Editing Affine transformation (section)

==Affine maps==
An affine map <math>f\colon\mathcal{A} \to \mathcal{B}</math> between two [[affine space]]s is a map on the points that acts [[Linear transformation|linearly]] on the vectors (that is, the vectors between points of the space). In symbols, ''<math>f</math>'' determines a linear transformation ''<math>\varphi</math>'' such that, for any pair of points <math>P, Q \in \mathcal{A}</math>:

:<math>\overrightarrow{f(P)~f(Q)} = \varphi(\overrightarrow{PQ})</math>
or
:<math>f(Q)-f(P) = \varphi(Q-P)</math>.

We can interpret this definition in a few other ways, as follows.

If an origin <math>O \in \mathcal{A}</math> is chosen, and <math>B</math> denotes its image <math>f(O) \in \mathcal{B}</math>, then this means that for any vector <math>\vec{x}</math>:

:<math>f\colon (O+\vec{x}) \mapsto (B+\varphi(\vec{x}))</math>.

If an origin <math>O' \in \mathcal{B}</math> is also chosen, this can be decomposed as an affine transformation <math>g\colon \mathcal{A} \to \mathcal{B}</math> that sends <math>O \mapsto O'</math>, namely

:<math>g\colon (O+\vec{x}) \mapsto (O'+\varphi(\vec{x}))</math>,

followed by the translation by a vector <math>\vec{b} = \overrightarrow{O'B}</math>.

The conclusion is that, intuitively, <math>f</math> consists of a translation and a linear map.

===Alternative definition===
Given two [[affine space]]s <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, over the same field, a function <math>f\colon \mathcal{A} \to \mathcal{B}</math> is an affine map [[if and only if]] for every family <math>\{(a_i, \lambda_i)\}_{i\in I}</math> of weighted points in <math>\mathcal{A}</math> such that 
: <math>\sum_{i\in I}\lambda_i = 1</math>,

we have<ref>
{{cite book|author1=Schneider, Philip K. |author2=Eberly, David H.|title=Geometric Tools for Computer Graphics|publisher=Morgan Kaufmann|year=2003|isbn=978-1-55860-594-7|page=98|url=https://books.google.com/books?id=3Q7HGBx1uLIC&pg=PA98}}</ref>

: <math>f\left(\sum_{i\in I}\lambda_i a_i\right)=\sum_{i\in I}\lambda_i f(a_i)</math>.

In other words, <math>f</math> preserves [[Barycentric_coordinate_system|barycenters]].