翻訳と辞書
Words near each other
・ Homostegia
・ Homosteus
・ Homostiidae
・ Homostinea
・ Homostolus acer
・ Homosynaptic plasticity
・ Homotaurine
・ Homotetramer
・ Homothallism
・ Homothecium
・ Homotherini
・ Homotherium
・ Homothetic
・ Homothetic center
・ Homothetic preferences
Homothetic transformation
・ Homothetic vector field
・ Homothorax
・ Homotima
・ Homotomidae
・ Homotonal
・ Homotopia
・ Homotopia (festival)
・ Homotopical algebra
・ Homotopotaxy
・ Homotopy
・ Homotopy analysis method
・ Homotopy category
・ Homotopy category of chain complexes
・ Homotopy colimit


Dictionary Lists
翻訳と辞書 辞書検索 [ 開発暫定版 ]
スポンサード リンク

Homothetic transformation : ウィキペディア英語版
Homothetic transformation

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
: M \mapsto S + \lambda \overrightarrow,
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
: \overrightarrow \mapsto \lambda \overrightarrow.
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
翻訳と辞書 : 翻訳のためのインターネットリソース

Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.