Geometric Transformations

Geometric transformations of points in the plane are bijective transformations that associate one and only one point in the plane to each point in the same plane.

Geometric transformations can be classified into

Affinity

In the Cartesian plane, the affine transformation is a bijective mapping that associates the point $\mathbf {p}$ to the point $\mathbf{p}'$ through a function of the form

$$\mathbf{p}' = \mathbf{A} \mathbf{p} + \mathbf{t}$$ (1.60)

An affinity enjoys the following properties:

• transforms lines into lines;
• preserves collinearity among points;
• preserves parallelism and incidence among lines;
• generally does not preserve shape or angles.

Being bijective, the affine transformation is invertible, and the inverse is also an affine transformation with parameters

$$\mathbf{p} = \mathbf{A}^{-1} \mathbf{p}' - \mathbf{A}^{-1} \mathbf{t} = \mathbf{A}' \mathbf{p}' + \mathbf{t}'$$ (1.61)

Similarity

A similarity is an affine transformation that preserves the ratio between dimensions and angles.

The form of the equation is identical to that of the affine transformation (1.60) but can only represent changes in scale, reflections, rotations, and translations. Depending on the sign of the determinant of $\mathbf{A}$, similarities are divided into direct (positive determinant) that preserve orientation or inverse (negative determinant) where the orientation is reversed.

Isometry

Isometries are similar transformations that preserve distances:

$$\Vert f(\mathbf{x}) - f(\mathbf{y}) \Vert = \Vert x - y \Vert$$ (1.62)

for every $x,y \in \mathbb{R}^n$.

Isometries between Euclidean spaces are written as in equation (1.60) where, however, $\mathbf{A}$, a necessary and sufficient condition for it to be an isometry, must be an orthogonal matrix.

Since the matrix $\mathbf{A}$ is orthogonal, it must have a determinant of $\pm1$. As with similarities, if $\det\mathbf{A}=1$ the isometry is said to be direct, while if $\det\mathbf{A}=-1$ the isometry is inverse.

Examples of isometries include

• translations;
• rotations;
• central and axial symmetries.