Steiner'schainhyperbolacase::usage="It was obtained from a symmetrical Steiner chain of 6 spheres by inversion.Reference: Wolfram MathWorld: Steiner chain, Inversion; Personal ";
Desarrollo:
The inversion sphere is centered in {7, 6, 4}, with radius k = 4 and the inversion distance was d = - 3. The center of inversion was chosen outside the minor sphere, so that the centers of the spheres of the chain are on a hyperbola in space.
In[]:=
​​Manipulate​​Showr=2;k=4;d=-3;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[0.8],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[0.3],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
,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,ImageSize900,PlotRange70,ViewPoint{-1.3,-2.4,5.},Style["Steiner's chain hyperbola case:",Bold,Medium],Delimiter,{{k1,0,"Valor (θ)"},0,6,0.2},ControlPlacementTop
Out[]=
​
Steiner's chain hyperbola case:
Valor (θ)
Explanation in 2D:
In[]:=
Manipulate​​Showr=2;k=4;d=-3;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;​​GraphicsBlue,Thick,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,Thick,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,Thick,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[0.005],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},t1,0,2π,
2π
n
,ParametricPlot
128
90
Cosh[t]+
139
45
,
64
3
15
Sinh[t]+6,{t,-π,π},PlotStyle->{Black,Thick},ParametricPlot-
128
90
Cosh[t]+
139
45
,
64
3
15
Sinh[t]+6,{t,-π,π},PlotStyle->{Black,Thick},GraphicsBlue,PointSize[0.005],Point
79
9
,6,GraphicsDarkYellow,PointSize[0.005],Point-
13
5
,6,AxesLabel->(Style[#,15,Blue]&/@{"X","Y"}),AxesOrigin{0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,ImageSize900,PlotRange70,Style["Steiner's chain hyperbola case:",Bold,Medium],Delimiter,{{k1,0,"Valor (θ)"},0,6,0.2},ControlPlacementTop
Out[]=
​
Steiner's chain hyperbola case:
Valor (θ)