guarantees that cross ratio is invariable even after conditioned (2).

Look at Fig. 2.

: It is an example of transformation matrix with condition (2). We can make many matrixes like this, but it is hard for us to find any common peculiarity of them.

: However, the condition (2) results that inside points of Klein disk never go outside, and outside points never get into the inside. In the figure points are given randomly and each point is moved by matrix that is chosen randomly. Lines are drawn between two points, one is given and another is after transformation. The lines are not loci but show connection of each transformation. Both end points of every red segment are inside of Klein disk, and those of every green segment are outide of the disk. A point on the circumference can move only on the circumference.

Let us leave the above as it is, and think about upper half plane . Because projective transformation has so many elements that we feel difficulties, and the action of projective transformation should appear on a model we are familiar with.

A figure on upper half plane is moved by

Mobius transformation forms group, too. Because Mobius transformation can be expressed with a matrix;

Symbol "" indicates formula (3). When we make Mobius transformation

The elements

We have not paid much attention to that the movement of a point is tied to such transformation. It was enough that we have known a hyperbolic straight line appears as a semicircle and how to measure its length. But now, we are going to see the root of the movement.

Look at Fig. 3.

Verious points on an ordinary plane are moved by Mobius transformation (3) with condition (4). A Mobius transformation is randomly chosen for each point. A point above X-axis (Y > 0) moves only above X-axis, and a point only below X-axis (Y < 0) moves only below X-axis. A point on X-axis moves only on X-axis. It never get out of X-axis. These movements correspond to Fig. 2, isn't it?

The expression, we often see, "upper half plane is transferred to its own" means that any point on upper half plane can move only on the upper half plane.

We can draw the same figure by using projective transformation (1) if we replace condition (2) by

Look at Fig. 4.

It is easy to connect upper half plane to Klein disk. We have drawn it before, but here we use a circle of inversion for drawing.

The small sky blue disk

Thus Klein disk becomes the half plane. Accordingly, projective transformation (1) with condition (2) should be replaced by Mobius transformation (1) with condition (2).

Then, it is enough for us to see

In other words, what we have seen on the upper half plane was what projective transformation (1) with condition (2) performed. It's OK, but it is not easy to formulate the relation between and

In Mobius transformation it is easy to make a formula with condition (4).

First, let

And we divide Mobius transformation into real and imaginary conponents:

Both

Here below let ad - bc = 1 .

If

If a = 1 and c = 0, then x' = x + b, y' = y, and shifts only right or left.

If

Formula (5'') is a formula of inversion except that

Let us see how Mobius transformation works.

There are two methods.

Fixed-abcd method: fix

and

Fixed-Z method: fix

is moved by either method. (Naming of methods is just for convenience of explanation.)

We adopt Fixed-abcd method, and apply Mobius transformation with condition (4) to an ordinary plane. We regard the lower half plane is a mirror copy of the upper half. We do not care about the upper half plane at this moment. We use different color on the upper and lower just for illustration.

Look at Fig. 5.

We move point

: None of

: Only

:

: Only

shall form a circle whenever

Look at Fig. 6.

We move point

: This figure is made with the matrix that is a product of three matrixes;

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is a red vertical straight line now. A matrix can be decomposed into some products of matrixes. One of decomposed matrixes we used was . The matrix (Mobius transformation) changes a circle centered at the origin with radias 1 into Y-axis that is a straight line. The center of circle

:

Suppose the plane above X-axis is the upper half plane and circle

: Ciercle

Next we adopt Fixed-Z method. This time we will see how Mobius transformation works on upper half plane .

Let us make a special agreement to apply Mobius transformation onto upper half plane . Usually we should not divide any number by zero. But here we do:

If you do not agree with this (6), think it as a limit . We may take the symbol "" for an infinity or a point at infinity, either will do. We do not care whether it positive or negative, or we choose either of them for convenience.

A hyperbolic plane model is open. How does it fit for the above agreement? Well, let us see it later and pass over it now.

Let fixed

Look at Fig. 7.

: Red circles are . The blue line is locus of its center. The arrow indicates the direction of locus when

The size of circle looks to chage but it is not true. All circles are the same in size. The plane we are using now is not an ordinary plane but the upper half plane that is different from Fig. 6. Though it is a matter of course, one that lost its size at the origin grows again after passing. How should we understand it? It is mysterious for us even it is mathematically correct.

have different

: The locus is a straight line. The arrow indicates the direction that

If we fix

Look at Fig. 8.

Hyperbolic segments are drawn on the upper half plane by using Fixed-abcd method. Black spots are corresponded end points.

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Sorry, under construction.

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