Affine transformation

A map between affine spaces that preserves affine combinations. In coordinates, it has the form of a linear transformation followed by a translation.

Linear transformation: A map between vector spaces that preserves vector addition and scalar multiplication. The linear part changes directions and scale before the translation shifts every point.
Affine space: A set of points equipped with a free and transitive action of a vector space. Affine transformations act on these spaces without requiring a distinguished origin.
Computer vision: A field that extracts information about scenes from images and video. Image registration and camera models use affine maps to relate image coordinates.
Euclidean transformation: A transformation preserving Euclidean distances and angles. Affine maps preserve parallelism but need not preserve distances or angles.
Translation (geometry): A transformation that moves every point by the same vector. The translation part shifts the output without changing relative positions.
Vector space: A set with vector addition and scalar multiplication satisfying the vector-space axioms. The linear part of an affine map acts on the displacement vectors of its domain.
Image registration: The alignment of images into a shared coordinate system. Affine registration handles shifts, rotations, scaling, and shearing between images.
Similarity transformation: A transformation that preserves angles and scales all distances by one common factor. It is a more restrictive affine map that cannot shear or distort angles.
Homogeneous coordinates: A coordinate system that represents points with an extra coordinate, enabling affine maps to be written as matrices. They combine the linear part and translation into one matrix multiplication.
Affine combination: A weighted sum of points whose coefficients sum to one. These combinations provide the point-based language for defining preservation.

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