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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:

enter image description here

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}]

enter image description here (this picture should not have a scratchy top, I don't why it does, please help) and:

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}]

enter image description here

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.

Please help. Thank You.

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1 Answer

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]]]]

Mathematica graphics

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 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 Sep 16 '13 at 5:11
1  
@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
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