This would be an exercise in spherical geometry (or, if one wants to be very detailed, oblate spheroidal geometry). The distance to the horizon of an object at some height above the earth's surface (if there is no atmospheric refraction) is the distance to the tangent point of a line drawn from the object to the surface. Take a look at this link for some insight:

http://mintaka.sdsu.edu/GF/explain/atmos_refr/horizon.html

Note that I am pretty sure that GeoVisibleRegion does not have a means of calculating that take refraction models into account and so it will generally give incorrect results for computations that involve the passage of the ray through significant portions of the earth's atmosphere.

A general rule of thumb for lowish altitudes (e.g. below a kilometer or so and for adiabatic atmospheres) is to replace the no-refraction calculation with the same calculation but using a radius for the earth (in a spherical approximation) that is 4/3 times the actual radius. GeoVisibleRegion does not have a means to do this.

This is an important issue as it renders GeoVisibleRegion's results generally incorrect for any situations in which atmospheric refraction is pertinent. For the area of the earth visible from an orbiting satellite this is generally not an issue, but for the horizon limitations for an visible, infrared, radar, or other sensor within the atmosphere, the result is often significantly off.