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 or both — 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]
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:
Each of these classes contains the previous one.[8]
Möbius transformations using complex coordinates on the plane (as well as
circle inversion) preserve the set of all lines and circles, but may interchange lines and circles.
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.
The
transpose of a row vector v is a column vector vT, and the transpose of the above equality is Here AT provides a left action on column vectors.
In transformation geometry there are
compositionsAB. Starting with a row vector v, the right action of the composed transformation is w = vAB. After transposition,
Thus for AB the associated left
group action is In the study of
opposite groups, the distinction is made between opposite group actions because
commutative groups are the only groups for which these opposites are equal.
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.
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).