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Earth Coverage by Satellites in Circular Orbit

The purpose of many satellites is to observe or communicate with points on Earth’s surface. Such functions require a line of sight that is neither too long nor too oblique, so only a certain segment of Earth will be covered by a satellite at any given time. Here the term “covered” is meant to include “can communicate with” if the purpose of the satellite is communications. The shape of this covered segment depends on circumstances — it might be a thin rectangle of width w for a satellite-borne side-looking radar, for example. The shape will always be taken to be a spherical cap here, but a rough equivalence with other shapes could be made by letting the largest dimension, (w, for the radar mentioned above) span the cap. The questions dealt with will be of the type “what fraction of Earth does a satellite system cover?” or “How long will it take a satellite to find something?” Both of those questions need to be made more precise before they can be answered.