Solid angle subtended by an ellipsoid

Let the the three semi-lengths be $(a,b,c)$ and let the corresponding unit vectors be $(\vec{u}_1,\vec{u}_2,\vec{u}_3)$ . Conceptualize a unit sphere around the given point $\vec{P}$ . Calculate the solid angle as follows:

$\large{\Omega = \int_0^{\pi} \int_0^{2 \pi} M(\theta,\phi) \, sin \phi \, d\theta \, d\phi}$

To calculate $M(\theta,\phi)$ , project the vector $\vec{u}$ (defined below) from $\vec{P}$ through space and see if it intersects the ellipsoid:

$\large{\vec{u}_x = cos \theta \, sin \phi \\ \vec{u}_y = sin \theta \, sin \phi \\ \vec{u}_z = cos \phi \\ \vec{P}_x + \alpha \, \vec{u}_x = \sigma \, \vec{u}_{1x} + \beta \, \vec{u}_{2x} + \gamma \, \vec{u}_{3x} \\ \vec{P}_y + \alpha \, \vec{u}_y = \sigma \, \vec{u}_{1y} + \beta \, \vec{u}_{2y} + \gamma \, \vec{u}_{3y} \\ \vec{P}_z + \alpha \, \vec{u}_z = \sigma \, \vec{u}_{1z} + \beta \, \vec{u}_{2z} + \gamma \, \vec{u}_{3z} \\ \frac{\sigma^2}{a^2} + \frac{\beta^2}{b^2} + \frac{\gamma^2}{c^2} = 1 }$

If, for a given $(\theta,\phi)$ , there exists a set of numbers $(\alpha, \sigma, \beta, \gamma)$ satisfying the last four equations above, $M(\theta,\phi) = 1$ . Otherwise, $M(\theta,\phi) = 0$ . This determination can be made analytically or through the use of search.