SteinersPorism::usage=" Steiner's Porism, says that given two non-concentric inner circumferences (spheres)​, regardless of where we start, either we always get a Steiner chain or we will never achieve it. A Steiner chain is a succession of circumferences tangent to the two initials and two of the sought after, surrounding the inside of the initials. Reference: Weisstein Eric W. Steiner's Porism from MathWorld; Personal. ";
2D
In[]:=
​​SwatchLegend{Purple,DarkGreen,Orange},"Chain starting at 0","Chain starting at
π
4
","Chain starting at
π
6
"​​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DarkGreen,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,
π
4
,2π+
π
4
,
2π
n
,​​​​TableGraphicsOrange,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,
π
6
,2π+
π
6
,
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
,TableGraphicsDarkGreen,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,
π
4
,2π+
π
4
,
2π
n
,TableGraphicsOrange,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,
π
6
,2π+
π
6
,
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 Porism:",Bold,Medium],{{k1,0,"Valor (k1)"},0,6,0.2},ControlPlacementTop
Out[]=
Chain starting at 0
Chain starting at
π
4
Chain starting at
π
6
Out[]=
​
Steiner's Porism:
Valor (k1)
In this animation, Steiner’s porism is verified. We can also observe that the centers of the circumferences (spheres) of each chain are on an ellipse. On the other hand, the centers of the base circumferences (spheres) , outer and inner circles (spheres), to which and every one of the circles (spheres) of the chains are tangent, are the foci of the ellipse.
3D
In[]:=