Sr Examen
xy+yz+zt+xt=0 forma canónica
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Solución
Ha introducido
[src]
t*x + t*z + x*y + y*z = 0
$$t x + t z + x y + y z = 0$$
t*x + t*z + x*y + y*z = 0
Método de invariantes
Se da la ecuación de superficie de 2 grado:
$$t x + t z + x y + y z = 0$$
Esta ecuación tiene la forma:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
donde
$$a_{11} = 0$$
$$a_{12} = \frac{1}{2}$$
$$a_{13} = 0$$
$$a_{14} = \frac{t}{2}$$
$$a_{22} = 0$$
$$a_{23} = \frac{1}{2}$$
$$a_{24} = 0$$
$$a_{33} = 0$$
$$a_{34} = \frac{t}{2}$$
$$a_{44} = 0$$
Las invariantes de esta ecuación al transformar las coordenadas son los determinantes:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$
sustituimos coeficientes
$$I_{1} = 0$$
$$I_{3} = \left|\begin{matrix}0 & \frac{1}{2} & 0\\\frac{1}{2} & 0 & \frac{1}{2}\\0 & \frac{1}{2} & 0\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}0 & \frac{1}{2} & 0 & \frac{t}{2}\\\frac{1}{2} & 0 & \frac{1}{2} & 0\\0 & \frac{1}{2} & 0 & \frac{t}{2}\\\frac{t}{2} & 0 & \frac{t}{2} & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda & \frac{1}{2} & 0\\\frac{1}{2} & - \lambda & \frac{1}{2}\\0 & \frac{1}{2} & - \lambda\end{matrix}\right|$$
$$I_{1} = 0$$
$$I_{2} = - \frac{1}{2}$$
$$I_{3} = 0$$
$$I_{4} = 0$$
$$I{\left(\lambda \right)} = - \lambda^{3} + \frac{\lambda}{2}$$
$$K_{2} = - \frac{t^{2}}{2}$$
$$K_{3} = 0$$
Como
$$I_{3} = 0 \wedge I_{4} = 0 \wedge I_{2} \neq 0$$
entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
o
$$\lambda^{3} - \frac{\lambda}{2} = 0$$
$$\lambda_{1} = - \frac{\sqrt{2}}{2}$$
$$\lambda_{2} = \frac{\sqrt{2}}{2}$$
$$\lambda_{3} = 0$$
entonces la forma canónica de la ecuación será
$$\left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) + \frac{K_{3}}{I_{2}} = 0$$
$$- \frac{\sqrt{2} \tilde x^{2}}{2} + \frac{\sqrt{2} \tilde y^{2}}{2} = 0$$
$$\frac{\tilde x^{2}}{\left(\sqrt[4]{2}\right)^{2}} - \frac{\tilde y^{2}}{\left(\sqrt[4]{2}\right)^{2}} = 0$$
es la ecuación para el tipo dos planos intersectantes
- está reducida a la forma canónica
$$t x + t z + x y + y z = 0$$
Esta ecuación tiene la forma:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
donde
$$a_{11} = 0$$
$$a_{12} = \frac{1}{2}$$
$$a_{13} = 0$$
$$a_{14} = \frac{t}{2}$$
$$a_{22} = 0$$
$$a_{23} = \frac{1}{2}$$
$$a_{24} = 0$$
$$a_{33} = 0$$
$$a_{34} = \frac{t}{2}$$
$$a_{44} = 0$$
Las invariantes de esta ecuación al transformar las coordenadas son los determinantes:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
|a11 a12| |a22 a23| |a11 a13|
I2 = | | + | | + | |
|a12 a22| |a23 a33| |a13 a33|$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$
|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
$$I_{1} = 0$$
| 0 1/2| | 0 1/2| |0 0|
I2 = | | + | | + | |
|1/2 0 | |1/2 0 | |0 0|$$I_{3} = \left|\begin{matrix}0 & \frac{1}{2} & 0\\\frac{1}{2} & 0 & \frac{1}{2}\\0 & \frac{1}{2} & 0\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}0 & \frac{1}{2} & 0 & \frac{t}{2}\\\frac{1}{2} & 0 & \frac{1}{2} & 0\\0 & \frac{1}{2} & 0 & \frac{t}{2}\\\frac{t}{2} & 0 & \frac{t}{2} & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda & \frac{1}{2} & 0\\\frac{1}{2} & - \lambda & \frac{1}{2}\\0 & \frac{1}{2} & - \lambda\end{matrix}\right|$$
| t| | t|
|0 -| |0 -|
| 2| |0 0| | 2|
K2 = | | + | | + | |
|t | |0 0| |t |
|- 0| |- 0|
|2 | |2 | | t|
| t| | 0 1/2 0| |0 0 -|
| 0 1/2 -| | | | 2|
| 2| | t| | |
| | |1/2 0 -| | t|
K3 = |1/2 0 0| + | 2| + |0 0 -|
| | | | | 2|
| t | | t | | |
| - 0 0| | 0 - 0| |t t |
| 2 | | 2 | |- - 0|
|2 2 |$$I_{1} = 0$$
$$I_{2} = - \frac{1}{2}$$
$$I_{3} = 0$$
$$I_{4} = 0$$
$$I{\left(\lambda \right)} = - \lambda^{3} + \frac{\lambda}{2}$$
$$K_{2} = - \frac{t^{2}}{2}$$
$$K_{3} = 0$$
Como
$$I_{3} = 0 \wedge I_{4} = 0 \wedge I_{2} \neq 0$$
entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
o
$$\lambda^{3} - \frac{\lambda}{2} = 0$$
$$\lambda_{1} = - \frac{\sqrt{2}}{2}$$
$$\lambda_{2} = \frac{\sqrt{2}}{2}$$
$$\lambda_{3} = 0$$
entonces la forma canónica de la ecuación será
$$\left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) + \frac{K_{3}}{I_{2}} = 0$$
$$- \frac{\sqrt{2} \tilde x^{2}}{2} + \frac{\sqrt{2} \tilde y^{2}}{2} = 0$$
$$\frac{\tilde x^{2}}{\left(\sqrt[4]{2}\right)^{2}} - \frac{\tilde y^{2}}{\left(\sqrt[4]{2}\right)^{2}} = 0$$
es la ecuación para el tipo dos planos intersectantes
- está reducida a la forma canónica