Tuesday, August 14, 2012

Navigating Near the Poles

If one looks at the Pythagorean Theorem for spherical coordinates which specifies the approximate distance between two points one will notice that it breaks down at the poles where changes in longitude do not correspond to a change in distance. There is also a problem with the formula for a heading since these are defined relative to the direction of north. The reason for this is that the spherical coordinate system is defined relative to the equator and could be called equatorial spherical coordinates in which the primary directions are along the equator and towards the pole. One could define a polar spherical coordinate system for use in a hemisphere. We would use the same meridians but we need another parameter to measure distance. The natural choice would be the distance along a meridian from the pole. The heading in this system would be defined as its relative direction to the that of the prime meridian. One could use parallel transport of the heading to the pole while keeping the angle relative to the meridian constant to more clearly specify the definition. The angle of the heading would then be its angle relative to the meridian of a point plus the angle of the meridian. So one sees that at the poles the longitudes become the headings. One could define Cartesian coordinates in a tangent plane at a pole by choosing 0° and 90° longitude as the two axes. In the tangent plane the Pythagorean Theorem for small changes becomes

Δs2 = Δr2 + (rΔλ)2    or    Δs2 = Δx2 + Δy2

assuming Δλ is measured in radians. Similar changes would need to be made for reference ellipsoids.

Supplemental (14 Aug): The azimuth for a direction might better be defined as the angle relative to a meridian passing through a point. The angle that defines latitude can be extended beyond the north pole along a great circle so there is still a reference direction to measure the azimuth relative to at the pole. It has to be clear what meridian one is using.