Also known as:
dilation.
A mapping
T on a
euclidian plane is a
dilatation iff it is a
similarity transformation that maps
every line to another line parallel to the original.
A
similarity transformation maps any shape to another that is similar to the original.
The set of all dilatations form a
group with composition of dilatations as the group operator.
The composition of dilatations is
associative but not
commutative, hence not
abelian.
If a dilatation has a
fixed point C, the dilatation is known as a
central dilatation and the point is known as the
dilatation center. The scale factor
r of a dilatation measures the change in perimeter length, and is known as the
dilatation ratio.
If a dilatation has a dilatation ratio of 1, then it is known as a
translation.
While the set of all translations form a group, the set of all central dilatations are not a group.
Equation for
translation:
τα(
x) =
x +
α
Equation for
central dilatation: δ
C,r(
x) =
r(
x - C) +
C =
rx + (1 -
r)
C
Source:
"Vectors and Transformations in Plane Geometry" by Philippe Tondeur, Publish or Perish, Inc. 1993