A transformation T (a.k.a., map, function) over a domain D takes the elements X in D to elements Y in T(D), where the range (a.k.a., image) of T is defined as

 Range(T)=T(D)={T(X):X in D}.

Note that when transformations are specified with respect to a coordinate system, it is important to specify whether the rotation takes place on the coordinate system, with space and objects embedded in it being viewed as fixed (a so-called alias transformation), or on the space itself relative to a fixed coordinate system (a so-called alibi transformation).

Examples of transformations are summarized in the following table.

dilationcenter of dilation, scale decrease factor
expansioncenter of expansion, scale increase factor
reflectionmirror line or plane
rotationcenter of rotation, rotation angle
shearinvariant line and shear factor
stretch (1-way)invariant line and scale factor
stretch (2-way)invariant lines and scale factors
translationdisplacement vector

See also

Affine Transformation, Alias Transformation, Alibi Transformation, Dilation, Expansion, Function, Map, Reflection, Rotation, Shear, Stretch, Transform, Translation

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Coxeter, H. S. M. and Greitzer, S. L. "Transformations." Ch. 4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 80-102, 1967.Graustein, W. C. "Transformation." Ch. 7 in Introduction to Higher Geometry. New York: Macmillan, pp. 84-114, 1930.Kapur, J. N. Transformation Geometry. New Delhi, India: Mathematical Sciences Trust Society, 1994-95.

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Cite this as:

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

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