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

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)?

• 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. – Christian Blatter Jan 20 '15 at 10:20
• 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). – john Jan 7 at 17:29
• 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. – john Jan 7 at 17:38

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.