Plot works while RevolutionPlot3D doesn't

Question

I am trying to draw a surface of revolution recovered from a differential equation with a singularity. In the simplest case, the answer is a sphere. You can check by hand. I manage to draw 2 dimensional plot but the 3 dimensional plot doesn't work.

e = 0.0001;
NDSolve[{Sqrt[1 + (u'[t])^2] == 1/(Sqrt[1 - t^2]), u[0.9999999] == 0}, u, {t, e, 1 - e}]
Plot[Evaluate[u[t] /. %], {t, e, 1 - e}, PlotRange -> All, AspectRatio -> Automatic]

--works

RevolutionPlot3D[Evaluate[u[t] /. %], {t, e, 1 - e}]

--doesn't work

What is the major difference? Do you have any idea how to fix it?

Thank you

Answer 1

The difference is that if you pass a list as the first argument to RevolutionPlot3D the list is interpreted as {f[x],f[z]} if the list has two elements, or as {f[x],f[y],f[z]} if it has three elements. On the contrary, Plot simply plots each function in the list separately. To fix it, you can do

e = 0.0001;
{u1,u2} = NDSolve[{Sqrt[1 + (u'[t])^2] == 1/(Sqrt[1 - t^2]), 
  u[0.9999999] == 0}, u, {t, e, 1 - e}];

and then either

RevolutionPlot3D[Evaluate[u[t] /. u1], {t, e, 1 - e}]

or

RevolutionPlot3D[Evaluate[u[t] /. u2], {t, e, 1 - e}]

The first one gives