Visualization

So, how can you visualize what these alternate coordinate systems "look like"?  This can be a bit tricky, so I will show you one way to do this.  

Consider a coordinate system {u,v,w}.  If you fix w and let u and v vary, you will get a surface (why?).  If you do this for different values of w, you will get a bunch of surfaces that are "parallel" in some sense (literally, in Cartesian coordinates).  These can be thought of as "level surfaces" for the coordinate system:  everything on one of these surfaces has the same w coordinate.  You can do the same thing for u and v by letting each of them stay constant.  Putting all these together give a pretty good picture of how the coordinate system behaves.  Let's take a look at this for Cartesian and Spherical coordinates:

First, you use the Table function to generate your "level surfaces" for each combination of variables.  Notice that the Table function works like:

Table[ Stuff to do for different values of k, {k, kStart, kEnd, kStep}]

where:
kStart = The beginning value to plug in for k
kEnd = The final value to plug in for k
kStep = The amount to increment k by each time

So, when I use the command:
Table[ParametricPlot3D[ stuff...], {k,-4,4,2}]

I am telling it to make a series of ParametricPlot3D's, letting k = -4, -2, 0, 2, 4.

(You can ignore the "spelling error" messages.  Mathematica is trying to be helpful, but not succeding very well here.)

zSlices = Table[ParametricPlot3D[{u, v, k}, {u, -5, 5}, {v, -5, 5}], {k, -4, 4, 2}]

-Graphics3D-

{⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃}

ySlices = Table[ParametricPlot3D[{u, k, v}, {u, -5, 5}, {v, -5, 5}], {k, -4, 4, 2}]

General :: spell1 : Possible spelling error: new symbol name \"ySlices\" is similar to existing symbol \"zSlices\".

-Graphics3D-

{⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃}

xSlices = Table[ParametricPlot3D[{k, u, v}, {u, -5, 5}, {v, -5, 5}], {k, -4, 4, 2}]

General :: spell : Possible spelling error: new symbol name \"xSlices\" is similar to existing symbols  {ySlices, zSlices} .

-Graphics3D-

{⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃}

Notice that in each of these, u and v are used as variables for graphing the surface, while k always stands for the variable that will get held constant for each surface.  Now, let's put all this together in a useful picture:

Show[xSlices, ySlices, zSlices]

-Graphics3D-

⁃Graphics3D⁃

Not surprisingly, we see that Cartesian coordinates can be thought of as a bunch of intersecting planes.  You locate points at the intersection of the three perpindicular planes.  This is boring, but makes for a good start.

A couple of things you should notice about how I set this up:  

I didn't let k run over as large a domain as I did u and v.  This was so that when I put everything together at the end (using the Show command), we could actually see the intersecting planes.  (If you let k run from -5 to 5, like u and v, all you see is a cube.  Perfectly correct, but not very informative.)  

You will probably have to experiment with the domain for u and v, as well as the range for k (and the number of slices) in different coordinate systems to come up with an informative final graph.

Let's do the same thing for Spherical coordinates.  Notices that it is very similar, though we have to convert to Cartesian coordinates before we can plot anything (ParametricPlot3D plots only in Cartesian).  Also, notice the use of Evaluate before the change of coordinates; this is just to stop Mathematica complaining (it compiles the function before plotting).

ρSlices = Table[ParametricPlot3D[Evaluate[CoordinatesToCartesian[{k, ϕ, θ}, Spherical]], {ϕ, 0, π}, {θ, π/6, (11π)/6}], {k, 1, 5, 2}]

-Graphics3D-

{⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃}

Notice that I didn't complete the spheres (I restricted θ).  I did this so that when we put the graphs together, you can get a "cut-away" view inside (see below).

θSlices = Table[ParametricPlot3D[Evaluate[CoordinatesToCartesian[{ρ, ϕ, k}, Spherical]], {ϕ, 0, π}, {ρ, 0, 6}], {k, π/3, (5π)/3, π/3}]

General :: spell1 : Possible spelling error: new symbol name \"θSlices\" is similar to existing symbol \"ρSlices\".

-Graphics3D-

{⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃}

ϕSlices = Table[ParametricPlot3D[Evaluate[CoordinatesToCartesian[{ρ, k, θ}, Spherical]], {θ, π/6, (11π)/6}, {ρ, 0, 6}], {k, 0, π, π/6}]

General :: spell : Possible spelling error: new symbol name \"ϕSlices\" is similar to existing symbols  {θSlices, ρSlices} .

-Graphics3D-

{⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃, ⁃Graphics3D⁃}

Show[ρSlices, θSlices, ϕSlices]

-Graphics3D-

⁃Graphics3D⁃

Again, you locate points at the intersection of the three types of surfaces (planes, spheres, and cones).

Problem:

Pick two other non-standard coordinate systems that Mathematica supports (listed in the Help Browser page I mention above; Cartesian, Cylindrical, and Spherical are all out) and generate visualizations of those systems like I did above.  Also, have Mathematica give you the formula to translate from these other coordinate systmes into Cartesian.  Choose your domains, etc., so that your final picture is easy to interpret.

Warning:  Some of the coordinate systems in the list seem to have more than 3 coordinates.  This isn't really true; the first 3 variables listed are always the coordinates.  The "extra" variables are actually constant parameters; you get a slightly different coordinate system for each choice of those parameters.  You can use the default values for each one for this exercise.


Created by Mathematica  (October 18, 2004)