A Non-singular Horizontal Position Representation

Position calculations, e.g. adding, subtracting, interpolating, and averaging positions, depend on the representation used, both with respect to simplicity of the written code and accuracy of the result. The latitude/longitude representation is widely used, but near the pole singularities, this representation has several complex properties, such as error in latitude leading to error in longitude. Longitude also has a discontinuity at ±180°. These properties may lead to large errors in many standard algorithms. Using an ellipsoidal Earth model also makes latitude/longitude calculations complex or approximate. Other common representations of horizontal position include UTM and local Cartesian ‘flat Earth’ approximations, but these usually only give approximate answers, and are complex to use over larger distances. The normal vector to the Earth ellipsoid (called n -vector) is a non-singular position representation that turns out to be very convenient for practical position calculations. This paper presents this representation, and compares it with other alternatives, showing that n -vector is simpler to use and gives exact answers for all global positions, and all distances, for both ellipsoidal and spherical Earth models. In addition, two functions based on n -vector are presented, that further simplify most practical position calculations, while ensuring full accuracy.