Why would a geosynchronous satellite have to be in one line over the equator? Why not a higher inclination?

Most people use geosynchronous as a synonym for geostationary.

https://en.wikipedia.org/wiki/Geosynchronous_orbit Popularly or loosely, the term geosynchronous may be used to mean geostationary.[2] Specifically, geosynchronous Earth orbit (GEO) may be a synonym for geosynchronous equatorial orbit,[3] or geostationary Earth orbit.[4] Communications satellites are often given geostationary or close to geostationary orbits so that the satellite antennas that communicate with them do not have to move, but can be pointed permanently at the fixed location in the sky where the satellite appears.

Certainly the comment I responded to (which has been edited to be non-sensical since) meant geostationary. In any case, GPS are not geosynchronous in the general sense of the term, but rather semi-synchronous (period of half a sidereal day). These orbits were historically chosen for convenience, but the syncronicity is not at all a requirement for global positioning satellites: e.g Galileo, GLONASS.

They can have inclination other than 0. If inclination is 0 thats a "geostationary" orbit, a special case of geosynchronous. In fact these are not stable and degrade into mere geosynchronous orbits without station keeping. In general though geostationary orbits are more useful since they are fixed competely in the sky. Ground antenna dont have to track the satellite as it moves north/south in the sky.

In this case a pair of satellites in a Tundra orbit [1] inclined to spend most of its time over this particular city seems like the right orbit for the job (as silly as it is to light a city by satellite), which is why I was curious to get an explanation behind the sweeping statement that geosynchronous (not geostationary in particular) orbits had to be in a line along the equator.

1. https://en.wikipedia.org/wiki/Tundra_orbit

Oh thats interesting. I hadn't heard of Tundra orbits. Wiki says 2 can provide continous coverage over an area. Wouldnt 1 be enough to act as a moon then? You only need it for half of the day and their period is 1 day right?

The problem with using just one is that geosynchronous orbital periods are one sidereal [1] day, not one solar day. So depending on the time of the year, the satellite would be in position in the wrong half of the day. On the other hand a satellite in Tundra orbit spends not half but more than half of its time in the designated area. Maybe a good compromise orbit could be found.

1. https://en.wikipedia.org/wiki/Sidereal_time

If the orbit is inclined the satellite will no longer stay above a single point it will move north and south throughout the day. All orbits pass over the equator twice during each period.

That's true, but it doesn't answer my question.

Geosynchronous satellites can and do modify their inclinations and eccentricities to increase their coverage beyond one spot over the equator.

All GPS satellites are in a geosynchronous orbit.

Citation needed?

You may find this helpful:

http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps....

The nominal GPS configuration consists of a network of 24 satellites in high orbits around the Earth, but up to 30 or so satellites may be on station at any given time. Each satellite in the GPS constellation orbits at an altitude of about 20,000 km from the ground, and has an orbital speed of about 14,000 km/hour (the orbital period is roughly 12 hours - contrary to popular belief, GPS satellites are not in geosynchronous or geostationary orbits). The satellite orbits are distributed so that at least 4 satellites are always visible from any point on the Earth at any given instant (with up to 12 visible at one time).

No, they are not.