Affine transformation

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Affine Transformation[edit | edit source]

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. An affine transformation can be constructed using a linear transformation followed by a translation.

Affine transformations are fundamental in computer graphics, image processing, and computer vision, as they can represent any combination of translation, scaling, rotation, and shearing.

Mathematical Definition[edit | edit source]

An affine transformation in two-dimensional space can be represented by the equation:

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

where \( \mathbf{A} \) is a linear transformation matrix, \( \mathbf{x} \) is the input vector, \( \mathbf{b} \) is the translation vector, and \( \mathbf{y} \) is the output vector.

In matrix form, this can be written as:

\[

\begin{bmatrix} y_1 \\ y_2 \\ 1 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & b_1 \\ a_{21} & a_{22} & b_2 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ 1 \end{bmatrix} \]

Types of Affine Transformations[edit | edit source]

  • Translation: Shifts every point of a shape in the same direction by the same distance.
  • Scaling: Enlarges or diminishes objects; the scale factor is the same in all directions.
  • Rotation: Rotates objects around a fixed point.
  • Shearing: Slants the shape of an object.
  • Reflection: Flips objects over a line.

Properties[edit | edit source]

Affine transformations preserve:

  • Collinearity: Points that lie on a line continue to be collinear.
  • Ratios of distances: The midpoint of a line segment remains the midpoint after transformation.
  • Parallelism: Parallel lines remain parallel.

Applications[edit | edit source]

Affine transformations are widely used in:

Related Pages[edit | edit source]

Gallery[edit | edit source]

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