In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a geometric transformation is any
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
of a
set to itself (or to another such set) with some salient
geometrical underpinning. More specifically, it is a
function whose
domain and
range are sets of points — most often both
or both
— such that the function is
bijective so that its
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when ad ...
exists.
The study of
geometry may be approached by the study of these transformations.
Classifications
Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations). They can also be classified according to the properties they preserve:
*
Displacements preserve
distances and
oriented angles (e.g.,
translations);
*
Isometries preserve angles and distances (e.g.,
Euclidean transformations);
*
Similarities preserve angles and ratios between distances (e.g., resizing);
*