An affine space is geometry after you forget where the origin is
Two people add arrows on the same map, but each secretly believes a different point is the centre. Their sums disagree, except in one special case: when the weights add up to exactly 1. That quiet agreement is the heart of affine space, a geometry with parallel lines but no favoured starting point.
Ordinary Euclidean geometry comes loaded with distances and angles. Affine geometry strips those away, keeping only parallelism and the ratios of lengths along parallel segments. Points remain, lines still run through any two of them, a plane through any three not in a row, and parallel lines in a shared plane never meet. Given a line and a point, you can always draw the parallel through that point, and all lines parallel to one another share a direction.
What goes missing is the origin. In a vector space, the zero vector sits at a privileged spot, and you can add or scale anything. In an affine space no point is special, so adding two points or doubling one makes no sense. What does make sense is the difference between two points, an arrow called a translation or displacement vector, and adding such an arrow to a point to reach a new point. The French mathematician Marcel Berger summed it up as a vector space whose origin you try to forget.
The thought experiment makes this concrete. Alice knows the true origin; Bob wrongly picks another point. When they combine two vectors with arbitrary weights, Bob's parallelogram lands somewhere different from Alice's. But whenever the weights sum to 1, the offsets cancel and both reach the identical point. Those weighted averages, called affine combinations, are exactly what survives without an origin, and they give barycentric coordinates for any flat through the chosen points.
Formally, mathematicians describe an affine space as a set of points acted on freely and transitively by the additions of a vector space, so each ordered pair of points determines exactly one translation between them; Weyl's axioms capture the same idea through subtraction. Any vector space becomes affine by forgetting its zero, and shifting a subspace off the origin gives an affine subspace, like the solutions of a linear system with a nonzero right-hand side.
Source: Affine space
