Ok, that seems clear. But, abstractly, without linking any figure, the cartesian coordinate system consists of the 3-tuples (x1,x2,x3). Well, and what about cilindrical coordinates? It consists of all 3-tuples as well. So, from a formal point of view, they both are the same. So, any of them are cartesian. Them how do we know which one is being refered? Only by the symbols used? (you know, x's for one and r,theta, phi for the other)

Moreover, if so, there is a prefered affin space among the infintely many "3-tuples"-spaces obtained by transformation?

Since 2 weeks ago all this is confusing for me (even when it'seems obvious things), we need a formal statement if possible. The problem is that I dont know yet if it can be done in a totally abstract way or if, in contrast, the meaning of "cartesian" always refer to a figure on a sheet, to geometry.

Help please.

Plus some affine and metric properties, e.g.

(x1,x2,x3) + (v1,v2,v3) = (x1+v1,x2+v2,x3+v3). or
dist((x1,x2,x3),(v1,v2,v3)) = sqrt(...

>Well, and what about cilindrical
>coordinates? It consists of all 3-tuples as well.

That's right, but how do you define distance between points in cylindrical coordinates?

>So, from a
>formal point of view, they both are the same.

Not at all. This is like saying that, since a postal address consists of (name, street, city) and a person's name consists of (given name, middle name, surname), every person is somehow identified with a postal address, and vice versa.