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. 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

Classifications

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:

• Displacements preserve distances and oriented angles (e.g., translations);^[3]
• Euclidean transformations);^[4][5]
• Similarities preserve angles and ratios between distances (e.g., resizing);^[6]
• parallelism (e.g., scaling, shear);^[5][7]
• collinearity;^[8]

Each of these classes contains the previous one.[8]

• circle inversion
  ) preserve the set of all lines and circles, but may interchange lines and circles.

• 
  Original image (based on the map of France)
• 
  Isometry
• 
  Similarity
• 
  Affine transformation
• 
  Projective transformation
• 
  Inversion

• 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.^[9] and are, in the first order, affine transformations of determinant 1.
• Homeomorphisms (bicontinuous transformations) preserve the neighborhoods of points.
• 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 transformation
• 
  Equiareal transformation
• 
  Homeomorphism
• 
  Diffeomorphism

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

Opposite group actions

Many geometric transformations are expressed with linear algebra. The bijective linear transformations are elements of a

vA gives another row vector w = vA.

The transpose of a row vector v is a column vector v^T, and the transpose of the above equality is ${\displaystyle w^{T}=(vA)^{T}=A^{T}v^{T}.}$ Here A^T provides a left action on column vectors.

In transformation geometry there are compositions AB. Starting with a row vector v, the right action of the composed transformation is w = vAB. After transposition,

${\displaystyle w^{T}=(vAB)^{T}=(AB)^{T}v^{T}=B^{T}A^{T}v^{T}.}$

Thus for AB the associated left group action is ${\displaystyle B^{T}A^{T}.}$ In the study of

are the only groups for which these opposites are equal.

See also

• Coordinate transformation
• Erlangen program
• Symmetry (geometry)
• Motion
• Reflection
• Rigid transformation
• Rotation
• Topology
• Transformation matrix

References

Further reading

Wikimedia Commons has media related to Transformations (geometry).

• ISBN 978-0-486-49851-5
• 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.
• David Gans – Transformations and geometries.
• ISBN 0-8284-1087-9
  .
• John McCleary (2013) Geometry from a Differentiable Viewpoint,
  ISBN 978-0-521-11607-7
• Modenov, P. S.; Parkhomenko, A. S. (1965) . Geometric Transformations (2 vols.): Euclidean and Affine Transformations, and Projective Transformations. New York: Academic Press.
• A. N. Pressley – Elementary Differential Geometry.
• Yaglom, I. M. (1962, 1968, 1973, 2009) . Geometric Transformations (4 vols.). Random House (I, II & III), MAA (I, II, III & IV).