A geometric transformation is any bijection of a set having some geometric structure to itself or another such set. In the book Mathematics for High School Teachers by Zalman Usiskin, Anthony L. Peressini and Elena Marchisotto, the following definition is given: "A geometric transformation is a function whose domain and range are sets of points. Most often the domain and range of a geometric transformation are both R2 or both R3. Often geometric transformations are required to be 1-1 functions, so that they have inverses."
Geometric transformations can be classified by the dimension of the sets they are acting on (thus distinguishing between transformations of the plane and transformations of space, for example). Additionally, they can also be classified according to the properties they preserve:
- displacements preserve distances and oriented angles;
- isometries preserve distances and angles;
- similarities preserve the ratios between distances;
- affine transformations preserve parallelism;
- projective transformations preserve collinearity;
Each of these classes contains the previous one.
- inversions preserve the set of all lines and circles in the planar case (but may interchange lines and circles), and Möbius transformations conserve all planes and spheres in dimension 3.
- 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 as:
- conformal transformations, preserving angles, and are, in the first order, similarities,
- equiareal transformations, preserving areas in the planar case or volumes in the three dimensional case, and are, in the first order, affine transformations of determinant 1.
And finally, including all the previous types:
- Homeomorphisms (bicontinuous transformations), preserve the neighborhoods of points.
Transformations of the same type form groups that may be sub-groups of other transformation groups.
The study of geometry is, in large part, the study of these transformations.
- Zalman Usiskin, Anthony L. Peressini, Elena Marchisotto – Mathematics for High School Teachers: An Advanced Perspective, page 84.
- Marcel Berger, Lester J. Senechal – Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry, page 131.
- Leland Wilkinson, D. Wills, D. Rope, A. Norton, R. Dubbs – The Grammar of Graphics, pages 181 and 182.
- Bruce E. Meserve – Fundamental Concepts of Geometry, page 191.
||This article cites its sources but does not provide page references. (July 2014)|
- John McCleary – Geometry from a Differentiable Viewpoint.
- A.N. Pressley – Elementary Differential Geometry.
- David Hilbert, Stephan Cohn-Vossen – Geometry and the Imagination.
- David Gans – Transformations and geometries.
- Irving Adler – A New Look at Geometry.
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