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(2x+y^2)/2=x^2+y^2+z^2 forma canónica

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y: [, ]
z: [, ]

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Solución

Ha introducido [src]
               2    
     2    2   y     
x - x  - z  - -- = 0
              2     
x2+xy22z2=0- x^{2} + x - \frac{y^{2}}{2} - z^{2} = 0
-x^2 + x - y^2/2 - z^2 = 0
Método de invariantes
Se da la ecuación de superficie de 2 grado:
x2+xy22z2=0- x^{2} + x - \frac{y^{2}}{2} - z^{2} = 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=1a_{11} = -1
a12=0a_{12} = 0
a13=0a_{13} = 0
a14=12a_{14} = \frac{1}{2}
a22=12a_{22} = - \frac{1}{2}
a23=0a_{23} = 0
a24=0a_{24} = 0
a33=1a_{33} = -1
a34=0a_{34} = 0
a44=0a_{44} = 0
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=a11a12a13a12a22a23a13a23a33I_{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=a11a12a13a14a12a22a23a24a13a23a33a34a14a24a34a44I_{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)=a11λa12a13a12a22λa23a13a23a33λI{\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=52I_{1} = - \frac{5}{2}
     |-1   0  |   |-1/2  0 |   |-1  0 |
I2 = |        | + |        | + |      |
     |0   -1/2|   | 0    -1|   |0   -1|

I3=1000120001I_{3} = \left|\begin{matrix}-1 & 0 & 0\\0 & - \frac{1}{2} & 0\\0 & 0 & -1\end{matrix}\right|
I4=1001201200001012000I_{4} = \left|\begin{matrix}-1 & 0 & 0 & \frac{1}{2}\\0 & - \frac{1}{2} & 0 & 0\\0 & 0 & -1 & 0\\\frac{1}{2} & 0 & 0 & 0\end{matrix}\right|
I(λ)=λ1000λ12000λ1I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda - 1 & 0 & 0\\0 & - \lambda - \frac{1}{2} & 0\\0 & 0 & - \lambda - 1\end{matrix}\right|
     |-1   1/2|   |-1/2  0|   |-1  0|
K2 = |        | + |       | + |     |
     |1/2   0 |   | 0    0|   |0   0|

     |-1    0    1/2|   |-1/2  0   0|   |-1   0   1/2|
     |              |   |           |   |            |
K3 = | 0   -1/2   0 | + | 0    -1  0| + | 0   -1   0 |
     |              |   |           |   |            |
     |1/2   0     0 |   | 0    0   0|   |1/2  0    0 |

I1=52I_{1} = - \frac{5}{2}
I2=2I_{2} = 2
I3=12I_{3} = - \frac{1}{2}
I4=18I_{4} = - \frac{1}{8}
I(λ)=λ35λ222λ12I{\left(\lambda \right)} = - \lambda^{3} - \frac{5 \lambda^{2}}{2} - 2 \lambda - \frac{1}{2}
K2=14K_{2} = - \frac{1}{4}
K3=38K_{3} = \frac{3}{8}
Como
I3 != 0

entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
I1λ2+I2λI3+λ3=0- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0
o
λ3+5λ22+2λ+12=0\lambda^{3} + \frac{5 \lambda^{2}}{2} + 2 \lambda + \frac{1}{2} = 0
λ1=12\lambda_{1} = - \frac{1}{2}
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
x~22y~2z~2+14=0- \frac{\tilde x^{2}}{2} - \tilde y^{2} - \tilde z^{2} + \frac{1}{4} = 0
z~2(12)2+(x~2(22)2+y~2(12)2)=1\frac{\tilde z^{2}}{\left(\frac{1}{2}\right)^{2}} + \left(\frac{\tilde x^{2}}{\left(\frac{\sqrt{2}}{2}\right)^{2}} + \frac{\tilde y^{2}}{\left(\frac{1}{2}\right)^{2}}\right) = 1
es la ecuación para el tipo elipsoide
- está reducida a la forma canónica