ParametricPlot3D Question

Question

When I graph:

ParametricPlot3D[
  If[11*Cos[t] <= 5, 
    {9*Cos[s]*Sin[t], 10*Sin[s]*Sin[t], 
    11*Cos[t]}, Null], 
{s, 0, 2*Pi}, {t, 0, Pi}]

I get

but I want to get:

Why does my code generate those spike and etches on top of the partially cut ellipsoid, instead of giving a smooth top like the second picture?

Answer 1

Better solution

Actually, you don't need a different parametrization as I suggest below. I haven't really looked at your formula before which I should have done. The probably best solution is to transform your condition into explicit values for t like this:

Reduce[11*Cos[t] <= 5 && 0 <= t <= Pi, t]

(* ArcCos[5/11] <= t <= Pi *)

And now you remove your if condition and adjust your time interval

ParametricPlot3D[{9*Cos[s]*Sin[t], 10*Sin[s]*Sin[t], 11*Cos[t]}, 
 {s, 0, 2*Pi}, {t, ArcCos[5/11], Pi}]

That's far easier.

Quick Hack

ParametricPlot3D expects 3d point for all input values of s and t. What you do by returning Null is kind of rude and most likely confuses the algorithm which tries to construct a polygon surface.

I would really recommend that you transform your expression into a better parameterization, but for a quick hack it seems sufficient to not return Null but the point on the ring at z=5.

expr = {9*Cos[s]*Sin[t], 10*Sin[s]*Sin[t], 11*Cos[t]}; 
sol = Last@Solve[11*Cos[t] == 5, t];
With[{surf = expr, ring = expr /. sol},
 ParametricPlot3D[
  If[11*Cos[t] <= 5, surf, ring], {s, 0, 2*Pi}, {t, 0, Pi}]
]

Answer 2

You can also do this with RegionFunction:

ParametricPlot3D[{9*Cos[s]*Sin[t], 10*Sin[s]*Sin[t], 11*Cos[t]}, 
 {s, 0, 2*Pi}, {t, 0, Pi}, RegionFunction -> (11 Cos[#5] <= 5 &)]