Transformation (function)

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A composition of four mappings coded in SVG,
which transforms a rectangular repetitive pattern
into a rhombic pattern. The four transformations are linear.

In mathematics, particularly in semigroup theory, a transformation is a function f (usually with some geometrical underpinning) that maps a set X to itself, i.e. f : XX[1][2][3][4]. In other areas of mathematics, a transformation may simply refer to any function, regardless of domain and codomain[5]. For this wider sense of the term, see function (mathematics).

Examples include linear transformations and affine transformations, rotations, reflections and translations.[6][7] These can be carried out algebraically[8] or in Euclidean space, particularly in R2 (two dimensions) and R3 (three dimensions). They are also operations that can be performed using linear algebra, and described explicitly using matrices.

Translation[edit]

A translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In other words, if v is a fixed vector, then the translation Tv (translation by the vector v) can be described by the rule Tv(p) = p + v.

The two interpretations of a translation lead to two related but different coordinate transformations. To illustrate this, the examples will be restricted to the two dimensional case (although the same argument can also be generalized to any dimension).

Let P(xy) be a point in the plane and apply the translation (hk) to obtain a new point P' with coordinates (XY). It follows from the definition that[9]

X = x + h or x = Xh

and

Y = y + k or y = Yk.

Now consider a point P(xy) in the plane, whose coordinates are determined with respect to a given pair of axes. Suppose the axes are shifted from their original position by (hk), and that the shifted axes are taken as the new reference axes. The point P now has coordinates (XY) with respect to the new reference axes. To obtain the coordinates of P with respect to the new reference axes (from the coordinates of P with respect to the original reference axes), these formulas of translation are used (for more details, see Translation of axes):

X = xh or x = X + h

and

Y = yk or y = Y + k.

Replacing the original coordinates, that is, x and y, with these expressions in an equation of an object in the original coordinates, will produce the transformed equation for the same object—with respect to the new reference axes.[10]

The relationship that holds here is that each of the coordinate transformations is the inverse function of the other.

Reflection[edit]

A reflection is a map that transforms an object into its mirror image with respect to a "mirror", which is a hyperplane of fixed points in the geometry. For example, a reflection of the small Latin letter p with respect to a vertical line would look like a "q". In order to reflect a planar figure, one needs the "mirror" to be a line (axis of reflection or axis of symmetry), while for reflections in the three-dimensional space, one would use a plane (the plane of reflection or symmetry) for a mirror. Reflection may be considered as the limiting case of inversion—as the radius of the reference circle increases without bound.

Reflection is considered to be an opposite motion, since it changes the orientation of the figures it reflects.[7]

Glide reflection[edit]

Example of a glide reflection

A glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line. Reversing the order of combining gives the same result. Depending on context, we may consider a simple reflection (without translation) as a special case where the translation vector is the zero vector.

Rotation[edit]

A rotation is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation.[11] One can rotate the object at any degree measure, but 90° and 180° are two of the most common. Rotation by a positive angle rotates the object counterclockwise, whereas rotation by a negative angle rotates the object clockwise.

Scaling[edit]

Uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety or dilation. The result of uniform scaling is similar (in the geometric sense) to the original.

A more general concept is scaling with a separate scale factor for each axis direction; a special case is directional scaling (in one direction). Shapes not aligned with the axes may be subject to shear (see below) as a side effect: although the angles between lines parallel to the axes are preserved, other angles are not.

Shear[edit]

Shear is a transform that effectively rotates one axis so that the axes are no longer perpendicular. Under shear, a rectangle becomes a parallelogram, and a circle becomes an ellipse. Even if lines parallel to the axes stay the same length, others do not. As a mapping of the plane, it lies in the class of equi-areal mappings.

More generally[edit]

More generally, a transformation in mathematics means a mathematical function (synonyms: "map" or "mapping"). A transformation can be an invertible function from a set X to itself, or from X to another set Y. The choice of the term transformation may simply indicate that the geometric aspects of a function are being considered (for example, with respect to invariants).

A strong nonlinear transformation applied to a plane through the origin
Transformation (before).png Transformation (after).png
Before After

Partial transformations[edit]

While it is common to use the term transformation for any function of a set into itself (especially in terms like "transformation semigroup" and similar), there exists an alternative form of terminological convention in which the term "transformation" is reserved only for bijections. When such a narrow notion of transformation is generalized to partial functions, then a partial transformation is a function f: AB, where both A and B are subsets of some set X.[12]

Algebraic structures[edit]

The set of all transformations on a given base set, together with function composition, forms a regular semigroup.

Combinatorics[edit]

For a finite set of cardinality n, there are nn transformations and (n+1)n partial transformations.[13]

See also[edit]

References[edit]

  1. ^ "The Definitive Glossary of Higher Mathematical Jargon — Transformation". Math Vault. 2019-08-01. Retrieved 2019-12-13.
  2. ^ Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 1. ISBN 978-1-84800-281-4.
  3. ^ Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 2. ISBN 978-0-8247-9662-4.
  4. ^ Wilkinson, Leland & Graham (2005). The Grammar of Graphics (2nd ed.). Springer. p. 29. ISBN 978-0-387-24544-7.CS1 maint: uses authors parameter (link)
  5. ^ P. R. Halmos (1960). Naive Set Theory. Springer Science & Business Media. pp. 30–. ISBN 978-0-387-90092-6.
  6. ^ "Transformations". www.mathsisfun.com. Retrieved 2019-12-13.
  7. ^ a b "Types of Transformations in Math". Basic-mathematics.com. Retrieved 2019-12-13.
  8. ^ "Algebra - Transformations". tutorial.math.lamar.edu. Retrieved 2019-12-13.
  9. ^ Smart, James R. (1998), Modern Geometries (5th ed.), Brooks/Cole, p. 61, ISBN 0-534-35188-3
  10. ^ Wilson, W.A.; Tracey, J.I. (1925), Analytic Geometry (revised ed.), D.C. Heath and Co., pp. 135–136
  11. ^ "Geometry Rotation". www.mathsisfun.com. Retrieved 2019-12-13.
  12. ^ Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.
  13. ^ Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 2. ISBN 978-1-84800-281-4.