homothety

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Homothetic transformation

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

homothety (plural homotheties)

1. (mathematics, geometry) An isotropic scaling transformation of an affine space with a single fixed point.
   • 1927, Henry George Forder, The Foundations of Euclidean Geometry‎, 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_P,λ the homothety with center P and similitude ratio λ.

2. (commutative algebra, Bourbakist) A homomorphism from a module ${\displaystyle M}$ over a ring ${\displaystyle A}$ to itself of the form ${\displaystyle \nu :x\mapsto ax}$ for some fixed ${\displaystyle a\in A}$ (especially when ${\displaystyle M=A}$; ${\displaystyle a}$ is said to be the ratio of the homothety, by analogy with the geometric case).

• (isotropic scaling transformation with a fixed point): homothecy, homogeneous dilation, homothetic transformation

• homothetic