I was using Orbitron - Satellite Tracking software the other day and I saw that clicking on any of the satellites, you will see their footprint (they are usually not-ideal circles, usually ellipses), which are later used to calculate/predict where and when your location will be in the satellites coverage/footprint.

I was wondering, how is a satellite's footprint calculated mathematically and what needs to be considered when calculating it?

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    $\begingroup$ In general this is complicated. You would need to know the antenna's beam pattern, and then essentially you intersect the surface of the planet with that beam pattern. Typically you would also cut off the beam at some nonzero power. Analytically this might be practically impossible, numerically it's just the problem of intersecting two objects, which has long ago been solved in computer graphics $\endgroup$ – Michael Stachowsky Feb 3 at 18:08
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    $\begingroup$ Here: its.bldrdoc.gov/publications/download/80-51_ocr.pdf is a file that explains how to do it under some relatively mild simplifying assumptions $\endgroup$ – Michael Stachowsky Feb 3 at 18:29
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    $\begingroup$ You need to know the shape and angle of the beam. This defines a cone with circular or elliptical cross section. Then you estimate the intersection of the cone with the spherical surface of the Earth. $\endgroup$ – Uwe Feb 3 at 19:01
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    $\begingroup$ Is the Orbitron footprint based on a broadcast antenna pattern for the satellite, or just on whether the satellite is above the horizon? $\endgroup$ – DJohnM Feb 3 at 20:44
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    $\begingroup$ For the purpose of drawing contours on a map of the world, the comments made so far are sufficient. If you are interested in the exact shape of the edge at a high degree of zoom, then the satellite is necessarily very low on the horizon, and terrain masking becomes important. You then need to know the terrain elevation in detail, and have a good way of interpolating between points (B-splines have worked well for me). Then you take your antenna beam pattern contour of interest, send out lots of rays along it, and see where they hit. $\endgroup$ – Ryan C Feb 3 at 23:55

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