RevolutionSurfaceCircle::usage="The surface of revolution is based on a circle with radius r = 2 and center C{4, 4, -6} and it's on the plane x + 2 y + 2 z = 0. The circle will rotate around the line ℒ: t {4, -1, -1} to generate the toroid.Reference: Personal.";
Development:
Parametric equations of the circumference:
x=-2y-2z;​​y=u;​​z=v;​​{-2,1,0}{1,2,2}
In[]:=
{2,4,-5};​​{4,4,-6}+
2Cos[t]
Norm[{-2,1,0}]
{-2,1,0}+
2Sin[t]
Norm[{2,4,-5}]
{2,4,-5}
4-
4Cos[t]
5
+
4Sin[t]
3
5
,4+
2Cos[t]
5
+
8Sin[t]
3
5
,-6-
2
3
5
Sin[t];
In[]:=
Showζ[t_]:=4-
4Cos[t]
5
+
4Sin[t]
3
5
,4+
2Cos[t]
5
+
8Sin[t]
3
5
,-6-
2
3
5
Sin[t];{ParametricPlot3D[ζ[t],{t,0,2π},PlotStyle->{Black,Thick}],ContourPlot3D[x+2y+2z==0,{x,-6,6},{y,-6,6},{z,-10,10},ContourStyle{Yellow,Opacity[0.2]}],ParametricPlot3D[t{4,-1,-1},{t,-1,1},PlotStyle{Green,Thick}]},AxesLabel->{x,y,z},AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{15,15,10.},ImageSize500
Out[]=
Animation:
In[]:=
Manipulate​​ShowRevolutionPlot3D4-
4Cos[t]
5
+
4Sin[t]
3
5
,4+
2Cos[t]
5
+
8Sin[t]
3
5
,-6-
2
3
5
Sin[t],{t,0,2π},{θ,0,θ1},RevolutionAxis{4,-1,-1},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",PlotPoints->60,​​ParametricPlot3D[t{4,-1,-1},{t,-2,2},ColorFunction->"BlueGreenYellow",PlotStyleThick],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{15,15,10},ImageSize700,Style["Toroid:",Bold,Large],{{θ1,1,"Value (θ)"},0.1,2π,0.1},ControlPlacementTop
Out[]=
​
Toroid:
Value (θ)