Nvector

N-vector refers to a three-dimensional Cartesian representation of a position on or near the Earth, used primarily in navigation, geodesy, and related fields. In this system, a location is encoded by a unit vector from the Earth's center to the location, known as the n-vector. The coordinates (n_x, n_y, n_z) satisfy n_x^2 + n_y^2 + n_z^2 = 1, placing the point on the unit sphere. The n-vector provides a trigonometry-free or trigonometry-light means of describing position, which can simplify certain computations and filtering tasks.

For a spherical Earth model, the n-vector directly relates to geographic coordinates: longitude λ = atan2(n_y, n_x) and latitude φ = asin(n_z). Height above the reference surface is not part of the unit vector itself but can be incorporated by scaling the vector to intersect the reference surface (for a sphere, p = R n) or by projecting onto an ellipsoid and solving for the intersection point along the direction n.

For an ellipsoidal Earth, the n-vector identifies a direction from the center, and the surface point is obtained by intersecting the line through the origin in direction n with the reference ellipsoid. The resulting surface point p = η n lies on the ellipsoid x^2/a^2 + y^2/b^2 + z^2/c^2 = 1, where η is chosen accordingly. Height above ellipsoid can then be defined relative to that surface point.

Advantages of the n-vector include stable, well-conditioned operations for attitude estimation, cross-track calculations, and conjunctions, as well as avoiding singularities associated with some latitude/longitude representations. It remains one of several coordinate schemes used in modern navigation and geodetic computations.