# How does a satellite calculate a vector's intercept/path with Earth?

Let's say you have a satellite at altitude h, and you know the position of this satellite above earth in latitude and longitude.

You detect a signal with the satellite and want to know where on the Earth that signal is coming from. You know the direction you received the signal. So you have the satellite's origin, and a direction vector. How do you calculate what latitude and longitude that vector intercepts the Earth's surface at?

I'm trying to visualize the path of the vector, and because of the Earth's curvature its path (if lat/lon were projected into 2D space) would be curved, would it not? Is there a way to calculate by how much? Wouldn't the amount that it "dips" or curves also depend on where above the Earth you are (how far from the equator) and your altitude? Is there even a way to solve for this?

I'm imagining holding a straight line over the globe, and then shining a light down over it, to create a curved shadow on the surface of the globe, ie projecting the line onto the surface of the ellipsoid. Is there a way to analytically or numerically find this shadow's path?

What should I be googling in order to find more information on this topic? Thanks.

• Is this question about Mathematica or Wolfram Language, the software and programming language by Wolfram Research? – Carl Lange May 24 '19 at 16:40
• For a start see en.wikipedia.org/wiki/Satellite_geolocation – Bob Hanlon May 24 '19 at 18:51
• Very short answer is that it depends on the projection. It could be a geodesic if projected in one way and a non-geodesic if projected otherwise. – Radost May 29 '19 at 12:06
• try googling information about astrodynamics, orbital mechanics and remote sensing. There's a lot of information about doing these sort of projections for earth orbiting spacecraft. – JMJ May 29 '19 at 12:09