In mathematics, a homothety is a transformation of an affine spacedetermined by a point S called its center and a nonzero number λ called its ratio, which sends in other words it fixes S, and sends each M to another point N such that the segmentSN is on the same line as SM, but scaled by a factorλ. In Euclidean geometry homotheties are the similarities that fix a point and either preserve or reverse the direction of all vectors. Together with the translations, all homotheties of an affinespace forma group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a lineparallel to L. In projective geometry, a homothetic transformation is a similarity transformation that leaves the line at infinity pointwise invariant. In Euclidean geometry, a homothety of ratio λ multiplies distances between points by |λ| and all areas by λ2. Here |λ| is the ratio of magnification or dilationfactor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1. The above-mentioned fixed pointS is called homothetic center or center of similarity or center of similitude. The term, coined by French mathematician Michel Chasles, is derived from two Greek elements: the prefix homo-, meaning "similar", and thesis, meaning "position". It describes the relationship between two figures of the same shape and orientation. For example, two Russian dolls looking in the same direction can be considered homothetic.
If the homothetic center S happens to coincide with the originO of the vector space, then every homothety with ratio λ is equivalent to a uniform scaling by the same factor, which sends As a consequence, in the specific case in which S ≡ O, the homothety becomes a linear transformation, which preserves not only the collinearity of points, but also vector addition and scalar multiplication. The image of a point after a homothety with center and ratio λ is given by, b + λ).
