BicornPlaneSurfaceRevolution::usage="Given the bicorn  2 Sin[t]​, 2
2
Cos[t]
(2+Cos[t])
3+
2
Sin[t]
​, 0 follow these steps to obtain an interesting surface of revolution: 1 Project it orthogonally on the plane 3x - 2y + z = 0. 2 Stirn it with respect to the axis t{0, 1, 2}. 3 You get an interesting surface of revolution. Reference: Personal. ";
Step (1):
In[]:=
ParametricPlot3D2Sin[t],2
2
Cos[t]
(2+Cos[t])
3+
2
Sin[t]
,0,{t,-π,π},BoxedFalse,AxesTrue,AxesOrigin->{0,0,0},AxesLabel->{x,y,z},BoxRatios->Automatic
Out[]=
In[]:=
Solve3x-2y+z==0,
x-2Sin[t]
3
==
y-2
2
Cos[t]
(2+Cos[t])
3+
2
Sin[t]
-2
==
z
1
,{x,y,z}
x-
-12
2
Cos[t]
-6
3
Cos[t]
-15Sin[t]-5
3
Sin[t]
7(3+
2
Sin[t]
)
,y
2(10
2
Cos[t]
+5
3
Cos[t]
+9Sin[t]+3
3
Sin[t]
)
7(3+
2
Sin[t]
)
,z-
-4
2
Cos[t]
-2
3
Cos[t]
+9Sin[t]+3
3
Sin[t]
7(3+
2
Sin[t]
)
;
In[]:=
ShowParametricPlot3D-
-12
2
Cos[t]
-6
3
Cos[t]
-15Sin[t]-5
3
Sin[t]
7(3+
2
Sin[t]
)
,
2(10
2
Cos[t]
+5
3
Cos[t]
+9Sin[t]+3
3
Sin[t]
)
7(3+
2
Sin[t]
)
,-
-4
2
Cos[t]
-2
3
Cos[t]
+9Sin[t]+3
3
Sin[t]
7(3+
2
Sin[t]
)
,{t,-π,π},PlotStyle->{Black,Thick},ContourPlot3D[3x-2y+z==0,{x,-2,12},{y,-2,2},{z,-2,2},ContourStyle{Yellow,Opacity[0.2]}],ParametricPlot3D[t{0,1,2},{t,-1,1},PlotStyle{Green,Thick}],AxesLabel->{x,y,z},AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,-2.4,2.},ImageSize500
Out[]=
Step (2):
In[]:=
ManipulateShowRevolutionPlot3D-
-12
2
Cos[t]
-6
3
Cos[t]
-15Sin[t]-5
3
Sin[t]
7(3+
2
Sin[t]
)
,
2(10
2
Cos[t]
+5
3
Cos[t]
+9Sin[t]+3
3
Sin[t]
)
7(3+
2
Sin[t]
)
,-
-4
2
Cos[t]
-2
3
Cos[t]
+9Sin[t]+3
3
Sin[t]
7(3+
2
Sin[t]
)
,{t,-π,π},{θ,0,θ1},RevolutionAxis{0,1,2},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",PlotPoints->60,​​ParametricPlot3D[t{0,1,2},{t,-1,1},ColorFunction->"BlueGreenYellow",PlotStyleThick],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,-2.4,2.},ImageSize500,Style["Surface of revolution generated by a bicorn:",Bold,Large],{{θ1,1,"Value (θ)"},0.00001,2π,0.00001},ControlPlacementTop
Out[]=
​
Surface of revolution generated by a bicorn:
Value (θ)
Step (3): Watch the animation.