Image:Fractal fern explained.png|thumb|right|200px|An image of a fern which exhibits affine self-similarity In geometry, an affine transformation or affine map or an affinity (from the Latin, affinis, "connected with") between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, satisfying certain properties described below. Geometrically, an affine transformation in Euclidean space is one that preserves # The collinearity relation between points; i.e., three points which lie on a line continue to be collinear after the transformation # Ratios of distances along a line; i.e., for distinct collinear points, ,, the ratio is preserved In general, an affine transformation is composed of linear transformations (rotation, scaling or shear) and a translation (or "shift"). Several linear transformations can be combined into a single one, so that the general formula given above is still applicable. In the one-dimensional case, A and b are called, respectively, slope and intercept.

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