How Can There Be North Without Spin?

    Some of you may be thinking, "If we're ignoring the spin of the globe, how can we even have a North and South? Aren't those defined by the axis of rotation?" Well, it's easiest to define it that way, sure. But you don't need a spinning sphere in order to use a coordinate system that has a North Pole.
    Spherical Cartesian Coordinates (Wikipedia link) define position with three variables: ρ (radius from center), φ (latitude) and θ (longitude). The angle φ is defined as starting at some arbitrary direction, which is usually drawn as "up" on the page. Or, to relate it to a planet, it's measured as degrees down from the North Pole.
    When you have a spinning planet, it's convenient to use the axis of rotation to define the starting point for φ. However, the starting point for θ is totally arbitrary, even on a spinning sphere. By convention, on Earth we say θ=0° at the Prime Meridian (Wikipedia link), but it could be anywhere.
    On a non-spinning sphere, the starting point for φ is just as arbitrary as the starting point for θ. Once we've laid out our coordinates, moving in each cardinal direction is defined as follows:
  • Move North = Keep θ constant, decrease φ.
  • Move South = Keep θ constant, increase φ.
  • Move East = Increase θ, keep φ constant.
  • Move West = Decrease θ, keep φ constant.
    In any case, when you go back to the "Motion on a Globe" explanation, keep in mind that motion at constant θ results in a great circle, but motion at constant φ only gives you a great circle if you're at φ=90°. At any other value of φ, a free-flying object sent due "East" or "West" will soon change its value of φ. Geometrically speaking, a line at constant φ other than at the equator is not a "straight line" (or geodesic (Wikipedia link)) in the sense of being the shortest distance between two points. And free-flying objects will try to move along the shortest possible path.
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