Transformation

Transformation
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 which preserves this structure.
The process of changing a shape into another shape, or a set of numbers into another, different set of numbers. Transformations in geometry include: translation, reflection, rotation, glide, all of which leaves the shape unchanged while a stretch, shear, enlargement and a topological transformation alters the original shape.
A rule which is used to transform objects, number or shapes into other objects, numbers or shapes.
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.
3 types:-
1.Translation
Main article: Translation (geometry)
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 on 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 corner as the reference point or origin{dimensions-(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) be (x,y) and the point P be (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.


2.Rotation
Main article: Rotation (geometry)
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).
3. Scaling
Main article: Scaling (geometry)
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.
Scaling transformations stretch or shrink a given coordinate system and as a result change lengths and angles. So, scaling is not an Euclidean transformation. The meaning of scaling is making the new scale of a coordinate direction p times larger. In other words, the x coordinate is "enlarged" p times. This requirement satisfies x' = p x and therefore x = x'/p.
Scaling can be applied to all axes, each with a different scaling factor. For example, if the x-, y- and z-axis are scaled with scaling factors p, q and r, respectively.