- Transformation (geometry)
In

mathematics , a**transformation**could be anyfunction from a set "X" to itself. However, often the set "X" has some additionalalgebraic orgeometric structure and the term "transformation" refers to a function from "X" to itself which preserves this structure.Examples include

linear transformation s andaffine transformation s such asrotation s, reflections and translations. These can be carried out inEuclidean space , particularly in dimensions 2 and 3. They are also operations that can be performed usinglinear algebra , and described explicitly using matrices.**Translation**A

**translation**, or**translation operator**, is anaffine transformation ofEuclidean 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 thecoordinate system . In other words, if**v**is a fixed vector, then the translation "T"_{v}will work as "T"_{v}(**p**) =**p**+**v**.**Reflection**A

**reflection**is a map that transforms an object into itsmirror image . 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"), while for reflections in the three-dimensional space one would use a plane for a mirror. Reflection sometimes is considered as a special case of inversion with infinite radius of the reference circle.Or in easier terms a translation is on a coordinate grid you slide the figure over onto another coordinate plane.**Glide reflection**A

**glide reflection**is a type ofisometry of theEuclidean 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.In reflection all the coordinates becomes opposite.

**caling****Uniform scaling**is alinear transformation that enlarges or diminishes objects; thescale factor is the same in all directions; it is also called ahomothety . 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.**hear****Shear**is a transform that effectively rotates one axis so that the axes are no longer perpendicular. Under shear, arectangle becomes aparallelogram , and acircle becomes anellipse . 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 is one facet of the mathematical function; the term "mapping" is also used in ways that are quite close synonyms. A transformation can be an invertible function from a set "X" to itself, or from "X" to another set "Y". In a sense the term "transformation" only flags that a function's more geometric aspects are being considered (for example, with attention paid to invariants).**ee also***

Coordinate transformation

*Data transformation (statistics)

*Infinitesimal transformation

*Linear transformation

*Transformation geometry

*Transformation group

*Transformation matrix

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