In what kind of space is it possible to add points?

Question

When talking about affine space, we can subtract points and get a vector or add a point and a vector to get another point but in what kind of space is it possible to add points? How does that make sense?

Answer 1

It is not clear what kind of answer you are looking for. However, consider an affine space and pick an origin $O$. Then define the addition of points $P$ and $Q$ to be $R$ where $OPRQ$ is a parallelogram and $R$ is uniquely determined. This is just adding vectors $OP$ and $OQ$ using the parallelogram law. It seems that you always need something like an origin for addition of points. A possible alternative is using midpoints. That is, given points $P$ and $Q$ you define the "addition" to be the midpoint of line segment $PQ$. This is a special case of weighted average of two or more points in affine space.