Homothety

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Homothety

Dbpedia

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In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined 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 segment SN is on the same line as SM, but scaled by a factor λ. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a 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 line parallel to L. In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) 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 dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1. The above-mentioned fixed point S 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- (όμο), meaning "similar", and thesis (Θέσις), 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.

English Wikipedia http://en.wikipedia.org/wiki/Homothety

Spanish wikipedia http://es.wikipedia.org/wiki/Homotecia

Wiki page external link http://www.cut-the-knot.org/Curriculum/Geometry/Homothety.shtml