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# Transformation

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

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.

 Transformation Characterization dilation center of dilation, scale decrease factor expansion center of expansion, scale increase factor reflection mirror line or plane rotation center of rotation, rotation angle shear invariant line and shear factor stretch (1-way) invariant line and scale factor stretch (2-way) invariant lines and scale factors translation displacement vector

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

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## References

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.

Transformation

## Cite this as:

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