I have been going through Duke's series of lecture notes on computational topology, and on their formulation of simplicial complexes (https://www2.cs.duke.edu/courses/fall06/cps296.1/Lectures/sec-III-1.pdf), they discussed the concepts of the affine combination of points in an affine plane. From reading around the internet, it seems to me that an affine plane is a plane where:
a- Given any two points, there is a unique line through them
b- Given a line L and a point P not on L, there is a unique line goes through P that doesn't intersect L
c- There are four points where no three of which are colinear
However, I don't really understand the intuition behind these axioms, hence the concepts of affine combination of points where the coefficient of points (!?) sums up to 1? Any intuitive explanation of these concepts would be appreciated.
An affine plane (or space, more generally) is just a linear plane $V$ lifted by some support vector $x$ which is not necessarily in $V$:
$$A = x + V.$$
Now if you have a collection of points $a_1, \ldots, a_m \in A$, note you can rewrite them as $x + v_1, \ldots, x + v_m$. Then if $c_1, \ldots, c_m$ are coefficients such that $\sum c_i = 1$, we obtain
$$\sum_{i=1}^m c_i a_i = \sum_{i=1}^m c_i (x + v_i) = x + \sum_{i=1}^m c_i v_i,$$
which is an element of $A$.