The turning of an object or coordinate system by an
angle about a fixed point. A rotation is an orientation-preserving orthogonal transformation. Euler's
rotation theorem states that an arbitrary rotation can be parameterized using
three parameters. These parameters are commonly taken as the Euler
angles. Rotations can be implemented using rotation
Rotation in the plane can be concisely described in the
complex plane using multiplication of complex numbers with unit modulus such that the
resulting angle is given by .
For example, multiplication by represents a rotation to the right by and by represents rotation to the left by . So starting with and rotating left twice gives , which is the same as rotating right twice,
, and . For multiplication by multiples of , the possible positions are
then concisely represented by , ,
, and .
symmetry operation for rotation by is denoted " ." For periodic arrangements of points ("crystals"),
the crystallography restriction gives
the only allowable rotations as 1, 2, 3, 4, and 6.
See also Dilation
Euler's Rotation Theorem
Translation Explore this topic in the MathWorld classroom
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References Addington, S. "The Four Types of Symmetry in the Plane." http://mathforum.org/sum95/suzanne/symsusan.html. Beyer,
W. H. (Ed.). Boca Raton, FL: CRC Press, p. 211,
Standard Mathematical Tables, 28th ed. Coxeter, H. S. M. and Greitzer, S. L. "Rotation."
§4.2 in Washington, DC: Math. Assoc. Amer., pp. 82-85, 1967. Geometry
D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Rotations of
Coordinate Systems." §1.4 in Singapore: World Scientific, pp. 21-35,
Theory of Angular Momentum. Yates, R. C. "Instantaneous Center of Rotation and the
Construction of Some Tangents." Ann Arbor, MI: J. W. Edwards,
pp. 119-122, 1952. A
Handbook on Curves and Their Properties. Referenced on Wolfram|Alpha Rotation
Cite this as:
Weisstein, Eric W. "Rotation." From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/Rotation.html