User talk:Zero Memory

Équation du cylindre :



x^2 + y^2 = r^2 \, $$



-h \leq z \leq h  \, $$

Équation du rayon :

R(t) = R_0 + t\overrightarrow{R_d} \, $$

Ce rayon est substitué dans l'équation du cylindre :

(x_0 + tx_d)^2 + (y_0 + ty_d)^2 = 1 \, $$ Une fois développé :

A^2t + Bt + C = 0 \, $$



A = x_d^2 + y_d^2 \, $$



B = 2(x_0x_d + y_oy_d)\, $$



C = x_0^2 +y_0^2 -r^2 \, $$

On résoud ainsi avec l'équation de degrée 2

\frac{ -b \pm \sqrt{b^2 - 4ac} }{2a} \, $$



z_0 + tz_d = \pm 1 \, $$



x^2 + y^2 \leq r^2 $$



\Theta_1 = \frac{-2gsin(\Theta_1) - \Theta_2cos(\Theta_1 - \Theta_2) - (\Theta_2')^2sin(\Theta_1 - \Theta_2)}{2} $$

\Theta_2 = -gsin(\Theta_2) - \Theta_1cos(\Theta_1 - \Theta_2) + (\Theta_1')^2sin(\Theta_1 - \Theta_2) \, $$



\Theta_2 + (\frac{-2gsin(\Theta_1) - \Theta_2cos(\Theta_1 - \Theta_2) - (\Theta_2')^2sin(\Theta_1 - \Theta_2)}{2})cos(\Theta_1 - \Theta_2) - (\Theta_1')^2sin(\Theta_1-\Theta_2) = -gsin(\Theta_2) $$

\Theta_2'' = \frac{-2gsin(\Theta_1) + 2(\Theta_1')^2sin(\Theta_1 - \Theta_2) + 2gsin(\Theta_1)cos(\Theta_1-\Theta_2) + (\Theta_2')^2sin(\Theta_1-\Theta_2)cos(\Theta_1-\theta_2)}{2-(cos^2(\Theta_1-\Theta_2))} $$



2\Theta_1 + (-gsin(\Theta_2) - \Theta_1cos(\Theta_1-\Theta_2) + (\Theta_1')^2sin(\Theta_1-\Theta_2))cos(\Theta_1-\Theta_2) + (\Theta_2')^2sin(\Theta_1-\Theta_2) = -2gsin(\Theta_1)\, $$

\Theta_1'' = \frac{-2gsin(\Theta_1) + gsin(\Theta_2)cos(\Theta_1-\Theta_2) - (\Theta_1')^2sin(\Theta_1-\Theta_2)cos(\Theta_1-\Theta_2) - (\Theta_2')^2sin(\Theta_1-\Theta_2)}{2-cos^2(\Theta_1-\Theta_2)} $$



m_1 = 1 \, m_2 = 1 \, $$

L_1 = 1 \, L_2 = 1 \, $$



(m_1 + m_2)L_1^2 \Theta_1 + m_2L_1L_2\Theta_2cos(\Theta_1 - \Theta_2) + m_2L_1L_2(\Theta_2')^2 sin(\Theta_1-\Theta_2) = -(m_1+m_2)L_1gsin(\Theta_1) $$

m_2L_2^2\Theta_2 + m_2L_1L_2\Theta_1cos(\Theta_1-\Theta_2) - m_2L_1L_2(\Theta_1')^2sin(\Theta_1 - \Theta_2) = -(m_2)L_2gsin(\Theta_2) $$



2\Theta_1 + \Theta_2cos(\Theta_1 - \Theta_2) + (\Theta_2')^2 sin(\Theta_1-\Theta_2) = -2gsin(\Theta_1) \, $$

\Theta_2 + \Theta_1cos(\Theta_1-\Theta_2) - (\Theta_1')^2sin(\Theta_1 - \Theta_2) = -gsin(\Theta_2) \, $$



\Theta_1(0) = \frac{\pi}{4} $$

\Theta_1'(0) = 0\, $$

\Theta_2(0) = 0\, $$

\Theta_2'(0) = 0\, $$


 * $$\int_0^{1} \! {dx\over {1+x+x^2}}$$


 * $$ F(x) = \frac{2}{3} \sqrt{3} arctan (\frac{1}{3}(2x + 1)\sqrt{3} ) $$


 * $$ F(1) - F(0) = \frac{\pi}{9} \sqrt{3} = 0,60459978807807261686469275254739 $$

Simpson 1/3 :
 * $$ S_i = \frac{h}{3}( f_i + 4f_{i+1} + f_{i+2} ) $$

Simpson 3/8 :
 * $$ S_i = \frac{3h}{8} ( f_i + 3f_{i+1} + 3f_{i+2} + f_{i+3} ) $$



\begin{pmatrix} x^{[n+1]} \\ y^{[n+1]} \end{pmatrix} = \begin{pmatrix} x^{[n]} \\ y^{[n]} \end{pmatrix} + \begin{pmatrix} \nabla x^{[n]} \\ \nabla y^{[n]} \end{pmatrix} $$

\begin{pmatrix} \nabla x^{[n]} \\ \nabla y^{[n]} \end{pmatrix} = - \begin{pmatrix} f(x^{[n]},y^{n]}) \\ g(x^{[n]},y^{[n]}) \end{pmatrix} \begin{pmatrix} \frac{\partial f(x^{[n]},y^{[n]})}{\partial x} && \frac{\partial f(x^{[n]},y^{[n]})}{\partial y} \\ \frac{\partial g(x^{[n]},y^{[n]})}{\partial x} && \frac{\partial g(x^{[n]},y^{[n]})}{\partial y} \end{pmatrix}^{-1} $$
 * $$A^{-1} = \begin{bmatrix}

a & b \\ c & d \\ \end{bmatrix}^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix} $$