ParametricPlot3D of Boy's surface

Question

I'm trying to visualize Boy's surface using Bryant's parametrization, as per the MathWorld article. However, I'm not sure I understand the parametrization, and I don't know how to implement it when the parameter is complex.

z := u + vI
g1 := -(3/2) Im[(z (1 - z^4))/(z^6 + z^3 √5 - 1)]
g2 := -(3/2) Re[(z (1 + z^4))/(z^6 + z^3 √5 - 1)]
g3 := Im[(1 + z^6)/(z^6 + z^3 √5 - 1)] - 1/2
g = g1^2 + g2^2 + g3^2

ParametricPlot3D[{g1/g, g2/g, g3/g}, {u, 0, 0.5}, {v, 0, 0.5}]

Now, I understand that $z=u+v\mathrm{i}$ is the parameter, and that it must range over the closed unit disc, so $u^2+v^2\leq1$. But I don't know how do do this! (I let $u,v$ range between 0 and 0.5, for good measure, but the graphic appears empty.)

In the end, I want to end up with exactly this image, for which I've been unable to find the original code.

Edit: The 3 solutions provided here worked like a charm (with the caveat that the one using polar coordinates was weirdly sharped, and I couldn't fix that, but I'm a Mathematica newb), and I found another method:

ParametricPlot3D[Boys[u, v], {u, v} ∈ Disk[{0, 0}, 1]]

I don't know why, but using RegionFunction the surface rendered ~30% slower, and, using Piecewise, about %5 slower. The only issue is that the mesh is now weirdly triangulated.

Also, I compiled the actual parametric formula and performance increased threefold:

Boys = Compile[{u, 
v}, {(-(3/2)
    Im[((u + v I) (1 - (u + v I)^4))/((u + 
      v I)^6 + (u + v I)^3 Sqrt[5] - 1.)])/((-(3/2)
     Im[((u + v I) (1 - (u + v I)^4))/((u + 
       v I)^6 + (u + v I)^3 Sqrt[5] - 1.)])^2 + (-(3/2)
     Re[((u + v I) (1 + (u + v I)^4))/((u + 
       v I)^6 + (u + v I)^3 Sqrt[5] - 1.)])^2 + (Im[(
    1 + (u + v I)^6)/((u + v I)^6 + (u + v I)^3 Sqrt[5] - 1.)] - 
   1/2)^2), (-(3/2)
    Re[((u + v I) (1 + (u + v I)^4))/((u + 
      v I)^6 + (u + v I)^3 Sqrt[5] - 1.)])/((-(3/2)
     Im[((u + v I) (1 - (u + v I)^4))/((u + 
       v I)^6 + (u + v I)^3 Sqrt[5] - 1.)])^2 + (-(3/2)
     Re[((u + v I) (1 + (u + v I)^4))/((u + 
       v I)^6 + (u + v I)^3 Sqrt[5] - 1.)])^2 + (Im[(
    1 + (u + v I)^6)/((u + v I)^6 + (u + v I)^3 Sqrt[5] - 1.)] - 
   1/2)^2), (Im[(
   1 + (u + v I)^6)/((u + v I)^6 + (u + v I)^3 Sqrt[5] - 1.)] - 1/
  2)/((-(3/2)
     Im[((u + v I) (1 - (u + v I)^4))/((u + 
       v I)^6 + (u + v I)^3 Sqrt[5] - 1.)])^2 + (-(3/2)
     Re[((u + v I) (1 + (u + v I)^4))/((u + 
       v I)^6 + (u + v I)^3 Sqrt[5] - 1.)])^2 + (Im[(
    1 + (u + v I)^6)/((u + v I)^6 + (u + v I)^3 Sqrt[5] - 1.)] - 
   1/2)^2)}, Parallelization -> True];

I also tried compiling each of $g1,g2,g3,g$ separately, but performance was a bit worse.

Answer 1

z := u + v I
g1 := -(3/2) Im[(z (1 - z^4))/(z^6 + z^3 Sqrt[5] - 1)]
g2 := -(3/2) Re[(z (1 + z^4))/(z^6 + z^3 Sqrt[5] - 1)]
g3 := Im[(1 + z^6)/(z^6 + z^3 Sqrt[5] - 1)] - 1/2
g = g1^2 + g2^2 + g3^2;

You could use Piecewise here.

When Abs[z] > 1, the function is undefined and plots nothing.

ParametricPlot3D[Piecewise[{{{g1/g, g2/g, g3/g}, Abs[z] <= 1}}, Undefined],
                 {u, -1, 1}, {v, -1, 1}]

I'm not an expert on Graphics, but I hope that helps.

Answer 2

Altho traditionally one should use polar coordinates as I suggested in a comment above, I found it amusing that ParametricPlot3D[] can actually support a region specification for specifying the domain of the parameters. Thus here is a way to plot the Bryant-Kusner parametrization of the Boy surface:

ParametricPlot3D[With[{z = u + I v}, (#/(#.#)) &[Re[{0, 0, 1/2} +
                      {I z (1 - z^4), z (z^4 + 1), -(2 I/3) (z^6 + 1)}/
                      (z^6 + Sqrt[5] z^3 - 1)]]], {u, v} ∈ Disk[],
                 Axes -> None, Boxed -> False, Mesh -> True,
                 PlotPoints -> {20, 30}, PlotStyle -> Opacity[2/3]]

where I made the surface translucent to display the inner structure. Mathematically, the construction is an inversion of a certain minimal surface that is topologically equivalent to the projective plane, thrice-punctured. See the linked paper for more background.