What is the difference between coordinates transformation and change of coordinates?

Question

In the context on 3D computer graphics, what is the difference between coordinates transformation and change of coordinates?

It can just be a matter of notation, but my book makes a clear distinction between the 2 terms (that I do not understand completely).

As far as I understand, change of coordinates means to change the reference frame of all points expressed in a given frame and coordinates transformation means, given a set of points (in some reference frame), use one of them as the new origin (of a new reference frame) and express all the others in terms of that one.

It seems to me that the coordinates transformation concept is similar to global and object coordinates in 3D computer graphics applications (like openGL), but then again they seem highly similar.

Can you make some examples to point out the difference (if any)?

Answer 1

Coordinate transform is a technical term. It refers to the process of finding out the new coordinates of a point fixed in space when the coordinate system is changed. "Change of coordinates" is not really a technical term. When a point $P$ has its coordinates changes from $(x_1,y_1)$ to $(x_2,y_2)$, it could be that the point is physically moved in the space or we are simply moving the coordinate system while the point is fixed is space. Both actions will cause change of point $P$'s coordinates.

If your book is making a clear distinction between "coordinate transformation" and "change of coordinates", I think it is using "coordinate transformation" to refer to the action of moving the coordinate system and using "change of coordinates" for moving the object. These two are indeed distinct concepts.

Answer 2

You know, this is something that really annoys me about applied or non-mathematical subjects that invent superfluous terminology in order to clarify something to an audience whose background in careful mathematics is at best weak. Computer science texts are notorious for this, especially with the explosive growth of computational geometry the last decade or so. The amount of mathematical background that would be needed to make this definition precise really isn't much-basically some naive set theory and linear algebra, both of which are probably going to need to be mastered by any good computer scientist at some point anyway.

A coordinate transformation- or change of coordinates, if you must- is simply a mapping of an ordered basis to another in a vector space V where the coordinates are the scalars of the linear combinations of basis vectors that generate the space. In physics and geometry, where these transformations are so critical, the bases in question are functions that define a coordinate system on V. Therefore, the coordinate transformation is actually a linear operator from V onto V. A very nice and brief set of notes that gives examples and some more clarifying remarks can be found here.

To be honest, though, I was a bit baffled by Christian Blatter's comment above. I was always taught that a true coordinate transformation in the sense of linear algebra is always linear by definition. This extends also to manifolds, where coordinate transformations are local isomorphisms between tangent spaces. More general, nonlinear transformations of basis functions can be allowed, of course, as parametrizing families of functions by the Implicit Function Theorem. But in this case, we are referring to functions that are not necessarily basis functions in a vector space. Physicists tend to be a little more fast and loose with this distinction then mathematicians since both kinds of transformations are important in mechanics and it's underlying geometry. The distinction is important, though, and the student should beware. Technically, a nonlinear change of coordinates is not a coordinate transformation in the mathematical sense.