Mapping of the plane or of a space to itself.

Direct Isometry, flip, ratio of similarity and projections are examples of geometric transformations.

### Properties

- A continuous transformation is a geometric transformation such that no interruption or break is introduced in the geometric object; that is, all ordered pairs of Neighbouring Points in the images are images of the Neighbouring Points in the initial figure and all ordered pairs of neighbouring points in the initial figure have an ordered pair of neighbouring points as an image. Direct isometries, opposite isometries, similarities and projections are continuous geometric transformations.

- A
**transformation in the Cartesian plane**is a function of \(\mathbb{R} \times \mathbb{R}\) in \(\mathbb{R} \times \mathbb{R}\) whose Cartesian representation corresponds to a transformation of the geometric plane.

- An
**identity transformation**is a transformation that maps every point on the plane or space to itself.

### Examples

This is an illustration of a dilation h in the geometric plane:

This is an illustration of a translation t defined on \(\mathbb{R} \times \mathbb{R}\) by the relation \(\left(x,y\right) \longmapsto \left( x + 5, y + 1\right)\)