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Why does the ground track of orbits resemble sine waves?

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Why does the ground track of orbits resemble sine waves?

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The orbits of all satellites lie in a plane, whether the orbit is circular, elliptical or geosynchronous. Picture a giant piece of cardboard slicing through the equator of the Earth. A satellite circling the Earth at the equator would be constantly touching the cardboard and the orbit would always be above the equator. On a map, the orbit would be a straight line along the equator. Now picture the cardboard tipping so that one side moves north and the other side moves south. A satellite orbiting in that plane now would spend half its time above the equator and half its time below the equator when viewed on a map. So the track of the orbit on the map would go from the equator northward to say the 35th degree of latitude, then back down to the equator, then southward to the 35th degree of south latitude, then back up to the equator. When you look at this orbit on a flat map, the sine wave form appears. Yet all the time, the satellite is orbiting in a circle in a single plane. I hope that

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