Using Piecewise in RevolutionPlot3D

Question

The documentation for RevolutionPlot3D shows that you can specify a parametric curve like this:

RevolutionPlot3D[{u, u}, {u, 0, 1}]

I'd like to do this with a Piecewise function for {fx,fz}:

RevolutionPlot3D[Piecewise[{{{u, u}, u < 0.5}, {{u, 0.5}, u >= 0.5}}], {u, 0,
   1}]

but I get messages and a blank plot:

Dot::rect: Nonrectangular tensor encountered.

whereas a Piecewise function for {fx,fz} works in ParametricPlot, for example:

ParametricPlot[Piecewise[{{{u, u}, u < 0.5}, {{u, 0.5}, u >= 0.5}}], {u, 0, 1}]

By writing Piecewise expressions for fx and fz separately, it works:

RevolutionPlot3D[{Piecewise[{{u, u < 0.5}, {u, u >= 0.5}}], 
    Piecewise[{{u, u < 0.5}, {0.5, u >= 0.5}}]}, {u, 0, 1}]

How is it possible to piece together a function for {fx,fz} in RevolutionPlot3D?

Answer 1

As stated in the first note under More Information of the documentation, the form RevolutionPlot3D[fz,{t,...}] is short form for RevolutionPlot3D[{t,0,fz},{t,...}].

To invoke the form you want, you need to call RevolutionPlot3D with a list of two elements, avoiding triggering the short form above:

pw[u_] := Piecewise[{{{u, u}, u < 0.5}, {{u, 0.5}, u >= 0.5}}]

RevolutionPlot3D[{pw[u][[1]], pw[u][[2]]}, {u, 0, 1}]

This misses that there is a discontinuity at 0.5, so it does not exclude it from the plot. You can get the original plot by adding Exclusions -> {u == 0.5}:

RevolutionPlot3D[{pw[u][[1]], pw[u][[2]]}, {u, 0, 1}, Exclusions -> {u == 0.5}]

Update per comment from J. M.:

Evaluate[Boole[u < 0.5] {u, u} + Boole[u >= 0.5] {u, 0.5}]

works as well, because it evaluates to:

{u Boole[u >= 0.5] + u Boole[u < 0.5], 0.5 Boole[u >= 0.5] + u Boole[u < 0.5]}

This does detect the discontinuity, but of course has the drawback that the input had to be rewritten from a Piecewise:

RevolutionPlot3D[Evaluate[Boole[u < 0.5] {u, u} + Boole[u >= 0.5] {u, 0.5}], {u, 0, 1}]