**
Transformations of coordinates
**

*
Translation of axes.
*
*
Turning around origin of coordinates.
*

*
Central symmetry. Homothetic
transformation.
*
*
Affine
transformation.
*

Consider some transformations tied
with a transition from one coordinate system to another. Here (
*
õ
*
,
*
ó
*
**
**
)
è
(

*õ '*,

*ó*

**) are coordinates of arbitrary point**

*'**P*in old and new coordinate systems correspondingly.

**
Translation of axes.
**
Let's move the coordinate
system

*XOY*in a plane so, that the axes

*OX*and

*OY*are parallel to themselves, and the origin of coordinates

*Î*moves to the point

*Î '*(

*a*,

*b*). We'll receive the new coordinate system

*X'O'Y'*( Fig.1 ):

Coordinates of the point
*
P
*
in the new and old coordinate systems are
tied by the equations:

**
Turning around origin of
coordinates.
**
Let's
turn the coordinate system

*X*

*Î Y*in a plane by an angle

**( Fig.2 ).**

Now coordinates of the point
*
P
*
in the new and old coordinate systems are tied by
the
equations:

In the particular case:
=
we'll receive
**
a
central symmetry
**
relatively the origin
of
coordinates

*O*:

**
A
homothetic transformation
**

*with a center**O*(

*a*,

*b*)

*and a coefficient**k*0 :

**
An
affine transformation:
**

An affine transformation transfers straight lines to straight lines, intersecting lines to intersecting lines, parallel straight lines to parallel straight lines. All above mentioned transformations of coordinates are affine transformations.