Bourbaki in his Algebra, and MacLane & Birkhoff in their Algebra, have similar definitions. I do not know the one you cite. However, the definition I have given describes an affine space as an honest equationally presented algebraic structure; usually they just assert that given any two points a and b, there exists a unique vector v such that a + v = b. With the difference of points an explicitly given operation of the structure, the axiom a + (b - a) = b gives the existence, and the axiom (a + v) - a = v the uniqueness. Using an n-dimensional real space instead of beloved R^n gets rid of the `canonical' basis. Observing vectors as translations of the space of points does away with an a priory origin. My contribution was that to do without an a priory unit of length, let the metric of the Euclid's space be given by a class of mutually proportinal positive definite forms, instead of a single form.
"An affine space is, roughly speaking, a vector space, but without a particular vector being chosen as zero." Or shall we say that an affine space is a perfect geometric communism where all points are equal, while in a vector space one vector is more equal than others.
-- France Ellis D. Cooper wrote:
FYI, Hassler Whitney, "Geometric Information Theory", 1957, Appendix I, Section 10, presents the notion of "affine space" essentially as you describe: "An affine space is, roughly speaking, a vector space, but without a particular vector being chosen as zero."
Funny you should ask this just as I was trying to find a satisfying answer to this question. Now we are so accustomed that an Euclidean space has some orthonormal system plonked down somewhere in it, or that it has at least a fixed origin and the scalar product that defines lengths and angles. But, like Eve and Adam were created without navels, Euclid's space was created without an origin; also it was created completely flat (with curvature 0 in modern lingo), so that there is no 'canonical' unit of length; and it was created without a priory orientation. But you can measure one line segment with another in it, you can also mesure angles, and you can _choose_ one of the two possible orientations and call it positive. To capture all this I cooked up some structure; you decide if it does the job.
First, an Euclid's space (let's call it that, to avoid confusing it with an Euclidean space) is a real affine space (V,P), where V is a real vector space of _vectors_ in the space, and P is the set of its _points_. There are also two operations +: P >< V -> P and -: P >< P -> V; the first is an action of the additive group of V on P, so it satisfies a + 0 = a and (a + u) + v = a + (u + v) for all points a and all vectors u, v. Moreover, the two operations are 'local inverses' of each other in the sense that (a + u) - a = u and a + (b - a) = b for all points a, b and all vectors u. This definition eliminates the need for an a priory origin. To make life easier, assume V is finite dimensional.
Morphism (V,P) -> (U,Q) of affine spaces is a pair of maps h: V -> U and f: P -> Q, where h is a linear map and f(a + v) = f(a) + h(v) for all points a in P and all vectors v in V.
Now, the metric of the Euclid's space. There are positive-definite (symmetric bilinear) forms on V. Call two such forms similar if they differ by a constant positive factor; a similarity class is therefore a ray (open half-line with the endpoint the zero form) in the space of all symmetric bilinear forms on V. Now consider the structure E = (V, P, m), where (V,P) is a real affine space and m is a similarity class of positive-definite forms on V: this is my proposed structure of Euclid's space.
Let S be the group of all automorphisms (h,f) of the affine space (V,P) that preserve m: for every g in m, g(h(u),h(v)) = c g(u,v) for some positive constant c (depending on h, not depending on u and v and the choice of g in m) and all vectors u, v. These are the similarity trasformations of E, defining the similarity geometry of E. If A is any structure built from vectors and points (and lines etc) in E, then the orbit of A under S is the object studied in this geometry.
Let R be the group of all automorphisms (h,f) of the affine space (V,P) that preserve some form in m, and therefore preserve _every_ form in m: g(h(u),h(v)) = g(u,v) for all g in m and all vectors u, v. This is the group of rigid motions of E.
Orientation: if dim V = n, then there is a one-dimensional space of alternating n-linear forms on V; choosing one of the two rays (half-lines with the endpoint at the zero form) in this space chooses the orientation.
-- France
-- Dr. France Dacar Email: france.dacar@ijs.si Intelligent Systems Department Phone: +386 1 477-3813 Jozef Stefan Institute Fax: +386 1 425-3131 Jamova 39, 1000 Ljubljana, Slovenia