# Geometric transformation

In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning.[1] More specifically, it is a function whose domain and range are sets of points — most often both ${\displaystyle \mathbb {R} ^{2}}$ or both ${\displaystyle \mathbb {R} ^{3}}$ — such that the function is injective so that its inverse exists.[2] The study of geometry may be approached via the study of these transformations.[3]

Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations). They can also be classified according to the properties they preserve:

Each of these classes contains the previous one.[9]

• Diffeomorphisms (bidifferentiable transformations) are the transformations that are affine in the first order; they contain the preceding ones as special cases, and can be further refined.[10]
• Conformal transformations preserve angles, and are, in the first order, similarities.
• Equiareal transformations, preserve areas in the planar case or volumes in the three dimensional case.[11] and are, in the first order, affine transformations of determinant 1.
• Homeomorphisms (bicontinuous transformations) preserve the neighborhoods of points.

Transformations of the same type form groups that may be sub-groups of other transformation groups.

## References

1. ^ "The Definitive Glossary of Higher Mathematical Jargon — Transformation". Math Vault. 2019-08-01. Retrieved 2020-05-02.
2. ^ Zalman Usiskin, Anthony L. Peressini, Elena MarchisottoMathematics for High School Teachers: An Advanced Perspective, page 84.
3. ^ Venema, Gerard A. (2006), Foundations of Geometry, Pearson Prentice Hall, p. 285, ISBN 9780131437005
4. ^ "Geometry Translation". www.mathsisfun.com. Retrieved 2020-05-02.
5. ^ "Geometric Transformations — Euclidean Transformations". pages.mtu.edu. Retrieved 2020-05-02.
6. ^ a b Geometric transformation, p. 131, at Google Books
7. ^ "Transformations". www.mathsisfun.com. Retrieved 2020-05-02.
8. ^ "Geometric Transformations — Affine Transformations". pages.mtu.edu. Retrieved 2020-05-02.
9. ^ a b Leland Wilkinson, D. Wills, D. Rope, A. Norton, R. Dubbs – 'Geometric transformation, p. 182, at Google Books
10. ^ stevecheng (2013-03-13). "first fundamental form" (PDF). planetmath.org. Retrieved 2014-10-01.
11. ^ Geometric transformation, p. 191, at Google Books Bruce E. Meserve – Fundamental Concepts of Geometry, page 191.]