Transformation (geometry)


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 structure.

Examples include linear transformations and affine transformations such as 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 "T"v will work as "T"v(p) = p + v.

Reflection

A reflection is a map that transforms an object into its mirror 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 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.

In reflection all the coordinates becomes opposite.

caling

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.

hear

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 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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