SpecialToroid::usage="The plane y = x intersects to the sphere (x - 5
2
)
+ (y - 5
2
)
+
2
z
= 4 and the resulting circumference rotates around the line x = y = z. A very special inclined toroid is generated. Reference: Personal.";
Development:
In[]:=
Solve[{y==x,
2
(x-5)
+
2
(y-5)
+
2
z
==4},{x,y}]
Out[]=
x
1
2
10-
2
4-
2
z
,y
1
2
10-
2
4-
2
z
,x
1
2
10+
2
4-
2
z
,y5+
4-
2
z
2

In[]:=
ManipulateShowRevolutionPlot3D
1
2
10-
2
4-
2
z
,
1
2
10-
2
4-
2
z
,z,
1
2
10+
2
4-
2
z
,5+
4-
2
z
2
,z,{z,-2,2},{θ,0,θ1},RevolutionAxis{1,1,1},PlotStyle->Opacity[0.6],MeshFalse,PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",​​ParametricPlot3D[{z,z,z},{z,-8,8},ColorFunction->"BlueGreenYellow"],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-1,3,2.},ImageSize700,Style["Special toroid:",Bold,Large],{{θ1,0.00002,"Value (θ)"},0.00001,2π,0.00001},ControlPlacementTop
Out[]=
​
Special toroid:
Value (θ)