Relationships between $\widehat{\gamma}$, $\tilde{\gamma}$ and $\gamma$

The angle $\tilde{\gamma}$ can be calculated from $\widehat{\gamma}=\arctan \left.\frac{\partial z_\xi}{\partial h_\xi} \right\vert _{m_\xi=\widebar m_\xi}$ and $\widehat{\alpha_x}=\arctan \left.\frac{\partial z_\xi}{\partial m_\xi} \right\vert _{h_\xi=\widebar h_\xi}$ by solving the two quadratic equations given by equations (7) and (8):

$$\begin{eqnarray} \left[ \Delta_\widetilde{S} \tan \widehat{\alpha_x} - \left(\De... ...detilde{S} \tan \widehat{\gamma} - \tan \widehat{\alpha_x} &=&0.\end{eqnarray}$$ (15) (16)

The angle $\gamma$ can be estimated from $\tilde{\gamma}$ using

$$\begin{eqnarray} \tan \gamma = \frac {\tan \tilde{\gamma} + \frac{1}{\widetilde... ...e{V}}\frac{d \widetilde{V}}{d\tilde{\gamma}} \tan \tilde{\gamma}}.\end{eqnarray}$$ (17)