inplane_angle_jacobian Subroutine

public pure subroutine inplane_angle_jacobian(ca, cb, cc, cx, thet, J_a, J_b, J_c, J_x)

Computes the Jacobian matrix for the inplane angle definition. It computes the Jacobian for the normal angle using the projected point (R) as central point. Then projects $J_r = \frac{\partial \theta}{\partial \vec{R}}$ onto A, B, C, and X (auxiliary point). The projection is done computing the 3x3 matrices of partial derivative of $\vec{R}$ wrt any actual point and using them to project $J_r$. [\frac{\partial \vec{R}}{\partial \vec{A}} = \begin{bmatrix} \frac{\partial \vec{R}_x}{\partial \vec{A}_x} & \frac{\partial \vec{R}_y}{\partial \vec{A}_x} & \frac{\partial \vec{R}_z}{\partial \vec{A}_x} \ !! \frac{\partial \vec{R}_x}{\partial \vec{A}_y} &

     \frac{\partial \vec{R}_y}{\partial \vec{A}_y} & 
     \frac{\partial \vec{R}_z}{\partial \vec{A}_y} \        !!          \frac{\partial \vec{R}_x}{\partial \vec{A}_z} &

     \frac{\partial \vec{R}_y}{\partial \vec{A}_z} & 
     \frac{\partial \vec{R}_z}{\partial \vec{A}_z} \        !!      \end{bmatrix} \]

Those matrices are computed using the chain rule.

Exemple: $$\vec{V} = \vec{B} - \vec{X}$$ $$\vec{P} = (\vec{A} - \vec{X}) \times (\vec{C} - \vec{X})$$ $$\hat{P} = \frac{\vec{P}}{||\vec{P}||}$$ $$\vec{R} = \vec{B} - (\vec{V}\cdot\hat{P})\hat{P}$$ $$J_a = \frac{\partial \theta}{\partial \vec{A}} = \left( \frac{\partial \theta}{\partial \vec{A}} \right)_\vec{R} + \frac{\partial \vec{R}}{\partial \vec{A}} \times J_r = \left( \frac{\partial \theta}{\partial \vec{A}} \right)_\vec{R} + - \frac{\partial \vec{P}}{\partial \vec{A}} \times \frac{\partial \hat{P}}{\partial \vec{P}} \times \frac{\partial (\vec{V} \cdot \hat{P})}{\partial \hat{P}} \times J_r$$

Arguments