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:

*τ*_{α}(

*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