- Transformation (function)
In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.
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.
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.
Let us have a clear visualization of this. In day to day life we use computers in all fields. Let us consider this 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.
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 English letter p in respect to a vertical line would look like 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.
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 reflection a special case, where the translation vector is the zero vector.
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 your object at any degree measure, but 90° and 180° are two of the most common. Also, rotations are done counterclockwise (Anticlockwise).
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. 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 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, 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 non-linear transformation applied to a plane through the origin Before After
Wikimedia Foundation. 2010.
Look at other dictionaries:
Transformation matrix — In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping Rn to Rm and x is a column vector with n entries, then for some m×n matrix A, called the transformation matrix of T. There is an… … Wikipedia
Transformation de mellin — En mathématiques, la transformation de Mellin est une transformation intégrale qui peut être considérée comme la version multiplicative de la transformation de Laplace bilatérale. Cette transformation intégrale est fortement reliée à la théorie… … Wikipédia en Français
Transformation/transcription domain-associated protein — Transformation/transcription domain associated protein, also known as TRRAP, is a human gene.This is an adapter protein, which is found in various multiprotein chromatin complexes with histone acetyltransferase activity (HAT), which gives a… … Wikipedia
Transformation (geometry) — In mathematics, a transformation could be any function from a set X to itself. However, often the set X has some additional algebraic or geometric structure and the term transformation refers to a function from X to itself which preserves this… … Wikipedia
Function composition — For function composition in computer science, see function composition (computer science). g ∘ f, the composition of f and g. For example, (g ∘ f)(c) = #. In mathematics, function composition is the application of one function to the resul … Wikipedia
Transformation problem — In 20th century discussions of Karl Marx s economics the transformation problem is the problem of finding a general rule to transform the values of commodities (based on labour according to his labour theory of value) into the competitive prices… … Wikipedia
Transformation semigroup — In algebra, a transformation semigroup (or composition semigroup) is a collection of functions from a set to itself which is closed under function composition. If it includes the identity function, it is a transformation (or composition) monoid.… … Wikipedia
transformation — transformational, adj. /trans feuhr may sheuhn/, n. 1. the act or process of transforming. 2. the state of being transformed. 3. change in form, appearance, nature, or character. 4. Theat. a seemingly miraculous change in the appearance of… … Universalium
Transformation of the United States Army — Army Transformation describes the future concept of the United States Army s plan of modernization. Transformation is a generalized term for the integration of new concepts, organizations, and technology within the armed forces of the United… … Wikipedia
Function model — A function model or functional model is a structured representation of the functions, activities or processes within the modeled system or subject area. [http://www.itl.nist.gov/fipspubs/idef02.doc FIPS Publication 183] released of IDEFØ December … Wikipedia