A geometric transformation is any bijection of a portion of a geometric set to itself. 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 their dimension (thus distinguishing between transformations in the plane and transformations in space, for example). Additionally, they can also be classified according to the properties they conserve:
- displacements conserve distances and oriented angles;
- isometries conserve distances and angles;
- similarities conserve the ratios between distances;
- affine transformations conserve parallelism;
- projective transformations conserve collinearity;
Each of these classes contains the previous one.
- inversions conserve all the lines and circles in the planar case, and Möbius transformations retain all planes and spheres in dimension 3.
- bidifferentiable transformations or diffeomorphisms are the transformations that are affine in the first order; they contain the preceding ones as special cases, and also:
- conformal transformations conserve angles and are, in the first order, similarities
- equivalent or equiareal transformations conserve the areas in the planar case or the volumes in the 3D case, and are, in the first order, affine transformations of determinant 1
And finally, including all the previous ones:
- bicontinuous transformations or homeomorphisms, conserve the points' neighborhoods
They then create groups and sub-groups of transformations.
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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