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This question introduces a formula for the evaluation of the solid angle corresponding to an ellipsoid. However, it is an approximation. Instead, this paper represents a general surface formula for conical surfaces as follows

$$\Omega = 2\pi - \oint dl \sqrt{\vec{u}^2 - (\vec{s} \cdot \vec{u})^{2}},$$

where $s$ is a parametric curve, whose projected curve can be parameterized by the angle (being a linear distance on the surface of the unit sphere) $l$, so that the curve on the sphere is defined as $\vec{s}(l)$, and

$$\vec{u}:= \frac{d^{2}}{dl^{2}}\vec{s}.$$

The paper computed the solid angle corresponding to an sphere, and I need to use the formula for the case of an ellipse as below. However, I'm not clear about the $\vec{u}$ in this case. In particular, the parameterization of an ellipsoid is

$$\begin{align} x & = a \rho \sin(\theta) \sin(\varphi), \\ y & = b \rho \cos(\theta) \sin(\varphi), \\ z & = c \rho \cos(\varphi). \end{align}$$

So, one reads

$$\vec{u} = \frac{d^{2}\vec{s}(\varphi)}{dl^{2}} = \frac{d^{2}\vec{s}(\varphi)}{d\varphi^{2}}(\frac{d\varphi}{dl})^{2}$$

In the sphere case, we have $dl = \sin{\theta}d\varphi$. But what is the evaluation of $\frac{d\varphi}{dl}$ in the ellipsoid case?

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