2x^2+2y^2-5z^2+2xy+2*((2)^0.5)x-2*((2)^0.5)y+10z+74=0 forma canónica

Se da la ecuación de superficie de 2 grado:
$$2 x^{2} + 2 x y + 2 \sqrt{2} x + 2 y^{2} - 2 \sqrt{2} y - 5 z^{2} + 10 z + 74 = 0$$
Esta ecuación tiene la forma:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
donde
$$a_{11} = 2$$
$$a_{12} = 1$$
$$a_{13} = 0$$
$$a_{14} = \sqrt{2}$$
$$a_{22} = 2$$
$$a_{23} = 0$$
$$a_{24} = - \sqrt{2}$$
$$a_{33} = -5$$
$$a_{34} = 5$$
$$a_{44} = 74$$
Las invariantes de esta ecuación al transformar las coordenadas son los determinantes:
$$I_{1} = a_{11} + a_{22} + a_{33}$$

     |a11  a12|   |a22  a23|   |a11  a13|
I2 = |        | + |        | + |        |
     |a12  a22|   |a23  a33|   |a13  a33|

$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$

     |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
$$I_{1} = -1$$

     |2  1|   |2  0 |   |2  0 |
I2 = |    | + |     | + |     |
     |1  2|   |0  -5|   |0  -5|

$$I_{3} = \left|\begin{matrix}2 & 1 & 0\\1 & 2 & 0\\0 & 0 & -5\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}2 & 1 & 0 & \sqrt{2}\\1 & 2 & 0 & - \sqrt{2}\\0 & 0 & -5 & 5\\\sqrt{2} & - \sqrt{2} & 5 & 74\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}2 - \lambda & 1 & 0\\1 & 2 - \lambda & 0\\0 & 0 & - \lambda - 5\end{matrix}\right|$$

     |         ___|   |           ___|           
     |  2    \/ 2 |   |  2     -\/ 2 |   |-5  5 |
K2 = |            | + |              | + |      |
     |  ___       |   |   ___        |   |5   74|
     |\/ 2    74  |   |-\/ 2     74  |           

     |                 ___ |                                            
     |  2      1     \/ 2  |   |               ___|   |             ___|
     |                     |   |  2     0   -\/ 2 |   |  2    0   \/ 2 |
     |                  ___|   |                  |   |                |
K3 = |  1      2     -\/ 2 | + |  0     -5    5   | + |  0    -5    5  |
     |                     |   |                  |   |                |
     |  ___     ___        |   |   ___            |   |  ___           |
     |\/ 2   -\/ 2     74  |   |-\/ 2   5     74  |   |\/ 2   5    74  |
                                                 

$$I_{1} = -1$$
$$I_{2} = -17$$
$$I_{3} = -15$$
$$I_{4} = -1125$$
$$I{\left(\lambda \right)} = - \lambda^{3} - \lambda^{2} + 17 \lambda - 15$$
$$K_{2} = -103$$
$$K_{3} = -1350$$
Como

I3 != 0

entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
o
$$\lambda^{3} + \lambda^{2} - 17 \lambda + 15 = 0$$
$$\lambda_{1} = 3$$
$$\lambda_{2} = 1$$
$$\lambda_{3} = -5$$
entonces la forma canónica de la ecuación será
$$\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$3 \tilde x^{2} + \tilde y^{2} - 5 \tilde z^{2} + 75 = 0$$
$$- \frac{\tilde z^{2}}{\left(\frac{\frac{1}{5} \sqrt{5}}{\frac{1}{15} \sqrt{3}}\right)^{2}} + \left(\frac{\tilde x^{2}}{\left(\frac{\frac{1}{3} \sqrt{3}}{\frac{1}{15} \sqrt{3}}\right)^{2}} + \frac{\tilde y^{2}}{\left(\frac{1}{\frac{1}{15} \sqrt{3}}\right)^{2}}\right) = -1$$
es la ecuación para el tipo hiperboloide bilateral
- está reducida a la forma canónica