Affine transformation
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:
- Computer graphics: For rendering images and animations.
- Image processing: For image registration and alignment.
- Robotics: For coordinate transformations and motion planning.
- Geometric modeling: For transforming geometric shapes.
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