RevolutionSurfaceLemniscateBernoulli::usage="The surface of revolution is based on the Lemniscate of Bernoulli with cartesian equation:​(​
2
x
+
2
y
2
)
= 2
2
a
(​
2
x
-
2
y
​)​; z = 0. And it will be rotated around the X axis in the first example.In the second example you will see the envelope of a family of spheres with center on the axis:ℒ: {2, -3, 5} + t {1, 1, 1}.And the envelope will have the shape of the revoution surface based on the Lemniscate of Bernoulli obtained in example 1.Reference: Personal.";
Example 1:
Animation:
In[]:=
ManipulateShowa=3;RevolutionPlot3Dx,
-
2
a
-
2
x
+
2
a
(
2
a
+4
2
x
)
,0,x,-3
2
,3
2
,{θ,0,θ1},RevolutionAxis{1,0,0},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",PlotPoints->60,​​ParametricPlot3D[t{1,0,0},{t,-5,5},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700,Style["Revolution surface based on the Lemniscate of Bernoulli:",Bold,Large],{{θ1,1,"Value (θ)"},0.1,2π,0.1},ControlPlacementTop
Out[]=
​
Revolution surface based on the Lemniscate of Bernoulli:
Value (θ)
Example 2:
In[]:=
Out[]=