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Rotation (mathematics)
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<p>Rotation in <a href="/facts/Mathematics/pxTouaz4">mathematics</a> is a concept originating in <a href="/facts/Geometry/wTQf5ffQ">geometry</a>. Any rotation is a <a href="/facts/Motion_(geometry)/x5zUzg7l">motion</a> of a certain <a href="/facts/Space_(mathematics)/G6rt4bQ8">space</a> that preserves at least one <a href="/facts/Point_(geometry)/6YPAvX2a">point</a>. It can describe, for example, the motion of a <a href="/facts/Rigid_body/THezBlq2">rigid body</a> around a fixed point. Rotation can have a <a href="/facts/Sign_(mathematics)/z0xollg1">sign</a> (as in the <a href="/facts/Angle/gFUTEQYq">sign of an angle</a>): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: <a href="/facts/Translation_(geometry)/KlpWmnfX">translations</a>, which have no fixed points, and <a href="/facts/Reflection_(mathematics)/5JHWnrql">(hyperplane) reflections</a>, each of them having an entire (<i>n</i> − 1)-dimensional <a href="/facts/Flat_(geometry)/IlMjx7RK">flat</a> of fixed points in a n-<a href="/facts/Dimension_(mathematics)/0ktmUyac">dimensional</a> space. </p><p>Mathematically, a rotation is a <a href="/facts/Map_(mathematics)/FABF1l2f">map</a>. All rotations about a fixed point form a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> under <a href="/facts/Function_composition/r5Pvnmth">composition</a> called the rotation group (of a particular space). But in <a href="/facts/Mechanics/A3yHLQA5">mechanics</a> and, more generally, in <a href="/facts/Physics/a7aH2y08">physics</a>, this concept is frequently understood as a <a href="/facts/Coordinate_transformation/Mtd0q4SP">coordinate transformation</a> (importantly, a transformation of an <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a>), because for any motion of a body there is an inverse transformation which if applied to the <a href="/facts/Frame_of_reference/15yYfB9M">frame of reference</a> results in the body being at the same coordinates. For example, in two dimensions rotating a body <a href="/facts/Clockwise/3Npe4Swe">clockwise</a> 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 <a href="/facts/Active_and_passive_transformation/pxF9AaJ3">active and passive transformations</a>. </p>