Computing where line of sight intersects a given altitude

[DFEvans]

I'm putting this request in after finding this SO post from a few years ago, where there was interest in learning of possible use cases for "find where a line of sight intersects an altitude": https://stackoverflow.com/questions/62654081/path-between-two-topos-locations-determine-latitude-and-longitude-where-a-giv

My first use case is computing the coordinate that a satellite is actually observing on the Earth's surface when asked to image a location with an altitude significantly above or below the surface. The underlying idea is that by targeting a point under the Earth's surface, the actual target point will move slightly as the satellite moves, as it's not tracking the ground. This isn't a fully accurate solution as a DEM should ideally be involved, but for 20 lines of Python to play around with the idea, Skyfield is already enough.

The the image below, Green points are satellite positions, Red the target with some negative altitude, and yellow the 0m altitude point between the two (using the method in that SO post).

The other use case I can think of is a bit more hypothetical, but would be around computing the path length through the atmosphere below X km and the ground when observing a target from space (or a space target from the ground). Given ground target X, find the point Y on the LOS where the altitude is A, and then find the distance between X and Y. This would relate generally to atmospheric effects on light when observing. Again, Skyfield probably isn't the tool to do a full analysis of that type of problem, but it's an easy way to have an initial play around.