fom
Posts:
1,969
Registered:
12/4/12


Re: What does one call vector geometry without a coordinate system?
Posted:
Aug 31, 2013 6:28 PM


On 8/31/2013 1:57 PM, FredJeffries wrote: > On Saturday, August 31, 2013 7:46:12 AM UTC7, fom wrote: >> >> In the kind of geometric situation described here >> it is more than a vector space. >> >> Vectors in this situation are equivalence classes >> of directed line segments (magnitude and direction). >> The original poster is discussing specific, labeled >> directed line segments. >> >> It would be called a Euclidean point space. The >> point difference (the additional algebraic structure) >> then becomes a ground for a distance function. > > I could be mistaken, but I believe that what you (and > Professor Bowen) call a Euclidean point space is referred > to by many others as a real affine space. > > Another reference: > http://math.sjtu.edu.cn/course/gdds/%E6%95%99%E5%AD%A6%E8%AF%BE%E4%BB%B6/affinegeometry.pdf > > Note the fourth paragraph: > "Use coordinate systems only when needed!" >
They are closely related.
See Definition 2.1.1.
First, thank you very much. You have brought to my attention the meaning of a detail from Bowen's remarks:
"If V does not have an inner product, the set E defined above is called an affine space"
So, I suppose the answer to the original poster's question actually depends on whether that intuitive presentation included any treatment of angles in addition to directed line segments.
I like what is in the link you posted. It clarifies many details for me. It differs from Bowen's presentation in that it presents an affine space in terms of a faithful group action.
f: ExV > E
where I have used V for the translation space.
In Bowen, one has
f: ExE > V
Although I would have to compare the presentations more carefully to appreciate any subtle differences, Bowen's view seems to be directed toward the topological constructions about to be discussed in his presentation.
A uniform topology can be defined with respect to a system of relations on the Cartesian product. Consequently, the uniform structure is related to ExE rather than ExV. I do not know if that really is a consideration for Bowen. But I find the possibility that the difference in presentation may have such an account to be interesting.
Thank you again. I knew that William's response was not quite right and only replied with what I knew in the language I had to say it. This link makes many things more clear for me and gives me the vocabulary for the future.

