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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)?

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  • $\begingroup$ They are the same. But note that there are many different kinds of coordinate transformations: they might or might not be linear, keep the origin fixed, etc. Whenever the geometric points ${\bf x}$ have old coordinates $(x_1,x_2,x_3)$ and new coordinates $(y_1,y_2,y_3)$ of some sort such that the $y_i$ can be computed from the $x_i$, and vice versa, in a definite way we have a coordinate transformation before us. $\endgroup$ – Christian Blatter Jan 20 '15 at 10:20
  • $\begingroup$ It would be clearer if they appended the word 'axes' to change of coordinates, e.g. change of coordinate axes. Then it is clear that we are keeping the points fixed in space but only changing the axes, and thus the coordinates of each point P. The new coordinate axes do not have to be orthogonal (for instance polar coordinates). $\endgroup$ – john Jan 7 at 17:29
  • $\begingroup$ A coordinate transformation or 'change of variables' is a dynamic mapping between two sets. Note that the graph of each set will use same coordinate system , typically orthogonal axes for simplicity. So we have equivalency principle here, (they are distinct conceptually though). We can keep points fixed and 'transform' just the coordinate axes, or keep the axes fixed and 'transform' the points. The latter approach is useful for evalutating the Jacobian, the former is useful in linear algebra in change of basis. $\endgroup$ – john Jan 7 at 17:38
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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.

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  • $\begingroup$ Regarding terminology, my research leads me to believe that terminology can go either way. Specifically a coordinate transformation can refer to changing the reference frame whilst keeping the points fixed- e.g. changing cartesian axes to polar axes or changing vector bases in linear algebra. And a coordinate transformation could refer to the action of moving the points of the object whilst keeping the axes fixed (typically the axes begin orthogonal). $\endgroup$ – john Jan 7 at 18:05
  • $\begingroup$ Here are more links for research and this , and this and this . Both approaches are useful. For example the 'keep axes fixed but only transform points' is useful for deriving the Jacobian factor in calculus integration, and 'keep points fixed but only transform axes' is useful for 3d graphing work. $\endgroup$ – john Jan 7 at 18:33
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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.

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