Affine transformation

In geometry, an affine transformation, affine map or an affinity (from the Latin, affinis, 'connected with') is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. In geometry, an affine transformation, affine map or an affinity (from the Latin, affinis, 'connected with') is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence. If X {displaystyle X} and Y {displaystyle Y} are affine spaces, then every affine transformation f : X → Y {displaystyle fcolon X o Y} is of the form x ↦ M x + b {displaystyle xmapsto Mx+b} , where M {displaystyle M} is a linear transformation on the space X {displaystyle X} , x {displaystyle x} is a vector in X {displaystyle X} , and b {displaystyle b} is a vector in Y {displaystyle Y} . Unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine, but not every affine transformation is linear. All Euclidean spaces are affine, but there are affine spaces that are non-Euclidean. In affine coordinates, which include Cartesian coordinates in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates. Another way to deal with affine transformations systematically is to select a point as the origin; then, any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation. An affine map f : A → B {displaystyle fcolon {mathcal {A}} o {mathcal {B}}} between two affine spaces is a map on the points that acts linearly on the vectors (that is, the vectors between points of the space). In symbols, f {displaystyle f} determines a linear transformation φ {displaystyle varphi } such that, for any pair of points P , Q ∈ A {displaystyle P,Qin {mathcal {A}}} :