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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 any ''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 ) or reverse (if ) 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 ''λ''2. The first number is called 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'' ==Homothety and uniform scaling== If the homothetic center ''S'' happens to coincide with the origin ''O'' of the vector space (''S'' ≡ ''O''), then every homothety with scale factor ''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 (straight lines are mapped to straight lines), but also vector addition and scalar multiplication. The image of a point (''x'', ''y'') after a homothety with center (''a'', ''b'') and scale factor ''λ'' is given by (''a'' + ''λ''(''x'' − ''a''), ''b'' + ''λ''(''y'' − ''b'')). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「homothetic transformation」の詳細全文を読む スポンサード リンク
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