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 $\mathbb {R} ^{2}$ or both $\mathbb {R} ^{3}$ — such that the function is bijective so that its inverse exists.^[1] The study of geometry may be approached by the study of these transformations.^[2]

Not to be confused with Transformation geometry.

preserve distances and oriented angles (e.g., translations);^[3]

Displacements

preserve angles and distances (e.g., Euclidean transformations);^[4]^[5]

Isometries

preserve angles and ratios between distances (e.g., resizing);^[6]

Similarities

preserve parallelism (e.g., scaling, shear);^[5]^[7]

Affine transformations

preserve collinearity;^[8]

Projective transformations

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.^[8]

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

Coordinate transformation

Erlangen program

Symmetry (geometry)

Motion

Reflection

Rigid transformation

Rotation

Topology

Transformation matrix

(2012) [1966], A New Look at Geometry, Dover, ISBN 978-0-486-49851-5

Adler, Irving

; Golding, E. W. (1967) . Geometry Through Transformations (3 vols.): Geometry of Distortion, Geometry of Congruence, and Groups and Coordinates. New York: Herder and Herder.

Dienes, Z. P.

– Transformations and geometries.

David Gans

; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed.). Chelsea. ISBN 0-8284-1087-9.

Hilbert, David

John McCleary (2013) Geometry from a Differentiable Viewpoint, ISBN 978-0-521-11607-7

Cambridge University Press

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.

(1962, 1968, 1973, 2009) . Geometric Transformations (4 vols.). Random House (I, II & III), MAA (I, II, III & IV).