The transformation is

This is an example of an

The transformation equations can be expressed through a matrix:

The product of two symmetries with different centers {C,C'} is a

This is the reason why the set of all possible rotations of the plane is NOT a group (see also section-5).

The set of rotations though about the

The important remark is that the two lines {CA,CB} can be rotated about C and give the same rotation as product of reflexions. For this it suffices to maintain the same oriented angle ω/2 between them.

By reversing the orientation of the angle i.e. taking the reflexions in the opposite order we define the inverse rotation.

In the above figure the first rotation R

Thus the result of the two rotations is the rotation about C'' by an angle equal to the sum of the two rotation angles.

This happens though in the

In case ω

The analysis gives a geometric procedure to construct the center (or translation vector) of the composition R.

2) The set of all reflexions is a

3) Every isometry can be expressed as the product of one, two or three reflexions. Hence the set of all reflexions

4) A thorough discussion of the group of isometries can be found in [Yaglom, vol. I].

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