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x^2+y^2+z^2=2x+2y-1 forma canónica

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Gráfico:

x: [, ]
y: [, ]
z: [, ]

Calidad:

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Tipo de trazado:

Solución

Ha introducido [src]
     2    2    2                
1 + x  + y  + z  - 2*x - 2*y = 0
x22x+y22y+z2+1=0x^{2} - 2 x + y^{2} - 2 y + z^{2} + 1 = 0
x^2 - 2*x + y^2 - 2*y + z^2 + 1 = 0
Método de invariantes
Se da la ecuación de superficie de 2 grado:
x22x+y22y+z2+1=0x^{2} - 2 x + y^{2} - 2 y + z^{2} + 1 = 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=1a_{14} = -1
a22=1a_{22} = 1
a23=0a_{23} = 0
a24=1a_{24} = -1
a33=1a_{33} = 1
a34=0a_{34} = 0
a44=1a_{44} = 1
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=3I_{1} = 3
     |1  0|   |1  0|   |1  0|
I2 = |    | + |    | + |    |
     |0  1|   |0  1|   |0  1|

I3=100010001I_{3} = \left|\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{matrix}\right|
I4=1001010100101101I_{4} = \left|\begin{matrix}1 & 0 & 0 & -1\\0 & 1 & 0 & -1\\0 & 0 & 1 & 0\\-1 & -1 & 0 & 1\end{matrix}\right|
I(λ)=1λ0001λ0001λI{\left(\lambda \right)} = \left|\begin{matrix}1 - \lambda & 0 & 0\\0 & 1 - \lambda & 0\\0 & 0 & 1 - \lambda\end{matrix}\right|
     |1   -1|   |1   -1|   |1  0|
K2 = |      | + |      | + |    |
     |-1  1 |   |-1  1 |   |0  1|

     |1   0   -1|   |1   0  -1|   |1   0  -1|
     |          |   |         |   |         |
K3 = |0   1   -1| + |0   1  0 | + |0   1  0 |
     |          |   |         |   |         |
     |-1  -1  1 |   |-1  0  1 |   |-1  0  1 |

I1=3I_{1} = 3
I2=3I_{2} = 3
I3=1I_{3} = 1
I4=1I_{4} = -1
I(λ)=λ3+3λ23λ+1I{\left(\lambda \right)} = - \lambda^{3} + 3 \lambda^{2} - 3 \lambda + 1
K2=1K_{2} = 1
K3=1K_{3} = -1
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
λ33λ2+3λ1=0\lambda^{3} - 3 \lambda^{2} + 3 \lambda - 1 = 0
λ1=1\lambda_{1} = 1
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~2+y~2+z~21=0\tilde x^{2} + \tilde y^{2} + \tilde z^{2} - 1 = 0
z~2(11)2+(x~2(11)2+y~2(11)2)=1\frac{\tilde z^{2}}{\left(1^{-1}\right)^{2}} + \left(\frac{\tilde x^{2}}{\left(1^{-1}\right)^{2}} + \frac{\tilde y^{2}}{\left(1^{-1}\right)^{2}}\right) = 1
es la ecuación para el tipo elipsoide
- está reducida a la forma canónica