, a homothety
) is a transformation
of space which takes each line into a parallel line (in essence, a similarity
that is similarly
arranged). All dilatations form a group
in either affine
or Euclidean geometry
. Typical examples of dilatations are translations, half-turns, and the identity transformation.
In Euclidean geometry, when not a translation, there is a unique number c by which distances in the dilatation are multiplied. It is called the ratio of magnification or dilation factor or similitude ratio. Such a transformation can be called an enlargement. More generally c can be negative; in that case it not only multiplies all distances by , but also inverts all points with respect to the fixed point.
Choose an origin or center A and a real number (possibly negative). The homothety maps any point M to a point such that
A homothety is an affine transformation (if the fixed point is the origin: a linear transformation) and also a similarity transformation. It multiplies all distances by , all surface areas by , etc.
One application is a homothetic relation R
, then, is homothetic if
An economic application of this is that a utility function which is homogeneous of degree one corresponds to a homothetic preference relation.
In economics a homothetic function that can be decomposed into two functions, the outer being a function U
) which is a homogeneous function
of degree one in x
, and an inner, f
), which is a monotonically increasing function
)) is a homothetic function.