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Topic: What does one call vector geometry without a coordinate system?
Replies: 32   Last Post: Sep 9, 2013 4:45 PM

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 lite.on.beta@gmail.com Posts: 134 Registered: 2/21/06
Re: What does one call vector geometry without a coordinate system?
Posted: Sep 1, 2013 12:27 AM

On Saturday, August 31, 2013 6:28:21 PM UTC-4, fom wrote:
> On 8/31/2013 1:57 PM, FredJeffries wrote:
>

> > On Saturday, August 31, 2013 7:46:12 AM UTC-7, 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:
>
>
> >
>
> > 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.

So are we able to define these:

- Euclidean plane (compass and straight edge only)
- Real affine plane (I'm guessing, adding machinery of vectors to above, but still no coordinates.. and the vectors do not have to start at origin ... equivalence classes of them exist like previously stated by someone)
- Euclidean space (same as above but with inner product, and thus a distance)
- Euclidean point set (2 dim.) (I'm guessing adding coordinates to above)
- Euclidean vector space R^2 (I'm guessing, same as above but the vectors have to start at origin ... so there is a 1-1 correspondence betweens points and vectors)