Mathematical Definition
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, f determines a linear transformation φ 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.
Read more about this topic: Affine Transformation
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