affineinvariant

Affineinvariant refers to a property or quantity that remains unchanged under affine transformations. An affine transformation is a map of the form x' = A x + b, where A is an invertible matrix and b is a vector. Such transformations preserve straight lines and parallelism, and they map polygons to polygons of the same combinatorial type, though not necessarily preserving lengths or angles.

Because affine transformations preserve collinearity and parallelism, several geometric relations are invariant. For example, points that lie on a single line remain collinear after an affine map, and two lines that are parallel remain parallel. Ratios of lengths along a fixed line are preserved: if four points lie on a line, the ratio of segment lengths determined by those points is unchanged by the transformation. The cross ratio of four collinear points is another fundamental affine invariant, surviving affine mapping and serving as a robust descriptor in projective geometry.

Area considerations are also tied to affine invariance. Under an affine transformation, areas of figures scale by the determinant of A, so the ratio of two areas (for figures subjected to the same affine map) is preserved. This yields affine-invariant comparisons of shapes in applications where perspective effects approximate affine distortions.

In practice, the concept is used in geometry, computer vision, and pattern recognition to compare shapes up to affine distortion. Affine-invariant descriptors aim to characterize objects in a way that is stable under viewpoint changes that can be modeled as affine transformations.