Se da la ecuación de superficie de 2 grado:
$$4 x_{1}^{2} - 4 x_{1} x_{2} + 2 x_{2}^{2} - 6 x_{2} x_{3} + 14 x_{3}^{2} = 0$$
Esta ecuación tiene la forma:
$$a_{11} x_{3}^{2} + 2 a_{12} x_{2} x_{3} + 2 a_{13} x_{1} x_{3} + 2 a_{14} x_{3} + a_{22} x_{2}^{2} + 2 a_{23} x_{1} x_{2} + 2 a_{24} x_{2} + a_{33} x_{1}^{2} + 2 a_{34} x_{1} + a_{44} = 0$$
donde
$$a_{11} = 14$$
$$a_{12} = -3$$
$$a_{13} = 0$$
$$a_{14} = 0$$
$$a_{22} = 2$$
$$a_{23} = -2$$
$$a_{24} = 0$$
$$a_{33} = 4$$
$$a_{34} = 0$$
$$a_{44} = 0$$
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
$$I_{1} = 20$$
|14 -3| |2 -2| |14 0|
I2 = | | + | | + | |
|-3 2 | |-2 4 | |0 4|
$$I_{3} = \left|\begin{matrix}14 & -3 & 0\\-3 & 2 & -2\\0 & -2 & 4\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}14 & -3 & 0 & 0\\-3 & 2 & -2 & 0\\0 & -2 & 4 & 0\\0 & 0 & 0 & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}14 - \lambda & -3 & 0\\-3 & 2 - \lambda & -2\\0 & -2 & 4 - \lambda\end{matrix}\right|$$
|14 0| |2 0| |4 0|
K2 = | | + | | + | |
|0 0| |0 0| |0 0|
|14 -3 0| |2 -2 0| |14 0 0|
| | | | | |
K3 = |-3 2 0| + |-2 4 0| + |0 4 0|
| | | | | |
|0 0 0| |0 0 0| |0 0 0|
$$I_{1} = 20$$
$$I_{2} = 79$$
$$I_{3} = 20$$
$$I_{4} = 0$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 20 \lambda^{2} - 79 \lambda + 20$$
$$K_{2} = 0$$
$$K_{3} = 0$$
Como
I3 != 0
entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
o
$$\lambda^{3} - 20 \lambda^{2} + 79 \lambda - 20 = 0$$
$$\lambda_{1} = 5$$
$$\lambda_{2} = \frac{15}{2} - \frac{\sqrt{209}}{2}$$
$$\lambda_{3} = \frac{\sqrt{209}}{2} + \frac{15}{2}$$
entonces la forma canónica de la ecuación será
$$\left(\tilde x1^{2} \lambda_{3} + \left(\tilde x2^{2} \lambda_{2} + \tilde x3^{2} \lambda_{1}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$\tilde x1^{2} \left(\frac{\sqrt{209}}{2} + \frac{15}{2}\right) + \tilde x2^{2} \left(\frac{15}{2} - \frac{\sqrt{209}}{2}\right) + 5 \tilde x3^{2} = 0$$
$$\frac{\tilde x1^{2}}{\left(\frac{1}{\sqrt{\frac{\sqrt{209}}{2} + \frac{15}{2}}}\right)^{2}} + \left(\frac{\tilde x2^{2}}{\left(\frac{1}{\sqrt{\frac{15}{2} - \frac{\sqrt{209}}{2}}}\right)^{2}} + \frac{\tilde x3^{2}}{\left(\frac{\sqrt{5}}{5}\right)^{2}}\right) = 0$$
es la ecuación para el tipo cono imaginario
- está reducida a la forma canónica