n-vector

It behaves smoothly at all Earth positions, and it holds the mathematical one-to-one property.

n-vector is an outward-pointing normal vector with unit length used as a position representation.

[1] For most applications the surface is the reference ellipsoid of the Earth, and thus n-vector is used to represent a horizontal position.

Hence, the angle between n-vector and the equatorial plane corresponds to geodetic latitude, as shown in the figure.

On the reference ellipsoid, latitude and longitude are common parameters for this purpose, but like all two-parameter representations, they have singularities.

This is similar to orientation, which has three degrees of freedom, but all three-parameter representations have singularities.

Note that the equation is exact both for spherical and ellipsoidal Earth model.

, latitude can be found by using: The rightmost expression is best suited for computer program implementation.

Finding the great circle distance between two horizontal positions (assuming spherical Earth) is usually done by means of latitude and longitude.

The expressions, which are successively more complex to avoid numerical instabilities, are not easy to find, and since they are based on latitude and longitude, the Pole singularities may become a problem.

Solving the same problem using n-vector is simpler due to the possibility of using vector algebra.

is the angular difference, and thus the great-circle distance is achieved by multiplying with the Earth radius.