steve waterman wrote:So a "coordinate transformation" equates the coordinates in one system to the coordinates in the other system?

No. It equates the point identified by the coordinates given in one system to the point identified by coordinates given in the other system.

First consider the pile of books again. Three different librarians came up with three different cataloging systems. Each slot (in any given catalog) identifies a specific book in the pile of books on the floor. Conversely, each book is associated with a specific slot in the catalog

for any given catalog. Given a

different catalog, the

same book will typically have a

different slot that points to it.

A "coordinate transformation" function for the library would, for any given slot number of

one catalog, figure out which slot number in

the other catalog it is that points to the

same actual book. So, if you wanted "book number seven" (Note - this is sloppy shorthand for "the book that slot number seven identifies) in the Julian system, and you applied the Julian to Gregorian transformation for this library, it would tell you that slot number forty-five is the slot you need to push in order to get that same book. You may be tempted to think of it as "Julian seven" = "Gregorian forty-five", but that would be incorrect.

What is correct is the book pointed to by Julian seven is the same as the book pointed to by Gregorian forty-five.The thing that is the same is the

book.

Something similar can be done with dates and calendar systems.

Now, consider points (in space) and coordinate systems (such as the Cartesian one).

"Space" is the manifold. It is the analog of the pile of books. DO NOT yield to the temptation of thinking of space as having a coordinate system of its own. That is what will lead you astray. Space, being the manifold, is just a pile of points.

If you want to identify these points with coordinates, you need to impose a coordinate system. This is the analog of the card catalog. There are many coordinate systems to choose from - you can choose rectangular or polar for example, you can orient the axes any way you like, and you can put the origin anywhere you like.

. I'll call the two systems R and P (rectangular and polar, just to keep things easy to differentiate).

Once you set up two different coordinate systems that each span the space (are capable of identifying any point in the manifold we call space), you will want to be able to find which coordinates of R refer to the same point that a given set of coordinates of P refers to. The function that does this is a coordinate transform.

You may have seen something written that looks like

R(1,1) = P(sqrt(2), pi/4)

This is sloppy notation which is usually harmless, but is what is throwing you. It is not the

coordinates that are equal, but the

points to which they refer.

The correct (albeit clumsier) way to say it is: The point identified by R(1,1) is the same as (equal to) the point identified by P(sqrt(2), pi/4).It is the

points that are equal, just like in the library, it's the

books that were equal.

Jose