Affine transformations

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 ${\displaystyle X}$ and ${\displaystyle Y}$ are affine spaces, then every affine transformation ${\displaystyle f\colon X\to Y}$ is of the form ${\displaystyle x\mapsto Mx+b}$, where ${\displaystyle M}$ is a linear transformation on ${\displaystyle X}$ and ${\displaystyle b}$ is a vector in ${\displaystyle Y}$. 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.

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