Steiner'schainParabolacase::usage="It was obtained from a symmetrical Steiner chain of 6 spheres by inversion.References: Wolfram MathWorld: Steiner chain, Inversion; ResourceFunction: InversionSurface, InversionCurve; Personal ";
Development:
The inversion sphere is centered in {6, 6, 4}, with radius k = 4 . The center of inversion was chosen over the minor sphere, so that the centers of the spheres of the chain are on a parabola in space.
In[]:=
​​Manipulate​​Showr=2;k=4;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;z1=4;​​ParametricPlot3D6+
16(-6+x1+(r+2ρ)Cos[u]Sin[v])
2
(-4+z1+(r+2ρ)Cos[v])
+
2
(-6+x1+(r+2ρ)Cos[u]Sin[v])
+
2
(-6+y1+(r+2ρ)Sin[u]Sin[v])
,6+
16(-6+y1+(r+2ρ)Sin[u]Sin[v])
2
(-4+z1+(r+2ρ)Cos[v])
+
2
(-6+x1+(r+2ρ)Cos[u]Sin[v])
+
2
(-6+y1+(r+2ρ)Sin[u]Sin[v])
,4+
16(-4+z1+(r+2ρ)Cos[v])
2
(-4+z1+(r+2ρ)Cos[v])
+
2
(-6+x1+(r+2ρ)Cos[u]Sin[v])
+
2
(-6+y1+(r+2ρ)Sin[u]Sin[v])
,{u,0,2π},{v,0,π},PlotStyleDirective[Yellow,Specularity[White,40],Opacity[0.8]],Mesh->None,PerformanceGoal->"Quality",PlotPoints->60,ParametricPlot3D6+
16(-6+x1+rCos[u]Sin[v])
2
(-4+z1+rCos[v])
+
2
(-6+x1+rCos[u]Sin[v])
+
2
(-6+y1+rSin[u]Sin[v])
,6+
16(-6+y1+rSin[u]Sin[v])
2
(-4+z1+rCos[v])
+
2
(-6+x1+rCos[u]Sin[v])
+
2
(-6+y1+rSin[u]Sin[v])
,4+
16(-4+z1+rCos[v])
2
(-4+z1+rCos[v])
+
2
(-6+x1+rCos[u]Sin[v])
+
2
(-6+y1+rSin[u]Sin[v])
,{u,0,2π},{v,0,π},PlotStyleDirective[Blue,Specularity[White,40],Opacity[0.3]],MeshNone,PerformanceGoal->"Quality",PlotPoints->60,​​TableParametricPlot3D6+(16(-6+x1+(r+ρ)Cos[2k1π+t1]+ρCos[u]Sin[v]))(
2
(-4+z1+ρCos[v])
+
2
(-6+x1+(r+ρ)Cos[2k1π+t1]+ρCos[u]Sin[v])
+
2
(-6+y1+(r+ρ)Sin[2k1π+t1]+ρSin[u]Sin[v])
),6+(16(-6+y1+(r+ρ)Sin[2k1π+t1]+ρSin[u]Sin[v]))(
2
(-4+z1+ρCos[v])
+
2
(-6+x1+(r+ρ)Cos[2k1π+t1]+ρCos[u]Sin[v])
+
2
(-6+y1+(r+ρ)Sin[2k1π+t1]+ρSin[u]Sin[v])
),4+(16(-4+z1+ρCos[v]))(
2
(-4+z1+ρCos[v])
+
2
(-6+x1+(r+ρ)Cos[2k1π+t1]+ρCos[u]Sin[v])
+
2
(-6+y1+(r+ρ)Sin[2k1π+t1]+ρSin[u]Sin[v])
),{u,0,2π},{v,0,π},PlotStyleDirective[Purple,Specularity[White,40],Opacity[0.8]],MeshNone,PerformanceGoal->"Quality",PlotPoints->60,t1,0,2π,
2π
n
,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,ImageSize900,PlotRange50,ViewPoint{-1.3,-2.4,5.},Style["Steiner's chain Parabola case:",Bold,Medium],Delimiter,{{k1,0,"Valor (θ)"},0,6,0.2},ControlPlacementTop
Out[]=
​
Steiner's chain Parabola case:
Valor (θ)
The blue sphere is transformed into a plane. At a certain point one of the spheres of the chain is very large and does not look complete. Therefore the centers of the spheres of the chain are on a parabola in space.
Explanation in 2D:
In[]:=
Manipulate​​Showr=2;k=4;d=-2;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;z1=4;​​ParametricPlot6+
16(-2+6Cos[t])
2
(-2+6Cos[t])
+36
2
Sin[t]
,6+
96Sin[t]
2
(-2+6Cos[t])
+36
2
Sin[t]
,{t,0,2π},PlotStyle{DarkYellow,Thick},ParametricPlot6+
16(-2+2Cos[t])
2
(-2+2Cos[t])
+4
2
Sin[t]
,6+
32Sin[t]
2
(-2+2Cos[t])
+4
2
Sin[t]
,{t,0,2π},PlotStyle{Blue,Thick},​​TableParametricPlot6+
16(-2+2Cos[t]+4Cos[2k1π+t1])
2
(-2+2Cos[t]+4Cos[2k1π+t1])
+
2
(2Sin[t]+4Sin[2k1π+t1])
,6+
16(2Sin[t]+4Sin[2k1π+t1])
2
(-2+2Cos[t]+4Cos[2k1π+t1])
+
2
(2Sin[t]+4Sin[2k1π+t1])
,{t,0,2π},PlotStyle{Purple,Thick},t1,0,2π,
2π
n
,TableGraphicsPurple,PointSize[0.005],Point{4-d,y1}+
2
k
2
((r+ρ)Cos[t1+2πk1]+x1-(4-d))
+
2
((r+ρ)Sin[t1+2πk1]+y1-6)
-
2
ρ
{(r+ρ)Cos[t1+2πk1]+x1-(4-d),(r+ρ)Sin[t1+2πk1]+y1-6},t1,0.00001,2π,
2π
n
,Graphics[{DarkYellow,PointSize[0.005],Point[{7,6}]}],ContourPlot[
2
y
-16x-12y+84==0,{x,0,30},{y,-20,30},ContourStyle->{Black,Thick},PlotPoints->50],AxesLabel->(Style[#,15,Blue]&/@{"X","Y"}),AxesOrigin{0,0},AxesTrue,BoxedFalse,AspectRatioAutomatic,ImageSize900,PlotRange20,Style["Steiner's chain Parabola case:",Bold,Medium],Delimiter,{{k1,0,"Valor (θ)"},0,6,0.2},ControlPlacementTop
Out[]=
​
Steiner's chain Parabola case:
Valor (θ)
The blue circle becomes a straight line (it is not the parabola’s directrix). The yellow circle is observed with its center (which is the focus of the parabola). The centers of the circles of the chain are on the parabola.