Familyofsphereswithinatoroid::usage="Find the envelope and characteristic of the family of spheres:​(x-9 Cos[θ]​
2​
)
+ (y-9 Sin[θ]​
2
)
+
2
z
= 9.Reference: Personal.";
Development:
In[]:=
Solve[
2
(x-9Cos[θ])
+
2
(y-9Sin[θ])
+
2
z
==9,
∂
θ
(
2
(x-9Cos[θ])
+
2
(y-9Sin[θ])
+
2
z
-9)==0,{x,y}]
In[]:=
Simplifyx
9
3
Cos[θ]
+9Cos[θ]
2
Sin[θ]
-
9
4
Cos[θ]
-
2
z
4
Cos[θ]
+9
2
Cos[θ]
2
Sin[θ]
-
2
z
2
Cos[θ]
2
Sin[θ]
2
Cos[θ]
+
2
Sin[θ]
,ySec[θ]
9
3
Cos[θ]
Sin[θ]
2
Cos[θ]
+
2
Sin[θ]
+
9Cos[θ]
3
Sin[θ]
2
Cos[θ]
+
2
Sin[θ]
-
Sin[θ]
-((-9+
2
z
)
2
Cos[θ]
(
2
Cos[θ]
+
2
Sin[θ]
))
2
Cos[θ]
+
2
Sin[θ]
,x
9
3
Cos[θ]
+9Cos[θ]
2
Sin[θ]
+
9
4
Cos[θ]
-
2
z
4
Cos[θ]
+9
2
Cos[θ]
2
Sin[θ]
-
2
z
2
Cos[θ]
2
Sin[θ]
2
Cos[θ]
+
2
Sin[θ]
,ySec[θ]
9
3
Cos[θ]
Sin[θ]
2
Cos[θ]
+
2
Sin[θ]
+
9Cos[θ]
3
Sin[θ]
2
Cos[θ]
+
2
Sin[θ]
+
Sin[θ]
-((-9+
2
z
)
2
Cos[θ]
(
2
Cos[θ]
+
2
Sin[θ]
))
2
Cos[θ]
+
2
Sin[θ]

Out[]=
x9Cos[θ]-
-((-9+
2
z
)
2
Cos[θ]
)
,y9Sin[θ]-
-((-9+
2
z
)
2
Cos[θ]
)
Tan[θ],x9Cos[θ]+
-((-9+
2
z
)
2
Cos[θ]
)
,y9Sin[θ]+
-((-9+
2
z
)
2
Cos[θ]
)
Tan[θ]
Envelope:
σ:9Cos[θ]-
-((-9+
2
z
)
2
Cos[θ]
)
,9Sin[θ]-
-((-9+
2
z
)
2
Cos[θ]
)
Tan[θ],z;​​
σ
1
:9Cos[θ]+
-((-9+
2
z
)
2
Cos[θ]
)
,9Sin[θ]+
-((-9+
2
z
)
2
Cos[θ]
)
Tan[θ],z;
For each value of θ there will be a characteristic.
In[]:=
​​ManipulateClearAll;σ[z_,θ_]:=9Cos[θ]-
-((-9+
2
z
)
2
Cos[θ]
)
,9Sin[θ]-
-((-9+
2
z
)
2
Cos[θ]
)
Tan[θ],z;
σ
1
[z_,θ_]:=9Cos[θ]+
-((-9+
2
z
)
2
Cos[θ]
)
,9Sin[θ]+
-((-9+
2
z
)
2
Cos[θ]
)
Tan[θ],z;​​ShowTableContourPlot3D[
2
(x-9Cos[t])
+
2
(y-9Sin[t])
+
2
z
==9,{x,-13,13},{y,-13,13},{z,-3,3},ContourStyle->{Opacity[0.6],Red},MeshFalse,PerformanceGoal"Quality"],t,0,2π,
π
6
,​​TableParametricPlot3D[σ[z,t],{z,-3,3},PlotStyle{Thick,Blue}],t,0,t1,
π
6
,​​TableParametricPlot3D[
σ
1
[z,t],{z,-3,3},PlotStyle{Thick,Blue}],t,0,t1,
π
6
,​​TableParametricPlot3D[σ[z,t],{z,-3,3},PlotStyle{Thick,Blue}],t,
π
2
+0.01,t2,
π
6
,​​TableParametricPlot3D[
σ
1
[z,t],{z,-3,3},PlotStyle{Thick,Blue}],t,
π
2
+0.01,t2,
π
6
,​​TableParametricPlot3D[σ[z,t],{z,-3,3},PlotStyle{Thick,Blue}],t,
3π
2
+0.01,t3,
π
6
,​​TableParametricPlot3D[
σ
1
[z,t],{z,-3,3},PlotStyle{Thick,Blue}],t,
3π
2
+0.01,t3,
π
6
,​​ParametricPlot3D[σ[z,θ],{z,-3,3},{θ,0,2π},PlotStyle->{Opacity[0.2],Yellow},MeshFalse,PerformanceGoal"Quality"],ParametricPlot3D[
σ
1
[z,θ],{z,-3,3},{θ,0,2π},PlotStyle->{Opacity[0.2],Yellow},MeshFalse,PerformanceGoal"Quality"],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,3,3},ImageSize500,Style["Envelope and characteristic of the sphere's family :",Bold,Medium],{t1,0.02,"Value (t1)"},0.01,
π
2
-0.01,
π
12
,Delimiter,{t2,1.59,"Value (t2)"},
π
2
+0.01,
3π
2
-0.01,
π
12
,Delimiter,{t3,4.73,"Value (t3)"},
3π
2
+0.01,2π,
π
12
,ControlPlacementTop
Out[]=
​
Envelope and characteristic of the sphere's family :
Value (t1)
Value (t2)
Value (t3)
In the animation appear some elements of the sphere family, the characteristic in each sphere appears in blue. The envelope is a yellow toroid.