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From: Val Samko (boost_at_[hidden])
Date: 20041106 08:12:14
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? :)
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.
MW> The point (pun intended) is, that the resulting type is NOT a point.
MW> "pointpoint" has a different meaning than "vectorvector".
In C++ sence  yes, it's a different point. In mathematical sense,
pointpoint 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.
If you need a distance, you need a norm, i.e. pointApointB.
>> RD> So in a mathematical sense, the difference between size and position is
>> RD> entirely relevant, it's just that some libraries (not mentioning names)
>> RD> do not adhere to the above mathematical rules.
>>
>> There are no mathematical rules, which say that a point in nD
>> Cartesian coordinate system is not just a set of corresponding
>> distances along each axe.
MW> A point can be *specified* by means of a vector, but it's not the same.
Please see
http://mathworld.wolfram.com/EuclideanSpace.html
"Euclidean nspace (Cartesian space) is commonly denoted R^n. ...
R^n is a vector space".
So, any point in nD Cartesian space IS a vector.
Valentin Samko
http://val.samko.info
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