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From: Michael Walter (michael.walter_at_[hidden])
Date: 2004-11-06 16:23:01

On Sat, 6 Nov 2004 13:12:14 +0000, Val Samko <boost_at_[hidden]> wrote:
> MW> On Fri, 5 Nov 2004 23:54:30 +0000, Val Samko <boost_at_[hidden]> wrote:
> >> Once again, in geometry, vector and point are practically the same thing.
> MW> Not really - you can only represent a point by its radius/position
> MW> vector wrt a given coordinate system.
> I thought we are only talking about Cartesian coordinates? Does anyone
> really need a gui library for radial coordinate system? :)
Apologies - I was apparently using the wrong term (not a native
English speaker), and was just seeing "the vector r from the origin to
the current position" on Mathworld. Googling a bit more it appears
that it is such a vector in a polar/spherical coordinate system only.

The word I was looking for is "Ortsvektor" in German, the vector from
the origin to a certain point. What would be the proper word for that
in English (tried googling, failed :)?

> MW> Radius vector and point are not the same thing, though. For instance,
> MW> there is no meaning behind multiplying a point by a scalar. More
> MW> formal, points are not members of any vector space, so you can't apply
> MW> operations which are only defined on vectors on them.
> We are not talking about abstract points here. They are points in nD
> space, and each of them is represented by a corresponding vector.
> You may apply any operations defined in your space to your
> points/vectors.
The Euclidean space is E^3, the vector space is R^3.

Please see:

Note also: "Not all books make this distinction so you need to be
prepared to encounter the unstated identification E^3 = R^n"

Mathworld is using this _unstated_ _identification_; it is imprecise at least.

> MW> The point (pun intended) is, that the resulting type is NOT a point.
> MW> "point-point" has a different meaning than "vector-vector".
> In C++ sence - yes, it's a different point. In mathematical sense,
> point-point depends on how this operation is defined in your
> particular space, and in Euclidean space, the result of this
> operations is the same as difference of corresponding vectors.
Yes. But the result is a *vector*. IOW, the difference between 2
points is a mapping from two points in Euclidean space to a vector in
a vector space:
 difference :: E^n x E^n -> R^n


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