The n-vector representation (also called geodetic normal or ellipsoid normal vector) is a three-parameter non-singular representation well-suited for replacing geodetic coordinates (latitude and longitude) for horizontal position representation in mathematical calculations and computer algorithms. Geometrically, the n-vector for a given position on an ellipsoid is the outward-pointing unit vector that is normal in that position to the ellipsoid. For representing horizontal positions on Earth, the ellipsoid is a reference ellipsoid and the vector is decomposed in an Earth-centered Earth-fixed coordinate system. It behaves smoothly at all Earth positions, and it holds the mathematical one-to-one property. More generally, the concept can be applied to representing positions on the boundary of a strictly convex bounded subset of k-dimensional Euclidean space, provided that that boundary is a differentiable manifold. In this general case, the n-vector consists of k parameters.
General properties A normal vector to a strictly convex surface can be used to uniquely define a surface position. n-vector is an outward-pointing normal vector with unit length used as a position representation.
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.
A surface position has two degrees of freedom, and thus two parameters are sufficient to represent any position on the surface. 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. In both cases the singularities are avoided by adding an extra parameter, i.e. to use n-vector (three parameters) to represent horizontal position and a unit quaternion (four parameters) to represent orientation. n-vector is a one-to-one representation, meaning that any surface position corresponds to one unique n-vector, and any n-vector corresponds to one unique surface position. As a Euclidean 3D vector, standard 3D vector algebra can be used for the position calculations, and this makes n-vector well-suited for most horizontal position calculations. For a general comparison of the various representations, see the horizontal position representations page.
Converting latitude/longitude to n-vector Based on the definition of the ECEF coordinate system, called e, it is clear that going from latitude/longitude to n-vector, is achieved by:
n e = [ cos ( l a t i t u d e ) cos ( l o n g i t u d e ) cos ( l a t i t u d e ) sin ( l o n g i t u d e ) sin ( l a t i t u d e ) ] {\displaystyle \mathbf {n} ^{e}=\left[{\begin{matrix}\cos(\mathrm {latitude} )\cos(\mathrm {longitude} )\\\cos(\mathrm {latitude} )\sin(\mathrm {longitude} )\\\sin(\mathrm {latitude} )\\\end{matrix}}\right]}
The superscript e means that n-vector is decomposed in the coordinate system e (i.e. the first component is the scalar projection of n-vector onto the x-axis of e, the second onto the y-axis of e etc.). Note that the equation is exact both for spherical and ellipsoidal Earth model.
Converting n-vector to latitude/longitude From the three components of n-vector, n x e {\displaystyle n_{x}^{e}} , n y e {\displaystyle n_{y}^{e}} , and n z e {\displaystyle n_{z}^{e}} , latitude can be found by using:
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