Mister Exam

Canonical form of a imaginary ellipsoid

For example, you have entered (calculator here):
2x2−4xy+4y2−4y+z2−2z+5=02 x^{2} - 4 x y + 4 y^{2} - 4 y + z^{2} - 2 z + 5 = 0

Detail solution (Invariants method)

Given equation of the surface of 2-order:
2x2−4xy+4y2−4y+z2−2z+5=02 x^{2} - 4 x y + 4 y^{2} - 4 y + z^{2} - 2 z + 5 = 0
This equation looks like:
a11x2+2a12xy+2a13xz+2a14x+a22y2+2a23yz+2a24y+a33z2+2a34z+a44=0a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0
where
a11=2a_{11} = 2
a12=−2a_{12} = -2
a13=0a_{13} = 0
a14=0a_{14} = 0
a22=4a_{22} = 4
a23=0a_{23} = 0
a24=−2a_{24} = -2
a33=1a_{33} = 1
a34=−1a_{34} = -1
a44=5a_{44} = 5
The invariants of the equation when converting coordinates are determinants:
I1=a11+a22+a33I_{1} = a_{11} + a_{22} + a_{33}
     |a11  a12|   |a22  a23|   |a11  a13|
I2 = |        | + |        | + |        |
     |a12  a22|   |a23  a33|   |a13  a33|

I3=∣a11a12a13a12a22a23a13a23a33∣I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|
I4=∣a11a12a13a14a12a22a23a24a13a23a33a34a14a24a34a44∣I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|
I(λ)=∣a11−λa12a13a12a22−λa23a13a23a33−λ∣I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|
     |a11  a14|   |a22  a24|   |a33  a34|
K2 = |        | + |        | + |        |
     |a14  a44|   |a24  a44|   |a34  a44|

     |a11  a12  a14|   |a22  a23  a24|   |a11  a13  a14|
     |             |   |             |   |             |
K3 = |a12  a22  a24| + |a23  a33  a34| + |a13  a33  a34|
     |             |   |             |   |             |
     |a14  a24  a44|   |a24  a34  a44|   |a14  a34  a44|

substitute coefficients
I1=7I_{1} = 7
     |2   -2|   |4  0|   |2  0|
I2 = |      | + |    | + |    |
     |-2  4 |   |0  1|   |0  1|

I3=∣2−20−240001∣I_{3} = \left|\begin{matrix}2 & -2 & 0\\-2 & 4 & 0\\0 & 0 & 1\end{matrix}\right|
I4=∣2−200−240−2001−10−2−15∣I_{4} = \left|\begin{matrix}2 & -2 & 0 & 0\\-2 & 4 & 0 & -2\\0 & 0 & 1 & -1\\0 & -2 & -1 & 5\end{matrix}\right|
I(λ)=∣2−λ−20−24−λ0001−λ∣I{\left(\lambda \right)} = \left|\begin{matrix}2 - \lambda & -2 & 0\\-2 & 4 - \lambda & 0\\0 & 0 & 1 - \lambda\end{matrix}\right|
     |2  0|   |4   -2|   |1   -1|
K2 = |    | + |      | + |      |
     |0  5|   |-2  5 |   |-1  5 |

     |2   -2  0 |   |4   0   -2|   |2  0   0 |
     |          |   |          |   |         |
K3 = |-2  4   -2| + |0   1   -1| + |0  1   -1|
     |          |   |          |   |         |
     |0   -2  5 |   |-2  -1  5 |   |0  -1  5 |

I1=7I_{1} = 7
I2=10I_{2} = 10
I3=4I_{3} = 4
I4=8I_{4} = 8
I(λ)=−λ3+7λ2−10λ+4I{\left(\lambda \right)} = - \lambda^{3} + 7 \lambda^{2} - 10 \lambda + 4
K2=30K_{2} = 30
K3=32K_{3} = 32
Because
I3 != 0

then by type of surface:
you need to
Make the characteristic equation for the surface:
−I1λ2+I2λ−I3+λ3=0- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0
or
λ3−7λ2+10λ−4=0\lambda^{3} - 7 \lambda^{2} + 10 \lambda - 4 = 0
λ1=1\lambda_{1} = 1
λ2=3−5\lambda_{2} = 3 - \sqrt{5}
λ3=5+3\lambda_{3} = \sqrt{5} + 3
then the canonical form of the equation will be
(z~2λ3+(x~2λ1+y~2λ2))+I4I3=0\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0
x~2+y~2(3−5)+z~2(5+3)+2=0\tilde x^{2} + \tilde y^{2} \left(3 - \sqrt{5}\right) + \tilde z^{2} \left(\sqrt{5} + 3\right) + 2 = 0
z~2(1225+3)2+(x~2(1122)2+y~2(1223−5)2)=−1\frac{\tilde z^{2}}{\left(\frac{1}{\frac{\sqrt{2}}{2} \sqrt{\sqrt{5} + 3}}\right)^{2}} + \left(\frac{\tilde x^{2}}{\left(\frac{1}{\frac{1}{2} \sqrt{2}}\right)^{2}} + \frac{\tilde y^{2}}{\left(\frac{1}{\frac{\sqrt{2}}{2} \sqrt{3 - \sqrt{5}}}\right)^{2}}\right) = -1
this equation is fora type imaginary ellipsoid
- reduced to canonical form
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