What is a transformation? And what are the four types of transformations?

In plane geometry a transformation is a process of changing the position of every point on a plane in a way that satisfies certain rules.

Transformations are usually symmetric in a sense that, if there is a transformation that transforms point #A# to point #B#, there is another transformation of the same type that transforms #B# to #A#.

For instance, translation (shift) by #5# of all points on a plane in certain direction has a symmetrical counterpart - shift by #5# in the opposite direction.
Reflection relative to a straight line is a counterpart to itself since the same reflection repeated again transforms a point back to its original position.

Transformations are usually transitive in a sense that, if one particular type of transformation of some type transforms point #A# to point #B# and another one of the same type transforms point #B# to point #C#, there is a transformation of the same type that combines the first two transformations and transforms point #A# into point #C#.

For instance, rotation of all points on a plane around some fixed point counterclockwise by #90^o# and another one, rotating around the same point by #30^o# clockwise can be combined into one rotation - rotation by #60^o# counterclockwise around the same point.

In every type of transformation we have the one that does nothing. For example, scaling by a factor of #1#, translation (shift) by a distance #0# or rotation- by an angle #0^o#. This property of transformations is called _reflexivity.