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Transformations of coordinates
Translation of axes.
Homothetic transformation.
Symmetry. Affine transformation.
Consider some transformations tied
with a transition from one coordinate system to another. Here (
õ,
ó
, z ) and (
õ',
ó',
z' ) are coordinates of arbitrary point P in old and new
coordinate systems correspondingly.
Translation of axes.
Let's move the coordinate
system XYZ
in a space so, that axes
OX, OY and
OZ
are parallel to themselves, and the origin of coordinates
Î
moves to the point
Î'
( a, b, ñ
).
We'll receive a new coordinate system X'Y'Z'
. Coordinates
of the point P in the new and old coordinate systems are tied by the
equations:
A
homothetic transformation
with a center O
(
a
,
b
,
c
)
and
a coefficient
k
 0 :
A
symmetry
relatively the plane XOY :
An
affine transformation:
An affine transformation transfers
straight lines to straight lines, intersecting lines to intersecting lines,
crossing straight lines to crossing straight lines, parallel straight lines
to
parallel straight lines.
All
above mentioned
transformations
of coordinates
are
affine transformations.
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