en forma matricial $$ \begin{equation*} v = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} + \alpha_{3} x^{3} = \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} \end{equation*} $$
la deformación angular es $$ \begin{equation*} \theta = \frac{d v}{ d x} = \alpha_{1} + 2 \alpha_{2} x + 3 \alpha_{3} x^{2} \end{equation*} $$
reemplazando \( x_{1} \) y \( x_{2} \) $$ \begin{align*} \alpha_{0} + \alpha_{1} x_{1} + \alpha_{2} x_{1}^{2} + \alpha_{3} x_{1}^{3} &= v_{1} \\ \alpha_{1} + 2 \alpha_{2} x_{1} + 3 \alpha_{3} x_{1}^{2} &= \theta_{1} \\ \alpha_{0} + \alpha_{1} x_{2} + \alpha_{2} x_{2}^{2} + \alpha_{3} x_{2}^{3} &= v_{2} \\ \alpha_{1} + 2 \alpha_{2} x_{2} + 3 \alpha_{3} x_{2}^{2} &= \theta_{2} \end{align*} $$
en forma matricial $$ \begin{equation*} \begin{bmatrix} 1 & x_{1} & x_{1}^{2} & x_{1}^{3} \\ 0 & 1 & 2 x_{1} & 3 x_{1}^{2} \\ 1 & x_{2} & x_{2}^{2} & x_{2}^{3} \\ 0 & 1 & 2 x_{2} & 3 x_{2}^{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} = \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{equation*} $$
resolviendo el sistema $$ \begin{equation*} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} = \begin{bmatrix} \frac{x_{2}^{2} (3 x_{1} - x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{2}^{2} x_{1}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{x_{1}^{2} (-3 x_{2} + x_{1})}{(-x_{2} + x_{1})(x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{1}^{2} x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ -\frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{2} (2 x_{1} + x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} -2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{1} (x_{1} + 2 x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ \frac{3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{1} + 2 x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & -\frac{3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{2 x_{1} + x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ -\frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{equation*} $$
reemplazando las incógnitas $$ \begin{align*} v &= \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} \\ &= \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} \frac{x_{2}^{2} (3 x_{1} - x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{2}^{2} x_{1}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{x_{1}^{2} (-3 x_{2} + x_{1})}{(-x_{2} + x_{1})(x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{1}^{2} x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ -\frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{2} (2 x_{1} + x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} -2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{1} (x_{1} + 2 x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ \frac{3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{1} + 2 x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & -\frac{3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{2 x_{1} + x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ -\frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \\ &= \begin{bmatrix} \frac{x_{2}^{2} (3 x_{1} - x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} - \frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x + \frac{3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{2} - \frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} {x}^{3} \\ - \frac{x_{2}^{2} x_{1}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} + \frac{x_{2} (2 x_{1} + x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} x - \frac{x_{1} + 2 x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} x^{2} + \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} {x}^{3} \\ \frac{x_{1}^{2} (-3 x_{2} + x_{1})}{(-x_{2} + x_{1}) ( x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} + \frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x - \frac {3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} {x}^{2} + \frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{3} \\ - \frac{x_{1}^{2} x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} + \frac{x_{1} (x_{1} + 2 x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} x - \frac{2 x_{1} + x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} x^{2} + \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} x^{3} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \\ &= \begin{bmatrix} N_{1} \\ N_{2} \\ N_{3} \\ N_{4} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{align*} $$