PencilOfQuadrics1::usage="All the steps will be followed to create a pencil of quadrics.Reference: Personal.";
Development:
1) Base quadrics:
Q
1
:23
2
x
-41
2
y
-31
2
z
+48xy-72xz-24yz=0;​​
Q
2
:8
2
x
+11
2
y
+8
2
z
-4xy+8xz+4yz-12=0;​​
Q
1
:(
x
y
z
1
)
23
24
-36
0
24
-41
-12
0
-36
-12
-31
0
0
0
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
23
24
-36
0
24
-41
-12
0
-36
-12
-31
0
0
0
0
0
+μ
8
-2
4
0
-2
11
2
0
4
2
8
0
0
0
0
-12
{{23λ+8μ,24λ-2μ,-36λ+4μ,0},{24λ-2μ,-41λ+11μ,-12λ+2μ,0},{-36λ+4μ,-12λ+2μ,-31λ+8μ,0},{0,0,0,-12μ}};
Quadric’s equation in the pencil is:
(
x
y
z
1
)
23λ+8μ
24λ-2μ
-36λ+4μ
0
24λ-2μ
-41λ+11μ
-12λ+2μ
0
-36λ+4μ
-12λ+2μ
-31λ+8μ
0
0
0
0
-12μ
x
y
z
1
=0
3) Base curve (Intersection, all the quadrics within the pencil pass through this curve):
In[]:=
Solve[{23
2
x
-41
2
y
-31
2
z
+48xy-72xz-24yz==0,8
2
x
+11
2
y
+8
2
z
-4xy+8xz+4yz-12==0},{x,y}]
Out[]=
x
312z
973
-
1
2
389376
2
z
946729
-
4
⋯1⋯
1532475
+
⋯1⋯
+
1/3
(
⋯1⋯
)
1532475
⋯1⋯
2
-
1
2
778752
2
z
946729
-
8(-390684+174287
2
z
)
1532475
-
16
1/3
2
(130897754064+35866596696
2
z
+4331763361
4
z
)
1532475
1/3
(
⋯1⋯
)
-
1/3
6002388322219892736+
⋯3⋯
+
⋯1⋯

1532475
1/3
2
-
1943764992
3
z
921167317
-
3328z-390684+174287
2
z

165677575
+
256-6448z+503
3
z

170275
4
389376
2
z
946729
-
4-390684+174287
2
z

1532475
+
⋯1⋯
1532475
1/3

⋯1⋯

+
1/3
6002388322219892736+
⋯3⋯
+
⋯1⋯
1532475
1/3
2
,y
⋯1⋯
⋯1⋯
,
⋯2⋯
,{
⋯1⋯
}
Full expression not available
(
original memory size:
0.8 MB)
4) Identify singular quadrics (the values of λ and μ for which result cones or cylinders ):
In[]:=
Det[{{23λ+8μ,24λ-2μ,-36λ+4μ,0},{24λ-2μ,-41λ+11μ,-12λ+2μ,0},{-36λ+4μ,-12λ+2μ,-31λ+8μ,0},{0,0,0,-12μ}}]==0
Out[]=
-1411788
3
λ
μ+556836
2
λ
2
μ
-30528λ
3
μ
-5184
4
μ
0
In[]:=
Solve[-1411788
2
λ
+556836λμ-30528
2
μ
-5184
3
μ
0,μ]
μ-
53
27
+(-931958784-8659913472λ)2239488
1/3
9528128+132805386λ+171532242
2
λ
+441
3
-4474006660
2
λ
-46754317113
3
λ
+50430479148
4
λ

-
1
108
1/3
9528128+132805386λ+171532242
2
λ
+441
3
-4474006660
2
λ
-46754317113
3
λ
+50430479148
4
λ

,μ-
53
27
-1+
3
(-931958784-8659913472λ)4478976
1/3
9528128+132805386λ+171532242
2
λ
+441
3
-4474006660
2
λ
-46754317113
3
λ
+50430479148
4
λ

+
1
216
1-
3

1/3
9528128+132805386λ+171532242
2
λ
+441
3
-4474006660
2
λ
-46754317113
3
λ
+50430479148
4
λ

,μ-
53
27
-1-
3
(-931958784-8659913472λ)4478976
1/3
9528128+132805386λ+171532242
2
λ
+441
3
-4474006660
2
λ
-46754317113
3
λ
+50430479148
4
λ

+
1
216
1+
3

1/3
9528128+132805386λ+171532242
2
λ
+441
3
-4474006660
2
λ
-46754317113
3
λ
+50430479148
4
λ

;
μ=-
53
27
+(-931958784-8659913472λ)2239488
1/3
9528128+132805386λ+171532242
2
λ
+441
3
-4474006660
2
λ
-46754317113
3
λ
+50430479148
4
λ

-
1
108
1/3
9528128+132805386λ+171532242
2
λ
+441
3
-4474006660
2
λ
-46754317113
3
λ
+50430479148
4
λ

;
In[]:=
Reduce[-4474006660-46754317113λ+50430479148
2
λ
>=0,λ]
λ≤
318056579-5751251
4321
686128968
||λ≥
318056579+5751251
4321
686128968
;
{-∞,-0.0874441]⋃{0}⋃[1.01455,∞);
For these values of λ, we would obtain the corresponding values of μ so that they result in cones or cylinders.
Animation:
In[]:=
Out[]=
Base curves
Original quadrics
Quadrics of the pencil
Out[]=
​
Pencil of quadrics 1:
Value (t1)
Value (t1)