Se da la ecuación de superficie de 2 grado:
$$4 x^{2} + 4 x y + 8 x z - 3 y^{2} + 3 z^{2} = 0$$
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
$$a_{11} = 4$$
$$a_{12} = 2$$
$$a_{13} = 4$$
$$a_{14} = 0$$
$$a_{22} = -3$$
$$a_{23} = 0$$
$$a_{24} = 0$$
$$a_{33} = 3$$
$$a_{34} = 0$$
$$a_{44} = 0$$
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
$$I_{1} = 4$$
|4 2 | |-3 0| |4 4|
I2 = | | + | | + | |
|2 -3| |0 3| |4 3|
$$I_{3} = \left|\begin{matrix}4 & 2 & 4\\2 & -3 & 0\\4 & 0 & 3\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}4 & 2 & 4 & 0\\2 & -3 & 0 & 0\\4 & 0 & 3 & 0\\0 & 0 & 0 & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}4 - \lambda & 2 & 4\\2 & - \lambda - 3 & 0\\4 & 0 & 3 - \lambda\end{matrix}\right|$$
|4 0| |-3 0| |3 0|
K2 = | | + | | + | |
|0 0| |0 0| |0 0|
|4 2 0| |-3 0 0| |4 4 0|
| | | | | |
K3 = |2 -3 0| + |0 3 0| + |4 3 0|
| | | | | |
|0 0 0| |0 0 0| |0 0 0|
$$I_{1} = 4$$
$$I_{2} = -29$$
$$I_{3} = 0$$
$$I_{4} = 0$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 4 \lambda^{2} + 29 \lambda$$
$$K_{2} = 0$$
$$K_{3} = 0$$
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
$$\lambda^{3} - 4 \lambda^{2} - 29 \lambda = 0$$
$$\lambda_{1} = 2 - \sqrt{33}$$
$$\lambda_{2} = 2 + \sqrt{33}$$
$$\lambda_{3} = 0$$
entonces la forma canónica de la ecuación será
$$\left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) + \frac{K_{3}}{I_{2}} = 0$$
$$\tilde x^{2} \left(2 - \sqrt{33}\right) + \tilde y^{2} \left(2 + \sqrt{33}\right) = 0$$
$$\frac{\tilde x^{2}}{\left(\frac{1}{\sqrt{-2 + \sqrt{33}}}\right)^{2}} - \frac{\tilde y^{2}}{\left(\frac{1}{\sqrt{2 + \sqrt{33}}}\right)^{2}} = 0$$
es la ecuación para el tipo dos planos intersectantes
- está reducida a la forma canónica