Transformation (mathematics)
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In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, 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 explicitly using matrix theory.
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[edit] Rotation
An arbitrary rotation with a given point fixed is given by the formula for a rotation about an axis through the origin; just add an arbitrary translation to get an arbitrary move of a rigid object. It can be decomposed into rotations about three fixed axes through that point, in terms of flight dynamics pitch, roll, and yaw. See also degrees of freedom (engineering).
[edit] 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.
[edit] 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.
[edit] 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.
[edit] Scaling
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.
[edit] Shear
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.
[edit] 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 is, most often, an invertible function from a set X to itself; but this is not always assumed. 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).
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