Inmathematics, ageometric transformation is anybijection of aset to itself (or to another such set) with some salientgeometrical underpinning, such as preserving distances,angles, or ratios (scale). More specifically, it is afunction whosedomain andrange are sets ofpoints – most often areal coordinate space, or – such that the function is bijective so that itsinverse exists.[1] The study of geometry may be approached by the study of these transformations, such as intransformation geometry.[2]
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 ascircle 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 ofdeterminant 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.
Thetranspose of a row vectorv is a column vectorvT, and the transpose of the above equality is HereAT provides a left action on column vectors.
In transformation geometry there arecompositionsAB. Starting with a row vectorv, the right action of the composed transformation isw =vAB. After transposition,
Thus forAB the associated leftgroup action is In the study ofopposite groups, the distinction is made between opposite group actions becausecommutative groups are the only groups for which these opposites are equal.
In the active transformation (left), a pointP is transformed to pointP′ by rotating clockwise byangleθ about theorigin of a fixed coordinate system. In the passive transformation (right), pointP stays fixed, while the coordinate system rotates counterclockwise by an angleθ about its origin. The coordinates ofP′ after the active transformation relative to the original coordinate system are the same as the coordinates ofP relative to the rotated coordinate system.
Geometric transformations can be distinguished into two types:active or alibi transformations which change the physical position of a set ofpoints relative to a fixedframe of reference orcoordinate system (alibi meaning "being somewhere else at the same time"); and passive or alias transformations which leave points fixed but change the frame of reference or coordinate system relative to which they are described (alias meaning "going under a different name").[10][11] Bytransformation,mathematicians usually refer to active transformations, whilephysicists andengineers could mean either.[citation needed]
For instance, active transformations are useful to describe successive positions of arigid body. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of thetibia relative to thefemur, that is, its motion relative to a (local) coordinate system which moves together with the femur, rather than a (global) coordinate system which is fixed to the floor.[11]
Dienes, Z. P.; Golding, E. W. (1967) .Geometry Through Transformations (3 vols.):Geometry of Distortion,Geometry of Congruence, andGroups and Coordinates. New York: Herder and Herder.
Modenov, P. S.; Parkhomenko, A. S. (1965) .Geometric Transformations (2 vols.):Euclidean and Affine Transformations, andProjective 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).
.gif)
