# Understanding line of sight between two satellite with oblate Earth using “scaling k-component”

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...

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.