PencilOfQuadrics4::usage="All the steps will be followed to create a pencil of quadrics.Reference: Personal.";
Development:
1) Base quadrics:
Q
1
:yz+2x-2z=0;​​
Q
2
:2
2
x
+
2
y
-2
2
z
+2xy-2xz-yz+3z-1=0;​​
Q
1
:(
x
y
z
1
)
0
0
0
1
0
0
1
2
0
0
1
2
0
-1
1
0
-1
0
x
y
z
1
=0​​
Q
2
:(
x
y
z
1
)
2
1
-1
0
1
1
-
1
2
0
-1
-
1
2
-2
3
2
0
0
3
2
-1
x
y
z
1
=0​​
2) Linear combination:
In[]:=
λ
0
0
0
1
0
0
1
2
0
0
1
2
0
-1
1
0
-1
0
+μ
2
1
-1
0
1
1
-
1
2
0
-1
-
1
2
-2
3
2
0
0
3
2
-1
{2μ,μ,-μ,λ},μ,μ,
λ
2
-
μ
2
,0,-μ,
λ
2
-
μ
2
,-2μ,-λ+
3μ
2
,λ,0,-λ+
3μ
2
,-μ;
Quadric’s equation in the pencil is:
(
x
y
z
1
)
2μ
μ
-μ
λ
μ
μ
λ
2
-
μ
2
0
-μ
λ
2
-
μ
2
-2μ
-λ+
3μ
2
λ
0
-λ+
3μ
2
-μ
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[{yz+2x-2z==0,2
2
x
+
2
y
-2
2
z
+2xy-2xz-yz+3z-1==0},{x,y}]
x
1
2
2z+
2
z
2-2z+
2
z
-
3
z
2-2z+
2
z
+
z
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,y
-z+
2
z
-
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,x
1
2
2z+
2
z
2-2z+
2
z
-
3
z
2-2z+
2
z
-
z
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,y
-z+
2
z
+
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
;
In[]:=
​​
ζ
1
[z_]:=
1
2
2z+
2
z
2-2z+
2
z
-
3
z
2-2z+
2
z
+
z
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,
-z+
2
z
-
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,z;
ζ
2
[z_]:=
1
2
2z+
2
z
2-2z+
2
z
-
3
z
2-2z+
2
z
-
z
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,
-z+
2
z
+
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,z;​​Reduce[4-16z+23
2
z
-16
3
z
+5
4
z
>=0,z]​​Reduce[2-2z+
2
z
>0,z]
z≤
3
0.527
…
||z≥1;
z∈;​​-
3
0.527
…
,1;
4) Identity singular quadrics (the values of λ and μ for which result cones or cylinders ):
In[]:=
Det{2μ,μ,-μ,λ},μ,μ,
λ
2
-
μ
2
,0,-μ,
λ
2
-
μ
2
,-2μ,-λ+
3μ
2
,λ,0,-λ+
3μ
2
,-μ==0
4
λ
4
+
3
λ
μ
2
+
5
2
λ
2
μ
4
+
3λ
3
μ
2
+
4
μ
4
0;​​
In[]:=
Solve
4
λ
4
+
3
λ
μ
2
+
5
2
λ
2
μ
4
+
3λ
3
μ
2
+
4
μ
4
0,λ
λ-
μ
2
+
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
2
3
-
1
2
-
14
2
μ
3
-
2
μ
3
1/3
215+12
321

-
1
3
1/3
215+12
321

2
μ
-
4
3
3
μ
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
,λ-
μ
2
+
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
2
3
+
1
2
-
14
2
μ
3
-
2
μ
3
1/3
215+12
321

-
1
3
1/3
215+12
321

2
μ
-
4
3
3
μ
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
,λ-
μ
2
-
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
2
3
-
1
2
-
14
2
μ
3
-
2
μ
3
1/3
215+12
321

-
1
3
1/3
215+12
321

2
μ
+
4
3
3
μ
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
,λ-
μ
2
-
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
2
3
+
1
2
-
14
2
μ
3
-
2
μ
3
1/3
215+12
321

-
1
3
1/3
215+12
321

2
μ
+
4
3
3
μ
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
;​​
In[]:=
Reduce-7
2
μ
+
2
μ
1/3
215+12
321

+
1/3
215+12
321

2
μ
>=0,μ​​Reduce-
14
2
μ
3
-
2
μ
3
1/3
215+12
321

-
1
3
1/3
215+12
321

2
μ
+
4
3
3
μ
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
>=0,μ
μ∈;
μ≥0;
Domain of μ: μ ≥ 0.
Animation:
In[]:=
SwatchLegend[{Yellow,Green,Orange},{"Base curve","Original quadrics","Quadrics of pencil"}]​​Manipulate
ζ
1
[z_]:=
1
2
2z+
2
z
2-2z+
2
z
-
3
z
2-2z+
2
z
+
z
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,
-z+
2
z
-
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,z;;
ζ
2
[z_]:=
1
2
2z+
2
z
2-2z+
2
z
-
3
z
2-2z+
2
z
-
z
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,
-z+
2
z
+
4-16z+23
2
z
-16
3
z
+5
4
z
2-2z+
2
z
,z;​​λ1=-
μ
2
+
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
2
3
-
1
2
-
14
2
μ
3
-
2
μ
3
1/3
215+12
321

-
1
3
1/3
215+12
321

2
μ
-
4
3
3
μ
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
;​​λ2=-
μ
2
-
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
2
3
-
1
2
-
14
2
μ
3
-
2
μ
3
1/3
215+12
321

-
1
3
1/3
215+12
321

2
μ
+
4
3
3
μ
-7
2
μ
+
2
μ
1/3
(215+12
321
)
+
1/3
215+12
321

2
μ
;​​​​Show[{Table[ContourPlot3D[λ1(yz+2x-2z)+μ(2
2
x
+
2
y
-2
2
z
+2xy-2xz-yz+3z-1)==0,{x,-10,10},{y,-10,10},{z,-5,4},ContourStyle{Orange,Opacity[0.2],Specularity[White,40]},MeshFalse,MaxRecursion->3],{μ,0.1,t1,1}],​​Table[ContourPlot3D[λ2(yz+2x-2z)+μ(2
2
x
+
2
y
-2
2
z
+2xy-2xz-yz+3z-1)==0,{x,-10,10},{y,-10,10},{z,-5,4},ContourStyle{Orange,Opacity[0.2],Specularity[White,40]},MeshFalse,MaxRecursion->3],{μ,0.1,t1,1}],​​ParametricPlot3D[
ζ
1
[t],{t,-5,0.8},PlotStyle->Yellow]/.Line[pts_,rest___]:>Tube[pts,0.1,rest],ParametricPlot3D[
ζ
2
[t],{t,-5,0.8},PlotStyle->Yellow]/.Line[pts_,rest___]:>Tube[pts,0.1,rest],​​ParametricPlot3D[
ζ
1
[t],{t,1,5},PlotStyle->Yellow]/.Line[pts_,rest___]:>Tube[pts,0.1,rest],ParametricPlot3D[
ζ
2
[t],{t,1,5},PlotStyle->Yellow]/.Line[pts_,rest___]:>Tube[pts,0.1,rest],ContourPlot3D[yz+2x-2z==0,{x,-10,10},{y,-10,10},{z,-5,5},ContourStyle->Directive[Green,Opacity[0.3],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality",PlotPoints->60],ContourPlot3D[2
2
x
+
2
y
-2
2
z
+2xy-2xz-yz+3z-1==0,{x,-10,10},{y,-10,10},{z,-5,5},ContourStyle->Directive[Green,Opacity[0.3],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality",PlotPoints->60]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,-2.4,0.5},ImageSize700],Style["Pencil of quadrics 4:",Bold,Large],{{t1,0.1+0.00002,"Value (t1)"},0.1+0.00001,10,1},SaveDefinitionsTrue,ControlPlacementTop
The pencil’s cones match.