Transformation (function)

For other uses, see Transformation.
A composition of four mappings coded in SVG,
which transforms a rectangular repetitive pattern
into a rhombic pattern.

In mathematics, particularly in semigroup theory, a transformation is any function f mapping a set X to itself, i.e. f:X→X.[1][2][3] In other areas of mathematics, a transformation may simply be any function, regardless of domain and codomain.[4] This wider sense shall not be considered in this article; refer instead to the article on function for that sense.

Examples include linear transformations and affine transformations, rotations, reflections and translations. These can be carried out in Euclidean space, particularly in dimensions 2 and 3. They are also operations that can be performed using linear algebra, and described explicitly using matrices.

Translation

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 will work as Tv(p) = p + v.

For the purpose of visualization, consider a browser window. This window, if maximized to full dimensions of the screen, is the reference plane. Imagine one of the corners as the reference point or origin (0, 0).

Consider a point P(x, y) in the corresponding plane. Now the axes are shifted from the original axes to a distance (h, k) and this is the corresponding reference axes. Now the origin (previous axes) is (x, y) and the point P is (X, Y) and therefore the equations are:

X = x − h or x = X + h or h = x − X and Y = y − k or y = Y + k or k = y − Y.

Replacing these values or using these equations in the respective equation we obtain the transformed equation or new reference axes, old reference axes, point lying on the plane.

Reflection

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.

Glide reflection

Example of a glide reflection
Main article: 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

Main article: Rotation (geometry)

A rotation is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation. You 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

Main article: Scaling (geometry)

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.

More general 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

Main article: Shear mapping

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

More generally, a transformation in mathematics means a mathematical function (synonyms: map and 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 flag that a function's more geometric aspects are being considered (for example, with attention paid to invariants).

A strong nonlinear transformation applied to a plane through the origin
Before After

Partial transformations

The notion of transformation generalized to partial functions. A partial transformation is a function f: A → B, where both A and B are subsets of some set X.[5]

Algebraic structures

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

Combinatorics

For a finite set of cardinality n, there are nn transformations and (n+1)n partial transformations.[6]

See also

References

  1. ↑ Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 1. ISBN 978-1-84800-281-4.
  2. ↑ Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 2. ISBN 978-0-8247-9662-4.
  3. ↑ Wilkinson, Leland & Graham (2005). The Grammar of Graphics (2nd ed.). Springer. p. 29. ISBN 978-0-387-24544-7.
  4. ↑ P. R. Halmos (1960). Naive Set Theory. Springer Science & Business Media. pp. 30–. ISBN 978-0-387-90092-6.
  5. ↑ 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.
  6. ↑ Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 2. ISBN 978-1-84800-281-4.
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