Homothetic transformation

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In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point A called the origin. The number c by which distances are multiplied is called the dilation factor or similitude ratio. Such a transformation is also 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 c², etc.

[edit] Homothetic relation

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

for .

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

[edit] See also

• dilation (mathematics)
• group with operators

[edit] External link

• Homothety