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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 translations, reflections, rotations, and scaling.
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 constantvector to every point, or as the shifting of the origin of the coordinate system.
In other words, if v is a fixed vector, then the translation Tv will work as $T_v(p) = p + v$.
A graphical representation of vertical translations can be viewed in .
A reflection is a map that transforms an object into its mirror image.
In geometry a "mirror" is a hyperplane of fixed points.
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.
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.
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.
Scaling can also be referred to as "stretching" or "shrinking" a function.
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