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 withot 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 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
If similarity geometry means similarity-invariant geometry, what are its objects? Google has a lot to say about similarity spaces, none of it relevant to similarity invariance.
Sticking to finite dimensions, a Euclidean space is standardly defined as an inner product space over the reals. As such it has predicates recognizing those line segments of unit length, and those lines passing through the origin, neither of which I remember from Euclid, who seems rather to be writing about similarity geometry.
Euclid does however talk about right triangles and line segments of equal length (and equal angles but that follows from the other two). So clearly Euclid is doing more than just affine geometry.
If Euclid's plane is not the Euclidean plane, what is it? If Euclid was doing similarity geometry it should be called a similarity space (certainly not a vector space or a Euclidean space or an affine space or a projective space or a metric space or a topological space). If it's called something else what is it? If Euclid was doing some other kind of geometry than similarity geometry what kind was it?
Vaughan
-- 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