Se da la ecuación de superficie de 2 grado:
$$64 x^{2} - 1536 x - 25 y^{2} + 1600 z + 8576 = 0$$
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
$$a_{11} = 64$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = -768$$
$$a_{22} = -25$$
$$a_{23} = 0$$
$$a_{24} = 0$$
$$a_{33} = 0$$
$$a_{34} = 800$$
$$a_{44} = 8576$$
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
$$I_{1} = 39$$
|64 0 | |-25 0| |64 0|
I2 = | | + | | + | |
|0 -25| | 0 0| |0 0|
$$I_{3} = \left|\begin{matrix}64 & 0 & 0\\0 & -25 & 0\\0 & 0 & 0\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}64 & 0 & 0 & -768\\0 & -25 & 0 & 0\\0 & 0 & 0 & 800\\-768 & 0 & 800 & 8576\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}64 - \lambda & 0 & 0\\0 & - \lambda - 25 & 0\\0 & 0 & - \lambda\end{matrix}\right|$$
| 64 -768| |-25 0 | | 0 800 |
K2 = | | + | | + | |
|-768 8576| | 0 8576| |800 8576|
| 64 0 -768| |-25 0 0 | | 64 0 -768|
| | | | | |
K3 = | 0 -25 0 | + | 0 0 800 | + | 0 0 800 |
| | | | | |
|-768 0 8576| | 0 800 8576| |-768 800 8576|
$$I_{1} = 39$$
$$I_{2} = -1600$$
$$I_{3} = 0$$
$$I_{4} = 1024000000$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 39 \lambda^{2} + 1600 \lambda$$
$$K_{2} = -895360$$
$$K_{3} = -23936000$$
Como
$$I_{3} = 0 \wedge I_{2} \neq 0 \wedge I_{4} \neq 0$$
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
$$\lambda^{3} - 39 \lambda^{2} - 1600 \lambda = 0$$
$$\lambda_{1} = 64$$
$$\lambda_{2} = -25$$
$$\lambda_{3} = 0$$
entonces la forma canónica de la ecuación será
$$\tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
y
$$- \tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
$$64 \tilde x^{2} - 25 \tilde y^{2} + 1600 \tilde z = 0$$
y
$$64 \tilde x^{2} - 25 \tilde y^{2} - 1600 \tilde z = 0$$
$$2 \tilde z + \left(\frac{\tilde x^{2}}{\frac{25}{2}} - \frac{\tilde y^{2}}{32}\right) = 0$$
y
$$- 2 \tilde z + \left(\frac{\tilde x^{2}}{\frac{25}{2}} - \frac{\tilde y^{2}}{32}\right) = 0$$
es la ecuación para el tipo paraboloide hiperbólico
- está reducida a la forma canónica