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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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 fom Posts: 1,968 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 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

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