angles¶
Module: core
Category: mat3dvaluator
Type string: "angles"
Parameters¶
| Name | Description | Default | Units |
|---|---|---|---|
theta |
theta | 0 | [] |
phi |
phi | 90 | [] |
Description¶
The angles mat3d valuator uses spherical angles to set up three perpendicular vectors to produce a orthonormal matrix.
theta: the azimuth angle (angle in xy plane and \(0\leqslant\theta<2\pi\))phi: declination angle (angle with respect to the z-axis and \(0\leqslant\phi\leqslant\pi\))
The first vector is defined via,
\[
\mathbf{e}_{r}=\cos\theta\sin\phi\,\mathbf{e}_{1}+\sin\theta\sin\phi\,\mathbf{e}_{2}+\cos\phi\,\mathbf{e}_{3}
\]
The second and third vectors are defined as follows,
\[
\mathbf{e}_{\theta}=-\sin\theta\,\mathbf{e}_{1}+\cos\theta\,\mathbf{e}_{2}
\]
\[
\mathbf{e}_{\phi}=-\cos\theta\cos\phi\,\mathbf{e}_{1}-\sin\theta\cos\phi\,\mathbf{e}_{2}+\sin\phi\,\mathbf{e}_{\phi}
\]
The matrix is formed by using these three vectors as columns.
\[
\mathbf{Q}=\left[ \mathbf{e}_{r} | \mathbf{e}_{\theta} | \mathbf{e}_{\phi} \right]
\]