PencilOfQuadrics2::usage="All the steps will be followed to create a pencil of quadrics.Reference: Personal.";
Development:
1) Base quadrics:
Q
1
:
2
x
+
2
y
+
2
z
-18x=0;​​
Q
2
:9
2
x
+9
2
y
-29
2
z
=0;​​
Q
1
:(
x
y
z
1
)
1
0
0
-9
0
1
0
0
0
0
1
0
-9
0
0
0
x
y
z
1
=0​​
Q
2
:(
x
y
z
1
)
9
0
0
0
0
9
0
0
0
0
-29
0
0
0
0
0
x
y
z
1
=0​​
2) Linear combination:
In[]:=
λ
1
0
0
-9
0
1
0
0
0
0
1
0
-9
0
0
0
+μ
9
0
0
0
0
9
0
0
0
0
-29
0
0
0
0
0
{{λ+9μ,0,0,-9λ},{0,λ+9μ,0,0},{0,0,λ-29μ,0},{-9λ,0,0,0}};
Quadric’s equation in the pencil is:
(
x
y
z
1
)
λ+9μ
0
0
-9λ
0
λ+9μ
0
0
0
0
λ-29μ
0
-9λ
0
0
0
x
y
z
1
=0
3) Base curve (Intersection, all the quadrics within the pencil pass through this curve, in this case Viviani’s curve):
In[]:=
Solve[{
2
x
+
2
y
+
2
z
-18x==0,9
2
x
+9
2
y
-29
2
z
==0},{x,y}]
In[]:=
x
19
2
z
81
,y-
1
81
21141
2
z
-361
4
z
,x
19
2
z
81
,y
1
81
21141
2
z
-361
4
z
;​​
ζ
1
[z_]:=
19
2
z
81
,-
1
81
21141
2
z
-361
4
z
,z;
ζ
2
[z_]:=
19
2
z
81
,
1
81
21141
2
z
-361
4
z
,z;​​Reduce[21141-361
2
z
>=0,z]
Out[]=
-
27
29
19
≤z≤
27
29
19
-
27
29
19
≤z≤
27
29
19
;
4) Identify singular quadrics (the values of λ and μ for which result cones or cylinders ):
In[]:=
Det[{{λ+9μ,0,0,-9λ},{0,λ+9μ,0,0},{0,0,λ-29μ,0},{-9λ,0,0,0}}]==0
In[]:=
Solve[-81
4
λ
+1620
3
λ
μ+21141
2
λ
2
μ
0,λ]
{{λ0},{λ0},{λ-9μ},{λ29μ}};
Animation:
In[]:=
SwatchLegend[{Yellow,Green,Orange},{"Base curve","Original quadrics","Quadrics of pencil"}]​​Manipulate
ζ
1
[z_]:=
19
2
z
81
,-
1
81
21141
2
z
-361
4
z
,z;
ζ
2
[z_]:=
19
2
z
81
,
1
81
21141
2
z
-361
4
z
,z;​​​​ShowTable[ContourPlot3D[-9μ(
2
x
+
2
y
+
2
z
-18x)+μ(9
2
x
+9
2
y
-29
2
z
)==0,{x,-20,20},{y,-20,20},{z,-10,10},ContourStyle{Orange,Opacity[0.2],Specularity[White,40]},MeshFalse,MaxRecursion->3],{μ,-5,t1,0.1}],​​Table[ContourPlot3D[29μ(
2
x
+
2
y
+
2
z
-18x)+μ(9
2
x
+9
2
y
-29
2
z
)==0,{x,-20,20},{y,-20,20},{z,-10,10},ContourStyle{Orange,Opacity[0.2],Specularity[White,40]},MeshFalse,MaxRecursion->3],{μ,-5,t1,0.1}],​​ParametricPlot3D
ζ
1
[t],t,-
27
29
19
,
27
29
19
,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.2,rest],ParametricPlot3D
ζ
2
[t],t,-
27
29
19
,
27
29
19
,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.2,rest],ParametricPlot3D[{9Cos[u]Sin[v]+9,9Sin[u]Sin[v],9Cos[v]},{u,0,2π},{v,0,π},PlotStyle->Directive[Opacity[0.7],Specularity[White,30]],MeshFalse,ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality",PlotPoints->40],ParametricPlot3D
29
3
vCos[u],
29
3
vSin[u],v,{u,0,2π},{v,-9,9},PlotStyle->Directive[Opacity[0.7],Specularity[White,30]],MeshFalse,ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality",PlotPoints->40,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,-2.4,2.},ImageSize700,Style["Pencil of quadrics 2:",Bold,Large],{{t1,-5+0.00002,"Value (t1)"},-5+0.00001,5,0.1},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
Base curve
Original quadrics
Quadrics of pencil
Out[]=
​
Pencil of quadrics 2:
Value (t1)
The pencil consists of cylinders and parabolic cylinders and they all coincide in the same positions.