MW> On Fri, 5 Nov 2004 23:54:30 +0000, Val Samko <boost@digiways.com> 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
On Sat, 6 Nov 2004 13:12:14 +0000, Val Samko <boost@digiways.com> wrote: 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.
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
Please see: http://www.ma.umist.ac.uk/kd/curves/node3.html 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. 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 Cheers, Michael