Math and science::Algebra

Affine function

Let $I$ be a real interval. A function $α : R \to R$ is affine iff

α (p x_{1} + (1 - p) x_{2}) = p α (x_{1}) + (1 - p) α (x_{2})

for all $x_{1}, x_{2} \in I$ and $p \in [0, 1]$ .

When coordinates from different origins are equal

The presence of $p$ and $1 - p$ in the affine criteria arises because this is the criteria for when two sets of coordinate systems have equal coordinates despite having different origins. If the function $α$ maps from one coordinate system to the other, then a weighted average of the coordinates in the first system will be equal to the weighted average of the coordinates in the second system. Why? Because when any two vectors in the first system are added with weights summing to 1, the "offset" of the resulting vector is the same as that coordinate system's origin offset (when viewed from the second coordinate system).

Wikipedia has a nice explanation:

Compared to ring homomorphism

The affine criteria is similar in form to the criteria for a ring homomorphism. The criteria for a ring homomorphism would look like:

\forall p, q, x_{1}, x_{2} \in G, α (p x_{1} + q x_{2}) = α (p) α (x_{1}) + α (q) α (x_{2})

The criteria for the affine function is more restrictive on $p$ and $q$ , and so it is less restrictive on the function, making affine functions more general.

Source

https://en.wikipedia.org/wiki/Affine_space