TangerineFamilyEnvelope::usage="References: A.S. Mishchenko, Yu. P. Solovyev, A.T. Fomenko, Problems in Differential Geometry and Topology. Mir Publishers: Moscow, p. 39, 1985; Personal.";
Development:
In[]:=
Solve[++==,(++-)==0,{x,y}]
2
(x-bCos[θ])
2
(y-bSin[θ])
2
z
2
a
∂
θ
2
(x-bCos[θ])
2
(y-bSin[θ])
2
z
2
a
In[]:=
Simplifyx+,ySec[θ]+++-+,x+,ySec[θ]+++++
b+bCos[θ]--+-
3
Cos[θ]
2
Sin[θ]
2
a
4
Cos[θ]
2
z
4
Cos[θ]
2
a
2
Cos[θ]
2
Sin[θ]
2
z
2
Cos[θ]
2
Sin[θ]
2
Cos[θ]
2
Sin[θ]
bSin[θ]
3
Cos[θ]
2
Cos[θ]
2
Sin[θ]
bCos[θ]
3
Sin[θ]
2
Cos[θ]
2
Sin[θ]
Sin[θ]
(-)(+)
2
a
2
z
2
Cos[θ]
2
Cos[θ]
2
Sin[θ]
2
Cos[θ]
2
Sin[θ]
b+bCos[θ]+-+-
3
Cos[θ]
2
Sin[θ]
2
a
4
Cos[θ]
2
z
4
Cos[θ]
2
a
2
Cos[θ]
2
Sin[θ]
2
z
2
Cos[θ]
2
Sin[θ]
2
Cos[θ]
2
Sin[θ]
bSin[θ]
3
Cos[θ]
2
Cos[θ]
2
Sin[θ]
bCos[θ]
3
Sin[θ]
2
Cos[θ]
2
Sin[θ]
Sin[θ]
(-)(+)
2
a
2
z
2
Cos[θ]
2
Cos[θ]
2
Sin[θ]
2
Cos[θ]
2
Sin[θ]
Out[]=
xbCos[θ]-
(-)
,ybSin[θ]-2
a
2
z
2
Cos[θ]
(-)
Tan[θ],xbCos[θ]+2
a
2
z
2
Cos[θ]
(-)
,ybSin[θ]+2
a
2
z
2
Cos[θ]
(-)
Tan[θ]2
a
2
z
2
Cos[θ]
Characteristic:
bCos[θ]-Cos[θ]-,bSin[θ]--Sin[θ],z;bCos[θ]+Cos[θ]-,bSin[θ]+-Sin[θ],z;
2
a
2
z
2
a
2
z
2
a
2
z
2
a
2
z
Envelope:
In[]:=
Eliminatex==bCos[θ]+
(-)
,y==bSin[θ]+2
a
2
z
2
Cos[θ]
(-)
Tan[θ],z==z,θ2
a
2
z
2
Cos[θ]
Out[]=
(-++++)(-x-3x++x+x)
2
a
2
b
2
x
2
y
2
z
2
a
2
b
3
x
2
y
2
z
2b(--)
2
a
2
b
2
z
2
x
2
y
2
(-2-2++2+2)
2
a
2
b
2
x
2
y
2
z
2
(-x-3x++x+x)
2
a
2
b
3
x
2
y
2
z
4
2
b
2
(--)
2
a
2
b
2
z
2
(-2+-2-2+-2+2+2+)
4
a
2
a
2
b
4
b
2
a
2
y
2
b
2
y
4
y
2
a
2
z
2
b
2
z
2
y
2
z
4
z
2
(-x-3x++x+x)
2
a
2
b
3
x
2
y
2
z
4
2
b
2
(--)
2
a
2
b
2
z
2
x
2
a
2
b
2
x
2
y
2
z
4
a
2
a
2
b
4
b
2
a
2
y
2
b
2
y
4
y
2
a
2
z
2
b
2
z
2
y
2
z
4
z
x(-2-2++2+2)(-x-3x++x+x)
2
y
2
a
2
b
2
x
2
y
2
z
2
a
2
b
3
x
2
y
2
z
(--)(-2+-2-2+-2+2+2+)
2
a
2
b
2
z
4
a
2
a
2
b
4
b
2
a
2
y
2
b
2
y
4
y
2
a
2
z
2
b
2
z
2
y
2
z
4
z
2
a
2
b
2
x
2
y
2
z
x(-x-3x++x+x)
2
a
2
b
3
x
2
y
2
z
2
a
2
b
2
z
xy(-+++)(-2-2++2+2)(-x-3x++x+x)
2
a
2
b
2
y
2
z
2
a
2
b
2
x
2
y
2
z
2
a
2
b
3
x
2
y
2
z
2b(--)(-2+-2-2+-2+2+2+)
2
a
2
b
2
z
4
a
2
a
2
b
4
b
2
a
2
y
2
b
2
y
4
y
2
a
2
z
2
b
2
z
2
y
2
z
4
z
x(-x-3x++x+x)
2
a
2
b
3
x
2
y
2
z
2b(--)
2
a
2
b
2
z
x(-2-2++2+2)
2
y
2
a
2
b
2
x
2
y
2
z
2
(-x-3x++x+x)
2
a
2
b
3
x
2
y
2
z
4(-2+-2-2+-2+2+2+)
2
b
2
(--)
2
a
2
b
2
z
4
a
2
a
2
b
4
b
2
a
2
y
2
b
2
y
4
y
2
a
2
z
2
b
2
z
2
y
2
z
4
z
2
(-x-3x++x+x)
2
a
2
b
3
x
2
y
2
z
4
2
b
2
(--)
2
a
2
b
2
z
4
x
2
x
2
a
2
b
2
y
2
z
4
a
2
a
2
b
4
b
2
a
2
y
2
b
2
y
4
y
2
a
2
z
2
b
2
z
2
y
2
z
4
z
x(-x-3x++x+x)
2
a
2
b
3
x
2
y
2
z
2
a
2
b
2
z
(-++)
2
a
2
b
2
z
2
(-x-3x++x+x)
2
a
2
b
3
x
2
y
2
z
4
2
b
2
(--)
2
a
2
b
2
z
2
x
4
x
2
x
2
a
2
b
2
y
2
z
4
a
2
a
2
b
4
b
2
a
2
y
2
b
2
y
4
y
2
a
2
z
2
b
2
z
2
y
2
z
4
z
Recoil edge:
In[]:=
Solve++==,(++-)==0,(++-)==0,{x,y,z}
2
(x-bCos[θ])
2
(y-bSin[θ])
2
z
2
a
∂
θ
2
(x-bCos[θ])
2
(y-bSin[θ])
2
z
2
a
∂
θ,θ
2
(x-bCos[θ])
2
(y-bSin[θ])
2
z
2
a
Out[]=
x0,y0,z---,x0,y0,z--
2
a
2
b
2
Cos[θ]
2
b
2
Sin[θ]
2
a
2
b
2
Cos[θ]
2
b
2
Sin[θ]
0,0,--;0,0,-;
2
a
2
b
2
a
2
b
In[]:=
ManipulateClearAll;a=3;b=2;ShowTableParametricPlot3D[{aCos[u]Sin[v]+bCos[θ],aSin[u]Sin[v]+bSin[θ],aCos[v]},{u,0,2π},{v,0,π},PlotStyleDirective[Orange,Opacity[0.3],Specularity[White,30]],MeshFalse,PerformanceGoal"Quality"],θ,0,t1,,TableParametricPlot3DbCos[θ]-Cos[θ]-,bSin[θ]--Sin[θ],z,{z,-a,a},PlotStyle{Blue,Thick},θ,0,t1,,TableParametricPlot3DbCos[θ]+Cos[θ]-,bSin[θ]+-Sin[θ],z,{z,-a,a},PlotStyle{Blue,Thick},θ,0,t1,,ContourPlot3D[+(-2-2+2+2)-+2-+2+2-+2-2-2-,{x,-5,5},{y,-5,5},{z,-5,5},ContourStyleDirective[Yellow,Opacity[0.2],Specularity[White,30]],MeshFalse,PerformanceGoal"Quality"],Graphics3DGreen,Thick,Ball0,0,--,0.15,Graphics3DGreen,Thick,Ball0,0,-,0.15,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,3,2},ImageSize900,Style["Tangerine-shapped sphere family envelope:",Bold,Large],{t1,0.002,"Value (t1)"},0.001,2π,,ControlPlacementTop
π
6
2
a
2
z
2
a
2
z
π
6
2
a
2
z
2
a
2
z
π
6
4
x
2
x
2
a
2
b
2
y
2
z
4
a
2
a
2
b
4
b
2
a
2
y
2
b
2
y
4
y
2
a
2
z
2
b
2
z
2
y
2
z
4
z
2
a
2
b
2
a
2
b
π
6
Out[]=
The envelope is in yellow, the spheres in orange, the characteristic in blue. It has two recoil edges, one point up and one down in green (run animation to observe all the details).