geometric transformation

{{Short description|Bijection of a set using properties of shapes in space}}

In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning, such as preserving distances, angles, or ratios (scale). More specifically, it is a function whose domain and range are sets of points – most often a real coordinate space, $\mathbb{R}^2$ or $\mathbb{R}^3$ – such that the function is bijective so that its inverse exists.{{cite book | last1=Usiskin | first1=Zalman | author-link1= Zalman Usiskin | first2=Anthony L. | last2=Peressini | first3 = Elena | last3 = Marchisotto |author-link3 = Elena Marchisotto | first4 = Dick | last4 = Stanley

| title=Mathematics for High School Teachers: An Advanced Perspective | publisher = Pearson Education | date=2003 | isbn=0-13-044941-5 | oclc=50004269 | page= 84}} The study of geometry may be approached by the study of these transformations, such as in transformation geometry.{{citation|first=Gerard A.|last=Venema|title=Foundations of Geometry|year=2006|publisher=Pearson Prentice Hall|isbn=9780131437005|page=285}}

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);{{Cite web|title=Geometry Translation|url=https://www.mathsisfun.com/geometry/translation.html|website=www.mathsisfun.com|access-date=2020-05-02}}
Isometries preserve angles and distances (e.g., Euclidean transformations);{{Cite web|title=Geometric Transformations — Euclidean Transformations|url=https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html#euclidean|website=pages.mtu.edu|access-date=2020-05-02}}{{google books |id=pN0iAVavPR8C|page=131}}
Similarities preserve angles and ratios between distances (e.g., resizing);{{Cite web|title=Transformations|url=https://www.mathsisfun.com/geometry/transformations.html|website=www.mathsisfun.com|access-date=2020-05-02}}
Affine transformations preserve parallelism (e.g., scaling, shear);{{Cite web|title=Geometric Transformations — Affine Transformations|url=https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html#affine|website=pages.mtu.edu|access-date=2020-05-02}}
Projective transformations preserve collinearity;Leland Wilkinson, D. Wills, D. Rope, A. Norton, R. Dubbs – {{google books|id=NRyGnjeNKJIC|page=182}}

Each of these classes contains the previous one.

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.

France identique.gif | Original image (based on the map of France)

France par rotation 180deg.gif | Isometry

France par similitude.gif | Similarity

France affine (1).gif | Affine transformation

France homographie.gif | Projective transformation

France circ.gif | 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.{{google books|id=Y6jDAgAAQBAJ|page=191}} Bruce E. Meserve – Fundamental Concepts of Geometry, page 191.] 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.

Fconf.gif | Conformal transformation

France aire.gif | Equiareal transformation

France homothetie.gif | Homeomorphism

France diff.gif | 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 general linear group. The linear transformation A is non-singular. For a row vector v, the matrix product 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 $w^T = (vA)^T = A^T v^T .$ 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,

: $w^T = (vAB)^T = (AB)^Tv^T = B^T A^T v^T .$

Thus for AB the associated left group action is $B^T A^T .$ 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.

geometric transformation

Classifications

Opposite group actions

Active and passive transformations

See also

References

Further reading