A bijective map between two metric spaces that preserves distances, i.e.,


where f is the map and d(a,b) is the distance function. Isometries are sometimes also called congruence transformations. Two figures that can be transformed into each other by an isometry are said to be congruent (Coxeter and Greitzer 1967, p. 80).

An isometry of the plane is a linear transformation which preserves length. Isometries include rotation, translation, reflection, glides, and the identity map. Two geometric figures related by an isometry are said to be geometrically congruent (Coxeter and Greitzer 1967, p. 80).

If a plane isometry has more than one fixed point, it must be either the identity transformation or a reflection. Every isometry of period two (two applications of the transformation preserving lengths in the original configuration) is either a reflection or a half-turn rotation. Every isometry in the plane is the product of at most three reflections (at most two if there is a fixed point). Every finite group of isometries has at least one fixed point.

See also

Congruent, Curve Length, Distance, Euclidean Motion, Glide, Hjelmslev's Theorem, Identity Map, Isometric, Map Fixed Point, Reflection, Rotation, Translation

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Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 80, 1967.Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991.Gray, A. "Isometries and Conformal Maps of Surfaces." §15.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 346-351, 1997.

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Cite this as:

Weisstein, Eric W. "Isometry." From MathWorld--A Wolfram Web Resource.

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