PencilOfQuadrics5::usage="All the steps will be followed to create a pencil of quadrics.Reference: Personal.";
Development:
1) Base quadrics:
Q
1
:
2
x
+2yz-2y=0;​​
Q
2
:8
2
x
+11
2
y
+8
2
z
-4xy+8xz+4yz-12=0;​​
Q
1
:(
x
y
z
1
)
1
0
0
0
0
0
1
-1
0
1
0
0
0
-1
0
0
x
y
z
1
=0​​
Q
2
:(
x
y
z
1
)
8
-2
4
0
-2
11
2
0
4
2
8
0
0
0
0
-12
x
y
z
1
=0​​
2) Linear combination:
In[]:=
λ
1
0
0
0
0
0
1
-1
0
1
0
0
0
-1
0
0
+μ
8
-2
4
0
-2
11
2
0
4
2
8
0
0
0
0
-12
{{λ+8μ,-2μ,4μ,0},{-2μ,11μ,λ+2μ,-λ},{4μ,λ+2μ,8μ,0},{0,-λ,0,-12μ}};
Quadric’s equation in the pencil is:
(
x
y
z
1
)
λ+8μ
-2μ
4μ
0
-2μ
11μ
λ+2μ
-λ
4μ
λ+2μ
8μ
0
0
-λ
0
-12μ
x
y
z
1
=0
3) Base curve (Intersection, all the quadrics within the pencil pass through this curve.):
In[]:=
Solve[{
2
x
+2yz-2y==0,8
2
x
+11
2
y
+8
2
z
-4xy+8xz+4yz-12==0},{x,z}]
x
1
2
y-
8y-3
2
y
-2
6
4
2
y
-4
3
y
-3
4
y
,z
2
y
+
6
-
2
y
(-4+4y+3
2
y
)
+y
8y-3
2
y
-2
6
-
2
y
(-4+4y+3
2
y
)
4y
,x
1
2
y+
8y-3
2
y
-2
6
4
2
y
-4
3
y
-3
4
y
,z
2
y
+
6
-
2
y
(-4+4y+3
2
y
)
-y
8y-3
2
y
-2
6
-
2
y
(-4+4y+3
2
y
)
4y
,x
1
2
y-
8y-3
2
y
+2
6
4
2
y
-4
3
y
-3
4
y
,z
2
y
-
6
-
2
y
(-4+4y+3
2
y
)
+y
8y-3
2
y
+2
6
-
2
y
(-4+4y+3
2
y
)
4y
,x
1
2
y+
8y-3
2
y
+2
6
4
2
y
-4
3
y
-3
4
y
,z
2
y
-
6
-
2
y
(-4+4y+3
2
y
)
-y
8y-3
2
y
+2
6
-
2
y
(-4+4y+3
2
y
)
4y
;
In[]:=
​​Reduce8y-3
2
y
-2
6
4
2
y
-4
3
y
-3
4
y
>=0,y​​Reduce[-
2
y
(-4+4y+3
2
y
)>=0,y]
y0||
1
27
-8+4
22
≤y≤
2
3
;
-2≤y≤
2
3
;
Domain (possible values of y for 1 and 2):
​​
1
27
-8+4
22
≤y≤
2
3
;​​
In[]:=
Reduce8y-3
2
y
+2
6
4
2
y
-4
3
y
-3
4
y
>=0,y​​
1
27
-8-4
22
≤y≤
2
3
;
Domain (possible values of y for 3 and 4):
1
27
-8-4
22
≤y≤
2
3
-{0};
4) Identity singular quadrics (the values of λ and μ for which result cones or cylinders ):
In[]:=
NSolve[Det[{{λ+8μ,-2μ,4μ,0},{-2μ,11μ,λ+2μ,-λ},{4μ,λ+2μ,8μ,0},{0,-λ,0,-12μ}}]==0,λ]
{{λ-6.`μ},{λ8.233687939614086`μ},{λ-26.233687939614086`μ}};
Animation:
In[]:=
ManipulateClearAll;
ζ
1
[y_]:=
1
2
y-
8y-3
2
y
-2
6
4
2
y
-4
3
y
-3
4
y
,y,
2
y
+
6
-
2
y
(-4+4y+3
2
y
)
+y
8y-3
2
y
-2
6
-
2
y
(-4+4y+3
2
y
)
4y
;​​
ζ
2
[y_]:=
1
2
y+
8y-3
2
y
-2
6
4
2
y
-4
3
y
-3
4
y
,y,
2
y
+
6
-
2
y
(-4+4y+3
2
y
)
-y
8y-3
2
y
-2
6
-
2
y
(-4+4y+3
2
y
)
4y
;​​
ζ
3
[y_]:=
1
2
y-
8y-3
2
y
+2
6
4
2
y
-4
3
y
-3
4
y
,y,
2
y
-
6
-
2
y
(-4+4y+3
2
y
)
+y
8y-3
2
y
+2
6
-
2
y
(-4+4y+3
2
y
)
4y
;​​
ζ
4
[y_]:=
1
2
y+
8y-3
2
y
+2
6
4
2
y
-4
3
y
-3
4
y
,y,
2
y
-
6
-
2
y
(-4+4y+3
2
y
)
-y
8y-3
2
y
+2
6
-
2
y
(-4+4y+3
2
y
)
4y
;​​​​λ1=-6.`μ;λ2=8.233687939614086`μ;λ3=-26.233687939614086`μ;​​​​ShowTable[ContourPlot3D[λ1(
2
x
+2yz-2y)+μ(8
2
x
+11
2
y
+8
2
z
-4xy+8xz+4yz-12)==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle{Orange,Opacity[0.2],Specularity[White,40]},MeshFalse,MaxRecursion->3],{μ,-5,t1,1}],​​Table[ContourPlot3D[λ2(
2
x
+2yz-2y)+μ(8
2
x
+11
2
y
+8
2
z
-4xy+8xz+4yz-12)==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle{Orange,Opacity[0.2],Specularity[White,40]},MeshFalse,MaxRecursion->3],{μ,-5,t1,1}],​​Table[ContourPlot3D[λ3(
2
x
+2yz-2y)+μ(8
2
x
+11
2
y
+8
2
z
-4xy+8xz+4yz-12)==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle{Orange,Opacity[0.2],Specularity[White,40]},MeshFalse,MaxRecursion->3],{μ,-5,t1,1}],​​ParametricPlot3D
ζ
1
[t],t,
1
27
-8+4
22
,
2
3
,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.1,rest],ParametricPlot3D
ζ
2
[t],t,
1
27
-8+4
22
,
2
3
,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.1,rest],​​ParametricPlot3D
ζ
3
[t],t,
1
27
-8-4
22
,-0.00001,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.1,rest],​​ParametricPlot3D
ζ
4
[t],t,
1
27
-8-4
22
,-0.00001,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.1,rest],​​ParametricPlot3D
ζ
3
[t],t,0.00001,
2
3
,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.1,rest],ParametricPlot3D
ζ
4
[t],t,0.00001,
2
3
,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.1,rest],​​ContourPlot3D[
2
x
+2yz-2y==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->Directive[Green,Opacity[0.3],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality",PlotPoints->60],ContourPlot3D[8
2
x
+11
2
y
+8
2
z
-4xy+8xz+4yz-12==0,{x,-10,10},{y,-10,10},{z,-10,10},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{1.3,2.4,2},ImageSize900,Style["Pencil of quadrics 5:",Bold,Large],{{t1,-5+0.00002,"Value (t1)"},-5+0.00001,5,1},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
​
Pencil of quadrics 5:
Value (t1)