I need advice regarding a geometry problem.
I illustrate this geometry in the figure below:
Background of question:
There are two points (represented by blue and orange stars) with associated 3D positions (x,y,z). Also, I have earth (as a perfect sphere) and a outer 'ring' defined at 350 km above earth surface.
The vector between the stars intersect the outer 'ring' at two points. One of the intersection points (point k) is nearer to the blue star (x1,y1,z1).
Actual question
A 3d vector drawn between k and the center of the earth results in the vector intersecting one particular point on the surface of the earth, which can be represented in a latitude, longitude coordinate.
My question is hence, how can I obtain the coordinates of that point?
To find the XYZ of point K I would use something like this:
https://en.wikipedia.org/wiki/Line%E2%80%93sphere_intersection
or
https://stackoverflow.com/questions/5883169/intersection-between-a-line-and-a-sphere
Now, let's say you solve that.
How to convert to longitude latitude?
Assume that the xy-plane is the same as the Equator.
Assume that the Greenwich Line (0 longitude) lines up with the xz-plane.
Now, note that the Long Lat on the 'great sphere' 350km is exactly the same as the Long Lat on Earth, since you are projecting straight down towards the Earth's center. So forget about Earth, just focus on the 350km sphere.
Now, check this pic out
The point forms a kind of tetrahedron in space. Check out two of the triangles... one in the x-y plane and another coming up from that first triangle.
Longitude will be the angle between the X axis and the black hypoteneuse...
The Black hypoteneuse is going to be found through Pythagoras theorem...
The blue leg is just $y$, and the green leg is $x$.
$x^2+y^2$ gives the square of the Black Hypoteneuse. So it's length is $\sqrt{x^2+y^2}$
The angle will be longitude.
$$longitude = arcsin(y/\sqrt{x^2+y^2})$$
Now, what is latitude? It's the angle between the black hypoteneuse and the pink hypoteneuse.
The Red leg is going to have length $z$. It goes up to point $x,y,z$.
The pink hypoteneuse is going to be found using Pythagoras' theorem with the black hypoteneuse and the red leg.
So. $(x^2+y^2)+(z^2)$ gives the square of the pink-hypoteneuse. It's length will then be $\sqrt{x^2+y^2+z^2}$
So... the angle between black hypoteneuse and the pink hypoteneuse will be latitude.
$$latitude = arcsin(z/\sqrt{x^2+y^2+z^2})$$
Of course if you write this into a computer program you need to deal with a lot of issues about avoiding division by zero, the input range of arcsin, etc.