Affine transformation

Affine Transformation

An affine transformation is a function between affine spaces which preserves points, straight lines, and planes. Sets of parallel lines remain parallel after an affine transformation. In the context of linear algebra, an affine transformation is a combination of linear transformations and translations.

Definition

Mathematically, an affine transformation can be represented as a function \( f: \mathbb{R}^n \to \mathbb{R}^m \) of the form:

\[

f(\mathbf{x}) = A\mathbf{x} + \mathbf{b}

\]

where \( A \) is a linear transformation represented by a matrix, \( \mathbf{x} \) is a vector in \( \mathbb{R}^n \), and \( \mathbf{b} \) is a translation vector in \( \mathbb{R}^m \).

Properties

Affine transformations have several important properties:

• Linearity: The linear part of the transformation, represented by the matrix \( A \), ensures that straight lines remain straight.
• Parallelism: Parallel lines remain parallel after the transformation.
• Midpoints: The midpoint of a line segment remains the midpoint after transformation.
• Ratios: Ratios of distances along parallel lines are preserved.

Types of Affine Transformations

Affine transformations include several specific types of transformations:

Translation

A translation moves every point a constant distance in a specified direction. It is represented by the transformation \( f(\mathbf{x}) = \mathbf{x} + \mathbf{b} \).

Scaling

Scaling changes the size of an object. Uniform scaling occurs when the scale factor is the same in all directions, while non-uniform scaling has different scale factors for different directions.

Rotation

Rotation turns an object around a fixed point, known as the center of rotation. In two dimensions, a rotation by an angle \( \theta \) is represented by the matrix:

\[

R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}

\]

Shearing

Shearing distorts the shape of an object such that the transformation moves each point in a fixed direction, proportional to its distance from a line that is parallel to that direction.

Reflection

Reflection is a transformation that "flips" an object over a line (in 2D) or a plane (in 3D).

Applications

Affine transformations are widely used in computer graphics, image processing, and geometric modeling. They are essential for tasks such as:

• Image registration: Aligning images from different sources.
• Geometric transformations: Modifying the geometry of objects in graphics applications.
• Fractals: Generating fractal patterns, such as the fractal fern.

Related Pages

• Linear transformation
• Matrix (mathematics)
• Vector space
• Euclidean space

Gallery

• 

  Fractal fern generated using affine transformations.

• 

  Video illustrating affine transformations.

• 

  Identity transformation on a checkerboard.

• 

  Reflection transformation on a checkerboard.

• 

  Scaling transformation on a checkerboard.

• 

  Rotation transformation on a checkerboard.

• 

  Shearing transformation on a checkerboard.

• 

  Original circle image.

• 

  Sheared circle image.

• 

  Central dilation example.

• 

  Example of geometric affine transformation.

• 

  2D affine transformation matrix.