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 XÎY
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.
Back
