Steiner'sChainEllipseCase::usage="It was obtained from a symmetrical Steiner chainof 6 spheres by inversion.References: Wolfram MathWorld: Steiner chain, Inversion; Personal ";
Development:
The inversion sphere is with center in {3, 6, 4}, which is within the minor sphere so that the centers of the spheres of the chain are on an ellipse in space. The inversion radius is k = 2 and the inversion distance was
d = 1.
In[]:=
​​Manipulate​​Showr=2;k=2;d=1;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;z1=4;​​Graphics3DBlue,Specularity[White,40],Opacity[0.8],Sphere{x1-d,y1,z1}+
2
k
2
d
+
2
(y1-6)
+
2
(z1-4)
-
2
(r+2ρ)
{d,0,0},Abs
2
k
2
d
+
2
(y1-6)
+
2
(z1-4)
-
2
(r+2ρ)
(r+2ρ),Graphics3DYellow,Specularity[White,40],Opacity[0.4],Sphere{x1-d,y1,z1}+
2
k
2
d
+
2
(y1-6)
+
2
(z1-4)
-
2
r
{d,0,0},Abs
2
k
2
d
+
2
(y1-6)
+
2
(z1-4)
-
2
r
r,​​TableGraphics3DPurple,Specularity[White,40],Opacity[0.8],Sphere{4-d,y1,z1}+
2
k
2
((r+ρ)Cos[t1+2πk1]+x1-(4-d))
+
2
((r+ρ)Sin[t1+2πk1]+y1-6)
+
2
(z1-4)
-
2
ρ
{(r+ρ)Cos[t1+2πk1]+x1-(4-d),(r+ρ)Sin[t1+2πk1]+y1-6,0},Abs
2
k
2
((r+ρ)Cos[t1+2πk1]+x1-(4-d))
+
2
((r+ρ)Sin[t1+2πk1]+y1-6)
+
2
(z1-4)
-
2
ρ
ρ,t1,0,2π,
2π
n
,ParametricPlot3DAbs
2
k
2
d
+
2
(y1-6)
+
2
(z1-4)
-
2
r
rCos[θ]+x1-d+
2
k
2
d
+
2
(y1-6)
+
2
(z1-4)
-
2
r
d,Abs
2
k
2
d
+
2
(y1-6)
+
2
(z1-4)
-
2
r
rSin[θ]+y1,z1,{θ,0,2π},PlotStyle->{Black,Thick},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,ImageSize900,ViewPoint{1.3,-2.4,4.},Style["Steiner's chain Ellipse Case:",Bold,Medium],Delimiter,{{k1,0,"Valor (θ)"},0,6,0.2},ControlPlacementTop
Out[]=
​
Steiner's chain Ellipse Case:
Valor (θ)
Explanation in 2D:
In[]:=
​​Manipulate​​Showr=2;k=2;d=1;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;​​​​ParametricPlot
176
105
Cos[θ]+
239
105
,
16
105
Sin[θ]+6,{θ,0,2π},ColorFunction->"DarkRainbow",​​GraphicsBlue,Circle{x1-d,y1}+
2
k
2
d
+
2
(y1-6)
-
2
(r+2ρ)
{d,0},Abs
2
k
2
d
+
2
(y1-6)
-
2
(r+2ρ)
(r+2ρ),GraphicsDarkYellow,Circle{x1-d,y1}+
2
k
2
d
+
2
(y1-6)
-
2
r
{d,0},Abs
2
k
2
d
+
2
(y1-6)
-
2
r
r,​​TableGraphicsPurple,Circle{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},Abs
2
k
2
((r+ρ)Cos[t1+2πk1]+x1-(4-d))
+
2
((r+ρ)Sin[t1+2πk1]+y1-6)
-
2
ρ
ρ,t1,0,2π,
2π
n
,​​​​TableGraphicsPurple,PointSize[0.01],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},t1,0,2π,
2π
n
,GraphicsBlue,PointSize[0.01],Point
101
35
,6,GraphicsDarkYellow,PointSize[0.01],Point
5
3
,6,AxesTrue,BoxRatiosAutomatic,AxesOrigin{0,0},ImageSize700,PlotRangeAll,Style["Steiner's Chain Ellipse Case:",Bold,Medium],{{k1,0,"Valor (k1)"},0,6,0.2},ControlPlacementTop
Out[]=
​
Steiner's Chain Ellipse Case:
Valor (k1)