PencilOfQuadrics3::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
-10x=0;​​
Q
2
:
2
z
-2x=0;​​
Q
1
:(
x
y
z
1
)
1
0
0
-5
0
1
0
0
0
0
1
0
-5
0
0
0
x
y
z
1
=0​​
Q
2
:(
x
y
z
1
)
0
0
0
-1
0
0
0
0
0
0
1
0
-1
0
0
0
x
y
z
1
=0​​
2) Linear combination:
In[]:=
λ
1
0
0
-5
0
1
0
0
0
0
1
0
-5
0
0
0
+μ
0
0
0
-1
0
0
0
0
0
0
1
0
-1
0
0
0
{{λ,0,0,-5λ-μ},{0,λ,0,0},{0,0,λ+μ,0},{-5λ-μ,0,0,0}};
Quadric’s equation in the pencil is:
(
x
y
z
1
)
λ
0
0
-5λ-μ
0
λ
0
0
0
0
λ+μ
0
-5λ-μ
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
-10x==0,
2
z
-2x==0},{x,y}]
x
2
z
2
,y-
1
2
16
2
z
-
4
z
,x
2
z
2
,y
1
2
16
2
z
-
4
z
;
In[]:=
​​
ζ
1
[z_]:=
2
z
2
,-
1
2
16
2
z
-
4
z
,z;
ζ
2
[z_]:=
2
z
2
,
1
2
16
2
z
-
4
z
,z;​​Reduce[16-
2
z
>=0,z]
-4≤z≤4;
4) Identify singular quadrics (the values of λ and μ for which result cones or cylinders ):
In[]:=
Det[{{λ,0,0,-5λ-μ},{0,λ,0,0},{0,0,λ+μ,0},{-5λ-μ,0,0,0}}]==0
In[]:=
-25
4
λ
-35
3
λ
μ-11
2
λ
2
μ
-λ
3
μ
0;​​Solve[-25
4
λ
-35
3
λ
μ-11
2
λ
2
μ
-λ
3
μ
0,λ]
{λ0},{λ-μ},λ-
μ
5
,λ-
μ
5
;
Animation:
In[]:=
SwatchLegend[{Yellow,Green,Orange},{"Base curve","Original quadrics","Quadrics of pencil"}]​​Manipulate
ζ
1
[z_]:=
2
z
2
,-
1
2
16
2
z
-
4
z
,z;
ζ
2
[z_]:=
2
z
2
,
1
2
16
2
z
-
4
z
,z;​​​​ShowTable[ContourPlot3D[-μ(
2
x
+
2
y
+
2
z
-18x)+μ(9
2
x
+9
2
y
-29
2
z
)==0,{x,-10,10},{y,-10,10},{z,-5,4},ContourStyle{Orange,Opacity[0.2],Specularity[White,40]},MeshFalse,MaxRecursion->3],{μ,-5,t1,0.1}],​​TableContourPlot3D-
μ
5
(
2
x
+
2
y
+
2
z
-18x)+μ(9
2
x
+9
2
y
-29
2
z
)==0,{x,-10,10},{y,-10,10},{z,-5,4},ContourStyle{Orange,Opacity[0.2],Specularity[White,40]},MeshFalse,MaxRecursion->3,{μ,-5,t1,0.1},​​ParametricPlot3D[
ζ
1
[t],{t,-4,4},PlotStyle->Yellow]/.Line[pts_,rest___]:>Tube[pts,0.1,rest],ParametricPlot3D[
ζ
2
[t],{t,-4,4},PlotStyle->Yellow]/.Line[pts_,rest___]:>Tube[pts,0.1,rest],ParametricPlot3D[{5Cos[u]Sin[v]+5,5Sin[u]Sin[v],5Cos[v]},{u,0,2π},{v,0,π},PlotStyle->Directive[Opacity[0.7],Specularity[White,30]],MeshFalse,ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality",PlotPoints->40],ContourPlot3D[
2
z
-2x==0,{x,-10,10},{y,-10,10},{z,-5,5},ContourStyle->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,-2.4,1.},ImageSize700,Style["Pencil of quadrics 3:",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 3:
Value (t1)
The pencil consists of cones and hyperboloids of one-leaf all coincide in the same positions.