Se da la ecuación de la línea de 2-o orden:
$$4 x^{2} + 4 x y + 16 x + y^{2} + 8 y + 16 = 0$$
Esta ecuación tiene la forma:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x + a_{22} y^{2} + 2 a_{23} y + a_{33} = 0$$
donde
$$a_{11} = 4$$
$$a_{12} = 2$$
$$a_{13} = 8$$
$$a_{22} = 1$$
$$a_{23} = 4$$
$$a_{33} = 16$$
Calculemos el determinante
$$\Delta = \left|\begin{matrix}a_{11} & a_{12}\\a_{12} & a_{22}\end{matrix}\right|$$
o, sustituimos
$$\Delta = \left|\begin{matrix}4 & 2\\2 & 1\end{matrix}\right|$$
$$\Delta = 0$$
Como
$$\Delta$$
es igual a 0, entonces
Hacemos el giro del sistema de coordenadas obtenido al ángulo de φ
$$x' = \tilde x \cos{\left(\phi \right)} - \tilde y \sin{\left(\phi \right)}$$
$$y' = \tilde x \sin{\left(\phi \right)} + \tilde y \cos{\left(\phi \right)}$$
φ - se define de la fórmula
$$\cot{\left(2 \phi \right)} = \frac{a_{11} - a_{22}}{2 a_{12}}$$
sustituimos coeficientes
$$\cot{\left(2 \phi \right)} = \frac{3}{4}$$
entonces
$$\phi = \frac{\operatorname{acot}{\left(\frac{3}{4} \right)}}{2}$$
$$\sin{\left(2 \phi \right)} = \frac{4}{5}$$
$$\cos{\left(2 \phi \right)} = \frac{3}{5}$$
$$\cos{\left(\phi \right)} = \sqrt{\frac{\cos{\left(2 \phi \right)}}{2} + \frac{1}{2}}$$
$$\sin{\left(\phi \right)} = \sqrt{1 - \cos^{2}{\left(\phi \right)}}$$
$$\cos{\left(\phi \right)} = \frac{2 \sqrt{5}}{5}$$
$$\sin{\left(\phi \right)} = \frac{\sqrt{5}}{5}$$
sustituimos coeficientes
$$x' = \frac{2 \sqrt{5} \tilde x}{5} - \frac{\sqrt{5} \tilde y}{5}$$
$$y' = \frac{\sqrt{5} \tilde x}{5} + \frac{2 \sqrt{5} \tilde y}{5}$$
entonces la ecuación se transformará de
$$4 x'^{2} + 4 x' y' + 16 x' + y'^{2} + 8 y' + 16 = 0$$
en
$$\left(\frac{\sqrt{5} \tilde x}{5} + \frac{2 \sqrt{5} \tilde y}{5}\right)^{2} + 4 \left(\frac{\sqrt{5} \tilde x}{5} + \frac{2 \sqrt{5} \tilde y}{5}\right) \left(\frac{2 \sqrt{5} \tilde x}{5} - \frac{\sqrt{5} \tilde y}{5}\right) + 8 \left(\frac{\sqrt{5} \tilde x}{5} + \frac{2 \sqrt{5} \tilde y}{5}\right) + 4 \left(\frac{2 \sqrt{5} \tilde x}{5} - \frac{\sqrt{5} \tilde y}{5}\right)^{2} + 16 \left(\frac{2 \sqrt{5} \tilde x}{5} - \frac{\sqrt{5} \tilde y}{5}\right) + 16 = 0$$
simplificamos
$$5 \tilde x^{2} + 8 \sqrt{5} \tilde x + 16 = 0$$
$$5 \tilde x^{2} + 8 \sqrt{5} \tilde x = -16$$
$$\left(\sqrt{5} \tilde x + 4\right)^{2} = 16$$
$$\left(\tilde x + \frac{4 \sqrt{5}}{5}\right)^{2} = \frac{16}{5}$$
$$\tilde x'^{2} = \frac{16}{5}$$
Esta ecuación es dos rectas paralelas
- está reducida a la forma canónica
donde se ha hecho la sustitución
$$\tilde x' = \tilde x + \frac{4 \sqrt{5}}{5}$$
$$\tilde y' = \tilde y$$
Centro de las coordenadas canónicas en Oxy
$$x_{0} = \tilde x \cos{\left(\phi \right)} - \tilde y \sin{\left(\phi \right)}$$
$$y_{0} = \tilde x \sin{\left(\phi \right)} + \tilde y \cos{\left(\phi \right)}$$
$$x_{0} = \frac{\left(-4\right) \sqrt{5}}{5} \frac{2 \sqrt{5}}{5} + 0 \frac{\sqrt{5}}{5}$$
$$y_{0} = \frac{\left(-4\right) \sqrt{5}}{5} \frac{\sqrt{5}}{5} + 0 \frac{2 \sqrt{5}}{5}$$
$$x_{0} = - \frac{8}{5}$$
$$y_{0} = - \frac{4}{5}$$
Centro de las coordenadas canónicas en el punto O
(-8/5, -4/5)
Base de las coordenadas canónicas
$$\vec e_1 = \left( \frac{2 \sqrt{5}}{5}, \ \frac{\sqrt{5}}{5}\right)$$
$$\vec e_2 = \left( - \frac{\sqrt{5}}{5}, \ \frac{2 \sqrt{5}}{5}\right)$$