Wolfram Cloud Document

DandelineSpheresEllipseCylinder::usage="Spheres tangent to a plane and a cylinder, their points of contact with the plane determine the foci of the ellipse in the space formed by the intersection of the plane and the cylinder. References: Wolfram MathWorld: Dandeline Spheres; Personal ";

In[]:=

Manipulatej[x_,y_]:=

x;γ[t_]:=Cos[t],Sin[t],

Cos[t];ζ[t_]:={Cos[t],Sin[t],2};

[t_]:={Cos[t],Sin[t],-2};d11[t_]:=NNorm-

,0,

+γ[t];d22[t_]:=NNorm--

,0,-

+γ[t];d33[t_]:=N[Norm[ζ[t]-γ[t]]];d44[t_]:=N[Norm[γ[t]-

[t]]];d3[t_]:=d11[t]+d22[t];d4[t_]:=d33[t]+d44[t];w=Grid[{{

,2a},{d11[t1],d22[t1],d3[t1]}},FrameAll,ItemSize->15,ItemStyle->15];o=Grid[{{

},{d33[t1],d44[t1],d4[t1]}},FrameAll,ItemSize->15,ItemStyle->15];ShowParametricPlot3D[{Cos[u],Sin[u],v},{u,0,2π},{v,-2,2},PlotStyleDirective[Green,Specularity[White,20],Opacity[0.1]],MeshFalse,PerformanceGoal"Quality"],ParametricPlot3D[{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]+2},{u,0,2π},{v,0,π},ColorFunction"LightTemperatureMap",PlotStyle->Directive[Opacity[0.7]],MeshFalse,PerformanceGoal"Quality"],ParametricPlot3D[{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]-2},{u,0,2π},{v,0,π},ColorFunction"LightTemperatureMap",PlotStyle->Directive[Opacity[0.7]],MeshFalse,PerformanceGoal"Quality"],Plot3D[j[x,y],{x,-5,5},{y,-5,5},PlotStyleDirective[Yellow,Opacity[0.1]],MeshFalse,PerformanceGoal"Quality"],ParametricPlot3D[ζ[θ],{θ,0,2π},PlotStyle{Green},PerformanceGoal"Quality"]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],ParametricPlot3D[

[θ],{θ,0,2π},PlotStyle{Green},PerformanceGoal"Quality"]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],ParametricPlot3D[γ[θ],{θ,0,t1},PlotStyle{Yellow},PerformanceGoal"Quality"]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],Graphics3DBlack,Ball

,0,

,0.04,Graphics3DBlack,Ball-

,0,-

,0.04,Graphics3D[{Yellow,Ball[N[γ[t1]],0.04]}],Graphics3D[{Green,Ball[N[ζ[t1]],0.04]}],Graphics3D[{Green,Ball[N[

[t1]],0.04]}],Graphics3DBlack,Dashed,Thick,LineN[γ[t1]],

,0,

,Graphics3DBlack,Dashed,Thick,LineN[γ[t1]],-

,0,-

,Graphics3D[{Green,Thick,Line[{ζ[t1],

[t1]}]}]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},MeshFalse,AxesTrue,BoxedFalse,PlotRange{{-2,2},{-2,2},{-3,3}},ViewPoint{1.3,-2.4,4.},ImageSize900,Style["Dandeline Spheres Ellipse Case (Cylinder):",Bold,Large],{{t1,0.00002,"Value (t)"},0.00001,2π,0.00001},Delimiter,w,2,"Vectors radii of Ellipse from point P to each focus, Sum",Delimiter,o,2,"Distance upper sphere-ellipse, Distance lower sphere-ellipse, Sum",ControlPlacementTop

Out[]=

Dandeline Spheres Ellipse Case (Cylinder):

Value (t)

Vectors radii of Ellipse from point P to each focus, Sum

PF 1	PF 2	2a
1.27415	2.72585	4.

Distance upper sphere-ellipse, Distance lower sphere-ellipse, Sum

QP	PT	PT + QP
1.27415	2.72585	4.