transformation

A transformation is any bijection of a set to itself, usually with some geometrical underpinning. More specifically, it is a function whose domain and range are sets of points such that the function is bijective so that its inverse function exists.

Its most basic exempla is as linear transformation of vector spaces.

Classifications of Transformations

Transformations with geometric underpinning can be classified by the properties they preserve:

• displacements preserve distances and oriented angles.
• isometries preserve angles and distances
• similarities preserve angles and ratios between distances
• affine transformations preserve parallelism
• projective transformations preserve collinearity
  where each of these classes contains the previous one.