Affine transformations keep straight lines straight and parallel lines parallel
Stretch, slide, rotate, reflect or shear a drawing, and lengths and angles may change, but straight lines stay straight and parallel lines never meet. That guarantee defines an affine transformation, and a neat trick of adding one extra coordinate lets a single matrix handle all of these moves at once.
The word affine comes from the Latin affinis, meaning connected with. In ordinary geometry, an affine transformation is any reshaping of space that sends points to points, lines to lines and planes to planes while keeping parallel things parallel. It does not have to preserve distances or angles, so a square can become a slanted parallelogram. What it does keep is proportion along a line: if a point sits one third of the way along a segment before the transformation, it still sits one third of the way along afterwards.
The familiar geometric moves all qualify: translation, rotation, reflection, uniform or uneven scaling, and shearing, along with any sequence of them. Mathematically, every affine transformation breaks down into two steps, a linear transformation followed by a translation. That second step matters. A purely linear map always leaves the origin where it is, but an affine one may shift it, so every linear map is affine while the reverse is not true.
In coordinates this becomes a compact formula: multiply the input vector by a matrix, then add a fixed vector. Mathematicians found a way to fold both operations into one multiplication. Append a 1 to the end of every vector, and enlarge the matrix with an extra column holding the translation, a bottom row of zeros, and a 1 in the corner. One matrix product then performs the whole transformation.
That enlarged matrix hints at a deeper connection. If the bottom row is allowed to hold values other than zeros and a final 1, the same machinery performs projective transformations, the more general maps of perspective. Seen from that angle, affine transformations are exactly the projective ones that leave the points at infinity in place, which is why parallel lines, meeting only at infinity, stay parallel.
Source: Affine transformation
