Se da la ecuación de superficie de 2 grado:
x 2 + x y − x z + 4 x + y 2 + y z + 2 y + z 2 + 4 z − 6 = 0 x^{2} + x y - x z + 4 x + y^{2} + y z + 2 y + z^{2} + 4 z - 6 = 0 x 2 + x y − x z + 4 x + y 2 + yz + 2 y + z 2 + 4 z − 6 = 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 = 1 2 a_{12} = \frac{1}{2} a 12 = 2 1 a 13 = − 1 2 a_{13} = - \frac{1}{2} a 13 = − 2 1 a 14 = 2 a_{14} = 2 a 14 = 2 a 22 = 1 a_{22} = 1 a 22 = 1 a 23 = 1 2 a_{23} = \frac{1}{2} a 23 = 2 1 a 24 = 1 a_{24} = 1 a 24 = 1 a 33 = 1 a_{33} = 1 a 33 = 1 a 34 = 2 a_{34} = 2 a 34 = 2 a 44 = − 6 a_{44} = -6 a 44 = − 6 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 1/2| | 1 1/2| | 1 -1/2|
I2 = | | + | | + | |
|1/2 1 | |1/2 1 | |-1/2 1 | I 3 = ∣ 1 1 2 − 1 2 1 2 1 1 2 − 1 2 1 2 1 ∣ I_{3} = \left|\begin{matrix}1 & \frac{1}{2} & - \frac{1}{2}\\\frac{1}{2} & 1 & \frac{1}{2}\\- \frac{1}{2} & \frac{1}{2} & 1\end{matrix}\right| I 3 = 1 2 1 − 2 1 2 1 1 2 1 − 2 1 2 1 1 I 4 = ∣ 1 1 2 − 1 2 2 1 2 1 1 2 1 − 1 2 1 2 1 2 2 1 2 − 6 ∣ I_{4} = \left|\begin{matrix}1 & \frac{1}{2} & - \frac{1}{2} & 2\\\frac{1}{2} & 1 & \frac{1}{2} & 1\\- \frac{1}{2} & \frac{1}{2} & 1 & 2\\2 & 1 & 2 & -6\end{matrix}\right| I 4 = 1 2 1 − 2 1 2 2 1 1 2 1 1 − 2 1 2 1 1 2 2 1 2 − 6 I ( λ ) = ∣ 1 − λ 1 2 − 1 2 1 2 1 − λ 1 2 − 1 2 1 2 1 − λ ∣ I{\left(\lambda \right)} = \left|\begin{matrix}1 - \lambda & \frac{1}{2} & - \frac{1}{2}\\\frac{1}{2} & 1 - \lambda & \frac{1}{2}\\- \frac{1}{2} & \frac{1}{2} & 1 - \lambda\end{matrix}\right| I ( λ ) = 1 − λ 2 1 − 2 1 2 1 1 − λ 2 1 − 2 1 2 1 1 − λ |1 2 | |1 1 | |1 2 |
K2 = | | + | | + | |
|2 -6| |1 -6| |2 -6| | 1 1/2 2 | | 1 1/2 1 | | 1 -1/2 2 |
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K3 = |1/2 1 1 | + |1/2 1 2 | + |-1/2 1 2 |
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| 2 1 -6| | 1 2 -6| | 2 2 -6| I 1 = 3 I_{1} = 3 I 1 = 3 I 2 = 9 4 I_{2} = \frac{9}{4} I 2 = 4 9 I 3 = 0 I_{3} = 0 I 3 = 0 I 4 = − 27 4 I_{4} = - \frac{27}{4} I 4 = − 4 27 I ( λ ) = − λ 3 + 3 λ 2 − 9 λ 4 I{\left(\lambda \right)} = - \lambda^{3} + 3 \lambda^{2} - \frac{9 \lambda}{4} I ( λ ) = − λ 3 + 3 λ 2 − 4 9 λ K 2 = − 27 K_{2} = -27 K 2 = − 27 K 3 = − 63 2 K_{3} = - \frac{63}{2} K 3 = − 2 63 Como
I 3 = 0 ∧ I 2 ≠ 0 ∧ I 4 ≠ 0 I_{3} = 0 \wedge I_{2} \neq 0 \wedge I_{4} \neq 0 I 3 = 0 ∧ I 2 = 0 ∧ I 4 = 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 + 9 λ 4 = 0 \lambda^{3} - 3 \lambda^{2} + \frac{9 \lambda}{4} = 0 λ 3 − 3 λ 2 + 4 9 λ = 0 λ 1 = 3 2 \lambda_{1} = \frac{3}{2} λ 1 = 2 3 λ 2 = 3 2 \lambda_{2} = \frac{3}{2} λ 2 = 2 3 λ 3 = 0 \lambda_{3} = 0 λ 3 = 0 entonces la forma canónica de la ecuación será
z ~ 2 ( − 1 ) I 4 I 2 + ( x ~ 2 λ 1 + y ~ 2 λ 2 ) = 0 \tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0 z ~ 2 I 2 ( − 1 ) I 4 + ( x ~ 2 λ 1 + y ~ 2 λ 2 ) = 0 y
− z ~ 2 ( − 1 ) I 4 I 2 + ( x ~ 2 λ 1 + y ~ 2 λ 2 ) = 0 - \tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0 − z ~ 2 I 2 ( − 1 ) I 4 + ( x ~ 2 λ 1 + y ~ 2 λ 2 ) = 0 3 x ~ 2 2 + 3 y ~ 2 2 + 2 3 z ~ = 0 \frac{3 \tilde x^{2}}{2} + \frac{3 \tilde y^{2}}{2} + 2 \sqrt{3} \tilde z = 0 2 3 x ~ 2 + 2 3 y ~ 2 + 2 3 z ~ = 0 y
3 x ~ 2 2 + 3 y ~ 2 2 − 2 3 z ~ = 0 \frac{3 \tilde x^{2}}{2} + \frac{3 \tilde y^{2}}{2} - 2 \sqrt{3} \tilde z = 0 2 3 x ~ 2 + 2 3 y ~ 2 − 2 3 z ~ = 0 2 z ~ + ( x ~ 2 2 3 3 + y ~ 2 2 3 3 ) = 0 2 \tilde z + \left(\frac{\tilde x^{2}}{\frac{2}{3} \sqrt{3}} + \frac{\tilde y^{2}}{\frac{2}{3} \sqrt{3}}\right) = 0 2 z ~ + ( 3 2 3 x ~ 2 + 3 2 3 y ~ 2 ) = 0 y
− 2 z ~ + ( x ~ 2 2 3 3 + y ~ 2 2 3 3 ) = 0 - 2 \tilde z + \left(\frac{\tilde x^{2}}{\frac{2}{3} \sqrt{3}} + \frac{\tilde y^{2}}{\frac{2}{3} \sqrt{3}}\right) = 0 − 2 z ~ + ( 3 2 3 x ~ 2 + 3 2 3 y ~ 2 ) = 0 es la ecuación para el tipo paraboloide elíptico
- está reducida a la forma canónica
