Date: Sep 1, 2013 12:49 PM
Author: fom
Subject: Re: What does one call vector geometry without a coordinate system?

On 9/1/2013 9:41 AM, Shmuel (Seymour J.) Metz wrote:
> In <WrudnVYbz_glnL_PnZ2dnUVZ_q6dnZ2d@giganews.com>, on 08/31/2013
> at 09:46 AM, fom <fomJUNK@nyms.net> said:
>

>> It would be called a Euclidean point space.
>
> I might believe affine.
>

>> The point difference (the additional algebraic structure)
>> then becomes a ground for a distance function.

>
> No; you can only correlate distances on parallel lines.
>


Thanks. The affine structure had been
clarified elsewhere.

Could you explain your second statement a
little more. Does it relate to the difference
between a vector space and an inner product
space?

That would make sense to me. In order to
correlate magnitudes with respect to non-parallel
directed line segments, one would need to have
a notion of rigid rotations. That would suggest
angle measure. In turn, that would suggest that
an inner product would be required.