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Forma canónica/ 2*x^2+2*y^2+2*z^2+2*x*y-2*x*z-2*y*z-14*x-14*y+4*z=26

2*x^2+2*y^2+2*z^2+2*x*y-2*x*z-2*y*z-14*x-14*y+4*z=26 forma canónica

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Solución

Ha introducido [src] 
                       2      2      2                                  
-26 - 14*x - 14*y + 2*x  + 2*y  + 2*z  + 4*z - 2*x*z - 2*y*z + 2*x*y = 0
2x2+2xy−2xz−14x+2y2−2yz−14y+2z2+4z−26=02 x^{2} + 2 x y - 2 x z - 14 x + 2 y^{2} - 2 y z - 14 y + 2 z^{2} + 4 z - 26 = 0 
2*x^2 + 2*x*y - 2*x*z - 14*x + 2*y^2 - 2*y*z - 14*y + 2*z^2 + 4*z - 26 = 0
Método de invariantes
Se da la ecuación de superficie de 2 grado:
2x2+2xy−2xz−14x+2y2−2yz−14y+2z2+4z−26=02 x^{2} + 2 x y - 2 x z - 14 x + 2 y^{2} - 2 y z - 14 y + 2 z^{2} + 4 z - 26 = 0
Esta ecuación tiene la forma:
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
donde
a11=2a_{11} = 2
a12=1a_{12} = 1
a13=−1a_{13} = -1
a14=−7a_{14} = -7
a22=2a_{22} = 2
a23=−1a_{23} = -1
a24=−7a_{24} = -7
a33=2a_{33} = 2
a34=2a_{34} = 2
a44=−26a_{44} = -26
Las invariantes de esta ecuación al transformar las coordenadas son los determinantes:
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|

sustituimos coeficientes
I1=6I_{1} = 6
     |2  1|   |2   -1|   |2   -1|
I2 = |    | + |      | + |      |
     |1  2|   |-1  2 |   |-1  2 |

I3=∣21−112−1−1−12∣I_{3} = \left|\begin{matrix}2 & 1 & -1\\1 & 2 & -1\\-1 & -1 & 2\end{matrix}\right|
I4=∣21−1−712−1−7−1−122−7−72−26∣I_{4} = \left|\begin{matrix}2 & 1 & -1 & -7\\1 & 2 & -1 & -7\\-1 & -1 & 2 & 2\\-7 & -7 & 2 & -26\end{matrix}\right|
I(λ)=∣2−λ1−112−λ−1−1−12−λ∣I{\left(\lambda \right)} = \left|\begin{matrix}2 - \lambda & 1 & -1\\1 & 2 - \lambda & -1\\-1 & -1 & 2 - \lambda\end{matrix}\right|
     |2   -7 |   |2   -7 |   |2   2 |
K2 = |       | + |       | + |      |
     |-7  -26|   |-7  -26|   |2  -26|

     |2   1   -7 |   |2   -1  -7 |   |2   -1  -7 |
     |           |   |           |   |           |
K3 = |1   2   -7 | + |-1  2    2 | + |-1  2    2 |
     |           |   |           |   |           |
     |-7  -7  -26|   |-7  2   -26|   |-7  2   -26|

I1=6I_{1} = 6
I2=9I_{2} = 9
I3=4I_{3} = 4
I4=−256I_{4} = -256
I(λ)=−λ3+6λ2−9λ+4I{\left(\lambda \right)} = - \lambda^{3} + 6 \lambda^{2} - 9 \lambda + 4
K2=−258K_{2} = -258
K3=−488K_{3} = -488
Como
I3 != 0

entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
−I1λ2+I2λ−I3+λ3=0- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0
o
λ3−6λ2+9λ−4=0\lambda^{3} - 6 \lambda^{2} + 9 \lambda - 4 = 0
λ1=4\lambda_{1} = 4
λ2=1\lambda_{2} = 1
λ3=1\lambda_{3} = 1
entonces la forma canónica de la ecuación será
(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
4x~2+y~2+z~2−64=04 \tilde x^{2} + \tilde y^{2} + \tilde z^{2} - 64 = 0
        2           2           2    
\tilde x    \tilde y    \tilde z     
--------- + --------- + --------- = 1
        2         2           2      
 /  1  \      / 1\        / 1\       
 |-----|      \8 /        \8 /       
 \2*1/8/

es la ecuación para el tipo elipsoide
- está reducida a la forma canónica
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