Geometric Transformations

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

Geometric transformations can be classified as follows:

Affine Transformations

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

$$\mathbf{p}' = \mathbf{A} \mathbf{p} + \mathbf{t}$$ (1.96)

An affine transformation has the following properties:

• maps lines to lines;
• preserves collinearity among points;
• preserves parallelism and incidence between lines;
• in general, does not preserve shape or angles.

Since it is bijective, an affine transformation is invertible, and its inverse is also an affine transformation with parameters

$$\mathbf{p} = \mathbf{A}^{-1} \mathbf{p}' - \mathbf{A}^{-1} \mathbf{t} = \mathbf{A}' \mathbf{p}' + \mathbf{t}'$$ (1.97)

Similarity Transformations

A similarity transformation is an affine transformation that preserves ratios of lengths and angles.

The equation has the same form as that of the affine transformation (1.96), but it can represent only changes of scale, reflections, rotations, and translations. Depending on the sign of the determinant of $\mathbf{A}$, similarities are divided into direct transformations (positive determinant), which preserve orientation, and inverse transformations (negative determinant), which reverse orientation.

Isometries

Isometries are similarity transformations that preserve distances:

$$\Vert f(\mathbf{x}) - f(\mathbf{y}) \Vert = \Vert x - y \Vert$$ (1.98)

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

Isometries between Euclidean spaces are written as in equation (1.96), where $\mathbf{A}$, a necessary and sufficient condition for the transformation to be an isometry, must be an orthogonal matrix.

Since it is orthogonal, matrix $\mathbf{A}$ must have determinant $\pm1$. As with similarities, if $\det\mathbf{A}=1$, the isometry is called direct, whereas if $\det\mathbf{A}=-1$, the isometry is inverse.

Examples of isometries include

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