TOPICS

# Rotation

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 matrices.

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 .

The rotation 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.

Dilation, Euclidean Group, Euler Angles, Euler Parameters, Euler's Rotation Theorem, Expansion, Half-Turn, Improper Rotation, Infinitesimal Rotation, Inversion Operation, Mirror Plane, Orientation-Preserving, Orthogonal Transformation, Reflection, Rotation Formula, Rotation Group, Rotation Matrix, Rotation Operator, Shift, Spiral Similarity, Translation Explore this topic in the MathWorld classroom

## Explore with Wolfram|Alpha

More things to try:

## References

Addington, S. "The Four Types of Symmetry in the Plane." http://mathforum.org/sum95/suzanne/symsusan.html.Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 211, 1987.Coxeter, H. S. M. and Greitzer, S. L. "Rotation." §4.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 82-85, 1967.Varshalovich, D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Rotations of Coordinate Systems." §1.4 in Quantum Theory of Angular Momentum. Singapore: World Scientific, pp. 21-35, 1988.Yates, R. C. "Instantaneous Center of Rotation and the Construction of Some Tangents." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 119-122, 1952.

Rotation

## Cite this as:

Weisstein, Eric W. "Rotation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Rotation.html