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 called the origin. The number 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 , but also inverses all points with respect to the fixed point.

Choose an origin or center and a real number (possibly negative). The homothety maps any point to a point such that (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 , all surfaces by , etc.

See also: dilation (mathematics).

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