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I have been doing some reading on accuracy of determining line-of-sight between two satellites give a WGS84 oblate earth. The paper Rapid Determination of Satellite Visibility Periods (Alfano, Negron Jr. and Moore, Journal of the Astronautical Sciences, Vol. 40, No. 2, Apr-Jun 92, pp. 281-296) talks about using the flatness/eccentricity of earth (from WGS84) to effectively scale the $\mathbf{\hat{k}}$ component and then perform the calculations of comparing central angles.

I have seen other equations based on ray-spheroid intersect calculations (mainly from video game design), and can't tell if this is a mathematical equivalent conversion or if there is error induced by the assumption. The paper doesn't seem to make any reference.


Figure 1 shows the geometry of the visibility function, with the position vectors forced to reside in an Earth-tangent plane. This function is determined by comparing the angle between $\mathbf{r_1}$ and $\mathbf{r_2}$ to the angles formed from the line-of-sight tangent to the spherical surface at $R_{Earth} + h$. To correct for oblateness, rescale the $\mathbf{\hat{k}}$ component of $\mathbf{r_1}$ and $\mathbf{r_2}$ by $1/\sqrt{1-\epsilon_e^2}$, where $\epsilon_e$ is the eccentricity of the Earth ellipsoid...

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The article assumes that the Earth's shape is an oblate spheroid, i.e., a "squashed" sphere. If we "unsquash" everything back, including satellites' positions (which means multiplying the $z$-coordinates of the satellites by $1/\sqrt{1-\epsilon_e^2}$), the line segment connecting the new positions would intersect the sphere if and only if the line segment connecting the actual positions intersects the spheroid.

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