Se da la ecuación de superficie de 2 grado:
x 1 2 − 2 x 1 x 2 + 3 x 2 2 + 3 x 3 2 = 0 x_{1}^{2} - 2 x_{1} x_{2} + 3 x_{2}^{2} + 3 x_{3}^{2} = 0 x 1 2 − 2 x 1 x 2 + 3 x 2 2 + 3 x 3 2 = 0 Esta ecuación tiene la forma:
a 11 x 3 2 + 2 a 12 x 2 x 3 + 2 a 13 x 1 x 3 + 2 a 14 x 3 + a 22 x 2 2 + 2 a 23 x 1 x 2 + 2 a 24 x 2 + a 33 x 1 2 + 2 a 34 x 1 + a 44 = 0 a_{11} x_{3}^{2} + 2 a_{12} x_{2} x_{3} + 2 a_{13} x_{1} x_{3} + 2 a_{14} x_{3} + a_{22} x_{2}^{2} + 2 a_{23} x_{1} x_{2} + 2 a_{24} x_{2} + a_{33} x_{1}^{2} + 2 a_{34} x_{1} + a_{44} = 0 a 11 x 3 2 + 2 a 12 x 2 x 3 + 2 a 13 x 1 x 3 + 2 a 14 x 3 + a 22 x 2 2 + 2 a 23 x 1 x 2 + 2 a 24 x 2 + a 33 x 1 2 + 2 a 34 x 1 + a 44 = 0 donde
a 11 = 3 a_{11} = 3 a 11 = 3 a 12 = 0 a_{12} = 0 a 12 = 0 a 13 = 0 a_{13} = 0 a 13 = 0 a 14 = 0 a_{14} = 0 a 14 = 0 a 22 = 3 a_{22} = 3 a 22 = 3 a 23 = − 1 a_{23} = -1 a 23 = − 1 a 24 = 0 a_{24} = 0 a 24 = 0 a 33 = 1 a_{33} = 1 a 33 = 1 a 34 = 0 a_{34} = 0 a 34 = 0 a 44 = 0 a_{44} = 0 a 44 = 0 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 = 7 I_{1} = 7 I 1 = 7 |3 0| |3 -1| |3 0|
I2 = | | + | | + | |
|0 3| |-1 1 | |0 1| I 3 = ∣ 3 0 0 0 3 − 1 0 − 1 1 ∣ I_{3} = \left|\begin{matrix}3 & 0 & 0\\0 & 3 & -1\\0 & -1 & 1\end{matrix}\right| I 3 = 3 0 0 0 3 − 1 0 − 1 1 I 4 = ∣ 3 0 0 0 0 3 − 1 0 0 − 1 1 0 0 0 0 0 ∣ I_{4} = \left|\begin{matrix}3 & 0 & 0 & 0\\0 & 3 & -1 & 0\\0 & -1 & 1 & 0\\0 & 0 & 0 & 0\end{matrix}\right| I 4 = 3 0 0 0 0 3 − 1 0 0 − 1 1 0 0 0 0 0 I ( λ ) = ∣ 3 − λ 0 0 0 3 − λ − 1 0 − 1 1 − λ ∣ I{\left(\lambda \right)} = \left|\begin{matrix}3 - \lambda & 0 & 0\\0 & 3 - \lambda & -1\\0 & -1 & 1 - \lambda\end{matrix}\right| I ( λ ) = 3 − λ 0 0 0 3 − λ − 1 0 − 1 1 − λ |3 0| |3 0| |1 0|
K2 = | | + | | + | |
|0 0| |0 0| |0 0| |3 0 0| |3 -1 0| |3 0 0|
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K3 = |0 3 0| + |-1 1 0| + |0 1 0|
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|0 0 0| |0 0 0| |0 0 0| I 1 = 7 I_{1} = 7 I 1 = 7 I 2 = 14 I_{2} = 14 I 2 = 14 I 3 = 6 I_{3} = 6 I 3 = 6 I 4 = 0 I_{4} = 0 I 4 = 0 I ( λ ) = − λ 3 + 7 λ 2 − 14 λ + 6 I{\left(\lambda \right)} = - \lambda^{3} + 7 \lambda^{2} - 14 \lambda + 6 I ( λ ) = − λ 3 + 7 λ 2 − 14 λ + 6 K 2 = 0 K_{2} = 0 K 2 = 0 K 3 = 0 K_{3} = 0 K 3 = 0 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 − 7 λ 2 + 14 λ − 6 = 0 \lambda^{3} - 7 \lambda^{2} + 14 \lambda - 6 = 0 λ 3 − 7 λ 2 + 14 λ − 6 = 0 λ 1 = 3 \lambda_{1} = 3 λ 1 = 3 λ 2 = 2 − 2 \lambda_{2} = 2 - \sqrt{2} λ 2 = 2 − 2 λ 3 = 2 + 2 \lambda_{3} = \sqrt{2} + 2 λ 3 = 2 + 2 entonces la forma canónica de la ecuación será
( x ~ 1 2 λ 3 + ( x ~ 2 2 λ 2 + x ~ 3 2 λ 1 ) ) + I 4 I 3 = 0 \left(\tilde x1^{2} \lambda_{3} + \left(\tilde x2^{2} \lambda_{2} + \tilde x3^{2} \lambda_{1}\right)\right) + \frac{I_{4}}{I_{3}} = 0 ( x ~ 1 2 λ 3 + ( x ~ 2 2 λ 2 + x ~ 3 2 λ 1 ) ) + I 3 I 4 = 0 x ~ 1 2 ( 2 + 2 ) + x ~ 2 2 ( 2 − 2 ) + 3 x ~ 3 2 = 0 \tilde x1^{2} \left(\sqrt{2} + 2\right) + \tilde x2^{2} \left(2 - \sqrt{2}\right) + 3 \tilde x3^{2} = 0 x ~ 1 2 ( 2 + 2 ) + x ~ 2 2 ( 2 − 2 ) + 3 x ~ 3 2 = 0 x ~ 1 2 ( 1 2 + 2 ) 2 + ( x ~ 2 2 ( 1 2 − 2 ) 2 + x ~ 3 2 ( 3 3 ) 2 ) = 0 \frac{\tilde x1^{2}}{\left(\frac{1}{\sqrt{\sqrt{2} + 2}}\right)^{2}} + \left(\frac{\tilde x2^{2}}{\left(\frac{1}{\sqrt{2 - \sqrt{2}}}\right)^{2}} + \frac{\tilde x3^{2}}{\left(\frac{\sqrt{3}}{3}\right)^{2}}\right) = 0 ( 2 + 2 1 ) 2 x ~ 1 2 + ( 2 − 2 1 ) 2 x ~ 2 2 + ( 3 3 ) 2 x ~ 3 2 = 0 es la ecuación para el tipo cono imaginario
- está reducida a la forma canónica