Talk:Change of Coordinate Systems
From Math Images
OK, here are some thoughts.
I really like the examples and other additions you made. I feel like the central thing that's missing here is the idea that a change in coordinate system is often a way of viewing the same set of points using a different naming scheme. This page brings out the other important interpretation of coordinate systems -- as a map from Rn to Rn with certain properties -- really well, but doesn't bring out the first part well. So for instance, you talk about a disk being mapped to a rectangle, which is true, but you can also think of the change in coordinates as being a new way to describe the same disk in a more mathematically convenient way.
This is a tricky fix to implement because usually when you use change of coordinates for a problem (say change of variables in integration), you're kind of working with both interpretations at the same time. But I think this, and maybe some examples of when changes of coordinates are used (though this might be opening a huge can of worms), are the most important changes.
- I like this idea, but I'm not quite sure how to implement it. If there's any way you could put in something really rough that I could build on, that would be great.
A couple of other thoughts:
The first example in the introductory part could use an image to go with it, unless this example is changed in light of the discussion above. I know that that's a pain not only because it means generating a new image but because the layout will be annoying.
In the coordinate vector mouse-over, I feel like a coordinate represents a component, not the other way around.
- Agreed, and fixed.
When you say "the ellipse that is tiled relative to the coordinate axes", what ellipse are you talking about?
- The one in the main image. Clarified.
I'm not sure that the explanation of going from cartesian to polar coordinates is quite flushed out enough to be helpful to anyone who doesn't already basically know what's going on. It may just be that the only problem is the line "Each ring at a different distance from the origin creates its own line in the polar system, and the collection of these lines creates a rectangle." An intermediate step explaining how the disk is a collection of rings, so its image is a collection of lines, would probably be helpful.
- I wrote a new little paragraph--tell me if it's better. I left the old paragraph in below it.
"each coordinate is a distance one a coordinate axis." it should be position, not distance, and maybe use terminology that is friendlier, like, "the three coordinates represent left-right position, up-down position, and forward-backward position, respectively." if you're concerned that this terminology is not sufficiently general, note that the current definition would also need to specify that the coordinate axes are pairwise perpendicular (and maybe even that they have a right-handed orientation).
- Your sentence is now being used.
I'd really appreciate some advice about what needs to be edited. I'm thinking about putting in the transformations for cylidrical and spherical coordinates, but other than that I'm not sure what else to do.
Hey Drexel folks,
An applet for this page would be nice. Perhaps the user could actively change parameters that change the coordinate system, and could see the effects this has on an object such as a circle. A 3D applet may be nice too since I don't have any 3D pictures or animations, though this would be more involved.
I'd like the first words a user encounters say a bit about what the image is; your observation at first is a worthwhile second observation.
Your More Math Exp requires vectors and matrices. Why not say? Why not have at least the title of 3-D Coords not hidden when More Math is? Might encourage people to go forward. For 3-D you could have simple non-math explanations of your examples, with nice images, too.
It would sure be nice to have some of this whole section interactive, so that users could see the effect of changing various parameters. Maybe a joint project with a Drexel person?