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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Coordinate rotations and reflections ： ウィキペディア英語版 Coordinate rotations and reflections (詳細はgeometry, two-dimensional coordinate rotations and reflections are two kinds of Euclidean plane isometries which are related to one another. A rotation in the plane can be formed by composing a pair of reflections. First reflect a point ''P'' to its image ''P′'' on the other side of line ''L1''. Then reflect ''P′'' to its image ''P′′'' on the other side of line ''L2''. If lines ''L1'' and ''L2'' make an angle ''θ'' with one another, then points ''P'' and ''P′′'' will make an angle ''2θ'' around point ''O'', the intersection of ''L1'' and ''L2''. I.e. angle ''POP′′'' will measure ''2θ''. A pair of rotations about the same point ''O'' will be equivalent to another rotation about point ''O''. On the other hand, the composition of a reflection and a rotation, or of a rotation and a reflection (composition is not commutative), will be equivalent to a reflection. The statements above can be expressed more mathematically. Let a rotation about the origin ''O'' by an angle ''θ'' be denoted as Rot(''θ''). Let a reflection about a line ''L'' through the origin which makes an angle ''θ'' with the ''x''-axis be denoted as Ref(''θ''). Let these rotations and reflections operate on all points on the plane, and let these points be represented by position vectors. Then a rotation can be represented as a matrix, : \sin \theta & \cos \theta \end, and likewise for a reflection, : \sin 2 \theta & - \cos 2 \theta \end. With these definitions of coordinate rotation and reflection, the following four equations are true: :$\backslash mathrm(\backslash theta)\; \backslash ,\; \backslash mathrm(\backslash phi)\; =\; \backslash mathrm(2(\backslash theta\; -\; \backslash phi)),\; \backslash$ :$\backslash mathrm(\backslash theta)\; \backslash ,\; \backslash mathrm(\backslash phi)\; =\; \backslash mathrm(\backslash theta\; +\; \backslash phi),\; \backslash$ :$\backslash mathrm(\backslash theta)\; \backslash ,\; \backslash mathrm(\backslash phi)\; =\; \backslash mathrm(\backslash phi\; +\; \backslash theta/2),\; \backslash$ :$\backslash mathrm(\backslash phi)\; \backslash ,\; \backslash mathrm(\backslash theta)\; =\; \backslash mathrm(\backslash phi\; -\; \backslash theta/2).\; \backslash$ These equations can be proved through straightforward matrix multiplication and application of trigonometric identities. The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group. The group has an identity: Rot(0). Every rotation Rot(''φ'') has an inverse Rot(−''φ''). Every reflection Ref(''θ'') is its own inverse. Composition has closure and is associative, since matrix multiplication is associative. Notice that both Ref(''θ'') and Rot(''θ'') have been represented with orthogonal matrices. These matrices all have a determinant whose absolute value is unity. Rotation matrices have a determinant of +1, and reflection matrices have a determinant of −1. The set of all orthogonal two-dimensional matrices together with matrix multiplication form the orthogonal group: ''O''(2). ==See also== *Euclidean plane isometry *dihedral group *Cartan–Dieudonné theorem *Rotation group SO(3) – 3 dimensions it:Isometria del piano 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「Coordinate rotations and reflections」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.