Alias or Alibi?

First, what is a coordinate system?
A coordinate system is about some space of points. The system determines an array of real numbers, called the ‘coordinates’ for each point. You might say that the coordinate system provides an address for each point in the space; in this context we can state the coordinates to specify the point. Generally the coordinates change smoothly as the point moves continuously, but witness what happens when the point moves thru the origin and we use polar coordinates.

Second, what is a transformation?
Here we claim that it is a set of rules to compute one set of coordinates from another. If we write Cartesian coordinates <x, y> then here is a transformation:
<x, y> → <x+1, y>.
For a fixed coordinate system this transformation is seen as moving points one unit to the right. The points have gone elsewhere and have an alibi!
If instead we have adopted another coordinate system for the 2nd expression, one displaced one unit to the left, we say instead that we have given the same place new coordinates, perhaps to compare them with another observer with whom we had been unable to coordinate coordinate systems. In such cases we have renamed the point, rather than moved it. The point has thus an alias!

These two perspectives on the same math may confuse and the alias-alibi distinction was taught in Berkeley in the 50’s. I have often wondered who came up with the clever terminology.

See alias and alibi.