I'm trying to figure out an expression for the fraction of a given latitude circle (i.e. a small circle parallel to the equator) that is visible from a given position above the Earth (or equivalent sphere).
For example, from above the north pole (latitude = $90^\circ$), all $360^\circ$ of each latitude line are visible for all northern latitudes down to the equator, and none of the southern latitudes are visible. Conversely, from a viewpoint above the equator (latitude = $0^\circ$), exactly half ($180^\circ$) of each latitude is visible, for all latitudes from $-90^\circ$ to $+90^\circ$. (In all cases I'm assuming that I'm far enough from the sphere for its size not to matter, e.g. many radii away, and [if we want to use the Earth as our model object] ignoring asphericity of the Earth.)
Now, if my viewpoint is above, say, a latitude of $+80^\circ$ (shifting down $10^\circ$ from the polar viewpoint), then I can still see 100% of all latitudes from $+90^\circ$ down to $+10^\circ$, but from $+10^\circ$ to $-10^\circ$ I only see a fraction of the latitude circle. What is that fraction as a function of latitude? It must be 100% for $+10^\circ$, 50% for $0^\circ$ and 0% for $-10^\circ$, but I don't think it's linear as a function of latitude, and I haven't come up with a formula for it.
One possible approach could be to rotate the coordinate system so that the vantage point becomes the new pole, and then see which of the transformed coordinates have positive latitudes, but I haven't quite worked out how to do that (and maybe there's a better approach). Thanks for any thoughts!