Affine transformation Totally Explained

Everything about Affine Transformation totally explained

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::

x mapsto A x+ b

In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as the matrix A with an extra column b.
Physically, an affine transform is one that preserves

Collinearity between points, for example, three points which lie on a line continue to be collinear after the transformation
Ratios of distances along a line, for example, for distinct colinear points $p_1$ , $p_2$ , $p_3$ , the ratio $|p_2-p_1| / |p_3-p_2|$ is preserved

In general, an affine transform is composed of zero or more linear transformations (rotation, scaling or shear) and translation (shift). Several linear transformations can be combined into a single matrix, thus the general formula given above is still applicable.

Representation of affine transformations

Ordinary vector algebra uses matrix multiplication to represent linear transformations, and vector addition to represent translations. Using an augmented matrix, it's possible to represent both using matrix multiplication. The technique requires that all vectors are augmented with a "1" at the end, and all matrices are augmented with an extra row of zeros at the bottom, an extra column — the translation vector — to the right, and a "1" in the lower right corner. If A is a matrix, ť

egin

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