# homothety

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## English

[edit]### Etymology

[edit]From Ancient Greek *ὁμο-* (homo-, “same”) + *θέσις* (thésis, “setting, placement, arrangement”).

### Noun

[edit]**homothety** (*plural* **homotheties**)

- (mathematics, geometry) An isotropic scaling transformation of an affine space with a single fixed point.
**1927**, Henry George Forder, The Foundations of Euclidean Geometry^{[1]}, page 178:- The product of two
**homotheties**with the same centre is a**homothety**with that centre.

**1972**, Clayton W. Dodge, Euclidean Geometry and Transformations, published 2004, page 106:- One cannot obtain all similarity mappings from products of
**homotheties**alone, but they are necessary and basic to similarities.

**2011**, Agustí Reventós Tarrida, Affine Maps, Euclidean Motions and Quadrics, Springer Undergraduate Mathematics Series, page 69:- Since
**homotheties**are determined by the fixed point, called the center of the**homothety**, and by the similitude ratio*λ*, we shall denote by*h*the_{P,λ}**homothety**with center*P*and similitude ratio*λ*.

- (commutative algebra, Bourbakist) A homomorphism from a module over a ring to itself of the form for some fixed (especially when ; is said to be the
**ratio**of the homothety, by analogy with the geometric case).

#### Synonyms

[edit]- (isotropic scaling transformation with a fixed point): homothecy, homogeneous dilation, homothetic transformation