Se da la ecuación de superficie de 2 grado:
x 2 − 2 x + y 2 − 2 y + z 2 + 1 = 0 x^{2} - 2 x + y^{2} - 2 y + z^{2} + 1 = 0 x 2 − 2 x + y 2 − 2 y + z 2 + 1 = 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 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 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 yz + 2 a 24 y + a 33 z 2 + 2 a 34 z + a 44 = 0 donde
a 11 = 1 a_{11} = 1 a 11 = 1 a 12 = 0 a_{12} = 0 a 12 = 0 a 13 = 0 a_{13} = 0 a 13 = 0 a 14 = − 1 a_{14} = -1 a 14 = − 1 a 22 = 1 a_{22} = 1 a 22 = 1 a 23 = 0 a_{23} = 0 a 23 = 0 a 24 = − 1 a_{24} = -1 a 24 = − 1 a 33 = 1 a_{33} = 1 a 33 = 1 a 34 = 0 a_{34} = 0 a 34 = 0 a 44 = 1 a_{44} = 1 a 44 = 1 Las invariantes de esta ecuación al transformar las coordenadas son los determinantes:
I 1 = a 11 + a 22 + a 33 I_{1} = a_{11} + a_{22} + a_{33} I 1 = a 11 + a 22 + a 33 |a11 a12| |a22 a23| |a11 a13|
I2 = | | + | | + | |
|a12 a22| |a23 a33| |a13 a33| I 3 = ∣ a 11 a 12 a 13 a 12 a 22 a 23 a 13 a 23 a 33 ∣ 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 3 = a 11 a 12 a 13 a 12 a 22 a 23 a 13 a 23 a 33 I 4 = ∣ 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 ∣ 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 4 = 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 I ( λ ) = ∣ a 11 − λ a 12 a 13 a 12 a 22 − λ a 23 a 13 a 23 a 33 − 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| I ( λ ) = a 11 − λ a 12 a 13 a 12 a 22 − λ a 23 a 13 a 23 a 33 − λ |a11 a14| |a22 a24| |a33 a34|
K2 = | | + | | + | |
|a14 a44| |a24 a44| |a34 a44| |a11 a12 a14| |a22 a23 a24| |a11 a13 a14|
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K3 = |a12 a22 a24| + |a23 a33 a34| + |a13 a33 a34|
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|a14 a24 a44| |a24 a34 a44| |a14 a34 a44| sustituimos coeficientes
I 1 = 3 I_{1} = 3 I 1 = 3 |1 0| |1 0| |1 0|
I2 = | | + | | + | |
|0 1| |0 1| |0 1| I 3 = ∣ 1 0 0 0 1 0 0 0 1 ∣ I_{3} = \left|\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{matrix}\right| I 3 = 1 0 0 0 1 0 0 0 1 I 4 = ∣ 1 0 0 − 1 0 1 0 − 1 0 0 1 0 − 1 − 1 0 1 ∣ I_{4} = \left|\begin{matrix}1 & 0 & 0 & -1\\0 & 1 & 0 & -1\\0 & 0 & 1 & 0\\-1 & -1 & 0 & 1\end{matrix}\right| I 4 = 1 0 0 − 1 0 1 0 − 1 0 0 1 0 − 1 − 1 0 1 I ( λ ) = ∣ 1 − λ 0 0 0 1 − λ 0 0 0 1 − λ ∣ I{\left(\lambda \right)} = \left|\begin{matrix}1 - \lambda & 0 & 0\\0 & 1 - \lambda & 0\\0 & 0 & 1 - \lambda\end{matrix}\right| I ( λ ) = 1 − λ 0 0 0 1 − λ 0 0 0 1 − λ |1 -1| |1 -1| |1 0|
K2 = | | + | | + | |
|-1 1 | |-1 1 | |0 1| |1 0 -1| |1 0 -1| |1 0 -1|
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K3 = |0 1 -1| + |0 1 0 | + |0 1 0 |
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|-1 -1 1 | |-1 0 1 | |-1 0 1 | I 1 = 3 I_{1} = 3 I 1 = 3 I 2 = 3 I_{2} = 3 I 2 = 3 I 3 = 1 I_{3} = 1 I 3 = 1 I 4 = − 1 I_{4} = -1 I 4 = − 1 I ( λ ) = − λ 3 + 3 λ 2 − 3 λ + 1 I{\left(\lambda \right)} = - \lambda^{3} + 3 \lambda^{2} - 3 \lambda + 1 I ( λ ) = − λ 3 + 3 λ 2 − 3 λ + 1 K 2 = 1 K_{2} = 1 K 2 = 1 K 3 = − 1 K_{3} = -1 K 3 = − 1 Como
I3 != 0 entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
− I 1 λ 2 + I 2 λ − I 3 + λ 3 = 0 - I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0 − I 1 λ 2 + I 2 λ − I 3 + λ 3 = 0 o
λ 3 − 3 λ 2 + 3 λ − 1 = 0 \lambda^{3} - 3 \lambda^{2} + 3 \lambda - 1 = 0 λ 3 − 3 λ 2 + 3 λ − 1 = 0 λ 1 = 1 \lambda_{1} = 1 λ 1 = 1 λ 2 = 1 \lambda_{2} = 1 λ 2 = 1 λ 3 = 1 \lambda_{3} = 1 λ 3 = 1 entonces la forma canónica de la ecuación será
( z ~ 2 λ 3 + ( x ~ 2 λ 1 + y ~ 2 λ 2 ) ) + I 4 I 3 = 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 ( z ~ 2 λ 3 + ( x ~ 2 λ 1 + y ~ 2 λ 2 ) ) + I 3 I 4 = 0 x ~ 2 + y ~ 2 + z ~ 2 − 1 = 0 \tilde x^{2} + \tilde y^{2} + \tilde z^{2} - 1 = 0 x ~ 2 + y ~ 2 + z ~ 2 − 1 = 0 z ~ 2 ( 1 − 1 ) 2 + ( x ~ 2 ( 1 − 1 ) 2 + y ~ 2 ( 1 − 1 ) 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 ( 1 − 1 ) 2 z ~ 2 + ( ( 1 − 1 ) 2 x ~ 2 + ( 1 − 1 ) 2 y ~ 2 ) = 1 es la ecuación para el tipo elipsoide
- está reducida a la forma canónica