Sorry if this is a beginner question but I have been trying to find a good definition of an affine space and can't seem to find one that makes intuitive sense. Hoping that one could help explain what an affine space is after defining it mathematically. I understand its relation to a Euclidean space at a high level at least.
Some resources I checked out include:
The last one is the closest to one that makes sense, but still not quite getting it:
A real affine space is a triple $(A, V, φ)$ where $A$ is a set of points, $V$ is a real vector space and $φ: A × A \rightarrow V$ is a map verifying:
- $∀P ∈ A$ and $∀u ∈ V$ there exists a unique $Q ∈ A$ such that $φ(P, Q) = u$.
- $φ(P, Q) + φ(Q, R) = φ(P, R)$ for every $P, Q, R ∈ A$.
My translation of that would be:
A real affine space is a set of points over a real vector space and a function mapping combinations of the points to the vector space, where the function satisfies:
- For any point in the vector space and point in the set of points there is another point in the set of points such that the function maps the vector space and the other point to the first point.
- The function is transitive.
That doesn't seem quite right, and still don't understand the meaning of that middle part.