transformation as a rotation by a certain angle with respect to a certain axis may give a clearer idea of the overall behavior of the transformation than describing it as a combination of a translation and a rotation. However, this depends on application and context. Affine maps An affine map 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, determines a linear transformation such that, for any pair of points : or . We can interpret this definition in a few other ways, as follows. If an origin is chosen, and denotes its image , then this means that for any vector : . If an origin is also chosen, this can be decomposed as an affine transformation that sends , namely , followed by the translation by a vector . The conclusion is that, intuitively, consists of a translation and a linear map. Alternative definition Given two affine spaces and , over the same field, a function is an affine map if and only if for every family of weighted points in such that , we have . In other words, preserves barycenters. History The word "affine" as a mathematical term is defined in connection with tangents to curves in Euler's 1748 Introductio in analysin infinitorum. Felix Klein attributes the term "affine transformation" to Möbius and Gauss. Image transformation In their applications to digital image processing, the affine transformations are analogous to printing on a sheet of rubber and stretching the sheet's edges parallel to the plane. This transform relocates pixels requiring intensity interpolation to approximate the value of moved pixels, bicubic interpolation is the standard for image transformations in image processing applications. Affine transformations scale, rotate, translate, mirror and shear images as shown in the following examples: The affine transforms are applicable to the registration process where two or more images are aligned (registered). An example of image registration is the generation of panoramic images that are the product of multiple images stitched together. Affine warping The affine transform preserves parallel lines. However, the stretching and shearing transformations warp shapes, as the following example shows: This is an example of image warping. However, the affine transformations do not facilitate projection onto a curved surface or radial distortions. In the plane Affine transformations in two real dimensions include: pure translations, scaling