Hi Harold,
The affine space is a space that preserves the angles of transformation. An affine structure is the generalized abstraction of a vector space - in that the affine space does not contain a unique element known as the "origin". In other words, affine spaces are average combinations - differences between two points. For example, the longitude on a globe is affine. There is no point of origin to tell us where to "begin," yet it preserves (angular) mapping onto the n-th dimension. (The "Prime Meridian" is arbitrary; any longitude can be considered the "origin".) Affinity doesn't depend on scaling.
A vector space is similar to an affine space, but it has the unique property of O - an object such that if we take two vectors v, w ∈ V such that v + w = O (the zero/origin vector). In other words, we actually evolve an origin by mapping to a vector space from an affine space. The origin becomes the point of reference in our affine space. As an example, if we take the same affine point on a longitude (call it the Prime Meridian), and we map to a point on Alpha Centauri, then the Prime Meridian becomes the origin of our vector space. The origin evolved only because of our mapping to a vector space, but in the affine space there is no "origin".
I hope this helps.
