Homothetic transformation

Homothetic transformation

In mathematics, a homothety (or homothecy or dilation) 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 |c|, but also inverts all points with respect to the fixed point.

Choose an origin or center A and a real number c (possibly negative). The homothety h_{A,c} maps any point M to a point M' such that

A-M'=c(A-M)!

(as vectors).

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 |c|, all surface areas by c^2, etc.

Homothetic relation

One application is a homothetic relation R. R, then, is homothetic if

text{for }a in mathbb{R}, a > 0, x R y Rightarrow ax R ay.

An economic application of this is that a utility function which is homogeneous of degree one corresponds to a homothetic preference relation.

In economics

In economics a homothetic function that can be decomposed into two functions, the outer being a function U(x) which is a homogeneous function of degree one in x, and an inner, f(y), which is a monotonically increasing function. U(f(y)) is a homothetic function.

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