How to highlight various parts of a plot

Question

I'd like to show students a graph of a function, say $(x,y)\mapsto\sqrt[3]{x^3+y^3}$, and then highlight its various properties connected with (non)differentiability at certain points etc. Having some lines included in the graph seems a good idea. Now I'd like to be able to highlight (by using different colors and/or thicker lines) some of these lines. It would be best if I could press or click something so that one or another line gets "selected" (highlighted). How to achieve something like this?

Note: I'll be fully satisfied with one-line, RTFM-style answers, if someone could provide me a hint where to look for the FM; I'm new to Mathematica and feel a bit overwhelmed by the multitude of documentation in various formats...

Answer 1

In addition to my comment, I guess I could also point out that Mathematica plots can be combined using the Show command. So you could define two different plots as follows and then combine them:

f[x_, y_] := (x^3 + y^3)^(1/3)

surface = Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}];

line = ParametricPlot3D[
   With[{x = t, y = t}, {x, y, f[x, y]}], {t, 0, 2}, 
   PlotStyle -> {Red, Tube[.03]}];

Show[surface, line]

Edit

Looks like you may also want to read about Manipulate, as shown in this example:

Manipulate[Show[surface,
  ParametricPlot3D[With[{x = t, y = t}, {x, y, f[x, y]}], {t, 0, 2}, 
   PlotStyle -> {style, Tube[.03]}]],
 {style, {Red, Directive[Opacity[.5], Blue]}}]

Here I just added the bare minimum to the previous example to make the color of the added line controllable by clicking a button.