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Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire -dimensional flat of fixed points in a -dimensional space. Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group (of a particular space). But in mechanics and, more generally, in physics, this concept is frequently understood as a coordinate transformation (importantly, a transformation of an orthonormal basis), because for any motion of a body there is an inverse transformation which if applied to the frame of reference results in the body being at the same coordinates. For example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations. ==Related definitions and terminology== The ''rotation group'' is a Lie group of rotations about a fixed point. This (common) fixed point is called the ''center of rotation'' and is usually identified with the origin. The rotation group is a ''point stabilizer'' in a broader group of (orientation-preserving) motions. For a particular rotation: * The ''axis of rotation'' is a line of its fixed points. They exist only in . * The ''plane of rotation'' is a plane that is invariant under the rotation. Unlike the axis, its points are not fixed themselves. The axis (where is present) and the plane of a rotation are orthogonal. A ''representation'' of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory. Rotations of (affine) spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as ''affine rotations'' (although the term is misleading), whereas the latter are ''vector rotations''. See the article below for details. 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「Rotation (mathematics)」の詳細全文を読む スポンサード リンク
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