# ParametricPlot3D[] Question

I know this is very similar to a previous question I asked, but there are some differences. When I graph:

 ParametricPlot3D[
If[Part[RotationMatrix[-ArcCos[7/Sqrt[83]], {-5, 3, 0}] . {9*Cos[s]*
Sin[(Pi/7) t], 10*Sin[s]*Sin[(Pi/7) t],
11*Cos[(Pi/7) t]}, 3] < 21/Sqrt[83],
RotationMatrix[-ArcCos[7/Sqrt[83]], {-5, 3, 0}] . {9*Cos[s]*
Sin[(Pi/7) t], 10*Sin[s]*Sin[(Pi/7) t],
11*Cos[(Pi/7) t]}, {7*(Cos[s]) (7 - t)/7 + 5,
4*(Sin[s]) (7 - t)/7 + 5, t + 5}], {s, 0, 2 Pi}, {t, 0, (7/Pi)*Pi}]


I get:

But I want to get a clean combination of:

  ParametricPlot3D[
If[Part[RotationMatrix[-ArcCos[7/Sqrt[83]], {-5, 3, 0}] . {9*Cos[s]*
Sin[(Pi/7) t], 10*Sin[s]*Sin[(Pi/7) t],
11*Cos[(Pi/7) t]}, 3] < 21/Sqrt[83],
RotationMatrix[-ArcCos[7/Sqrt[83]], {-5, 3, 0}] . {9*Cos[s]*
Sin[(Pi/7) t], 10*Sin[s]*Sin[(Pi/7) t],
11*Cos[(Pi/7) t]}], {s, 0, 2 Pi}, {t, 0, (7/Pi)*Pi}]


ParametricPlot3D[{7*(Cos[s]) (7 - t)/7 + 5, 4*(Sin[s]) (7 - t)/7 + 5,
t + 5}, {s, 0, 2 Pi}, {t, 0, (7/Pi)*Pi}]


When I say combination, I want to be able to have an equation that represents the combination of both graphs which is able to manipulated (rotated, shifted,etc...). If I combine them using Show[] this is quite difficult, because Show[], is not an actual math equation that is manipulable. I especially want to be able to apply RotationMatrix[] to the resulting combination equation.

-

There is a very good reason for Mathematica to display a weird plot: your surfaces aren't connected:

r[s_, t_] :=
Part[RotationMatrix[-ArcCos[7/Sqrt[83]], {-5, 3, 0}].{9*Cos[s]*
Sin[(Pi/7) t], 10*Sin[s]*Sin[(Pi/7) t], 11*Cos[(Pi/7) t]}, 3];

Show[
ParametricPlot3D[{9*Cos[s]*Sin[(Pi/7) t], 10*Sin[s]*Sin[(Pi/7) t], 11*Cos[(Pi/7) t]},
{s, 0, 2 Pi}, {t, 0, (7/Pi)*Pi},
RegionFunction -> Function[{x, y, z, s, t}, r[s, t] < 21/Sqrt[83]]],
ParametricPlot3D[{7*(Cos[s]) (7 - t)/7 + 5, 4*(Sin[s]) (7 - t)/7 + 5, t + 5},
{s, 0, 2 Pi}, {t, 0, (7/Pi)*Pi},
RegionFunction -> Function[{x, y, z, s, t}, r[s, t] > 21/Sqrt[83]]]]


You should work out the math.

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Why does the fact that my surfaces are disconnected, make it weird for Mathematica to graph? – user9197 Sep 16 '13 at 5:04
@user9197 Because Mma tries to connect them ... – belisarius has settled Sep 16 '13 at 5:07
Oh. Is there anyway to not use Show[] and implicitly graph the two surfaces without having Mathematica connected them? – user9197 Sep 16 '13 at 5:08
@user9197 Yes, but ... why don't you want to use Show[]? It is the natural way ... – belisarius has settled Sep 16 '13 at 5:11
@user9197 No need to complicate the question. Rotate[#, Pi, {1, 0, 0}] &@First@show // Graphics3D. Where show = Show[g1,g2] – Kuba Sep 16 '13 at 6:00