Geometric transformation

A geometric transformation is any bijection of a set having some geometric structure to itself or another such set. Specifically, "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." The study of geometry may be approached via the study of these transformations.

Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between planar transformations and those of space, for example). 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.

• Circle inversion 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.

• France identique.gif

  original image

• France par rotation.gif

  isometry

• France par similitude.gif

  similarity

• France affine (1).gif

  affine transformation

• France homographie.gif

  projective transformation

• France circ.gif

  inversion

• 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.
• Conformal transformations, preserving angles, are, in the first order, similarities.
• equiareal transformations, preserve areas in the planar case or volumes in the three dimensional case. and are, in the first order, affine transformations of determinant 1.
• Homeomorphisms (bicontinuous transformations), preserve the neighborhoods of points.

• Fconf.gif

  Conformal transformation

• France aire.gif

  Equiareal transformation

• France diff.gif

  Diffeomorphism

• France homothetie.gif

  Homeomorphism

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

See also

• Erlangen program
• Topology
• Rigid transformation

Further reading

• 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.
• Yaglom, I. M. (1962, 1968, 1973, 2009) . Geometric Transformations (4 vols.). Random House (I, II & III), MAA (I, II, III & IV).
• Dienes, Z. P.; Golding, E. W. (1967) . Geometry Through Transformations (3 vols.): Geometry of Distortion, Geometry of Congruence, and Groups and Coordinates. New York: Herder and Herder.
• Modenov, P. S.; Parkhomenko, A. S. (1965) . Geometric Transformations (2 vols.): Euclidean and Affine Transformations, and Projective Transformations. New York: Academic Press.