N-vector

Last updated November 10, 2021

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 in general, 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. ^[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.

The direction of n-vector corresponds to geodetic latitude N vector and geodetic latitude.png — The direction of n-vector corresponds to geodetic latitude

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.^[2] 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.

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:

\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, $n_{x}^{e}$ , $n_{y}^{e}$ , and $n_{z}^{e}$ , latitude can be found by using:

\mathrm {latitude} =\arcsin \left(n_{z}^{e}\right)=\arctan \left({\frac {n_{z}^{e}}{\sqrt {{n_{x}^{e}}^{2}+{n_{y}^{e}}^{2}}}}\right)

The rightmost expression is best suited for computer program implementation.^[1]

Longitude is found using:

\mathrm {longitude} =\arctan \left({\frac {n_{y}^{e}}{n_{x}^{e}}}\right)

In these expressions $\arctan(y/x)$ should be implemented using a call to atan2(y,x). The Pole singularity of longitude is evident as atan2(0,0) is undefined. Note that the equations are exact both for spherical and ellipsoidal Earth model.

Example: Great circle distance

Finding the great circle distance between two horizontal positions (assuming spherical Earth) is usually done by means of latitude and longitude. Three different expressions for this distance are common; the first is based on arccos, the second is based on arcsin, and the final is based on arctan. 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. They also contain deltas of latitude and longitude, which in general should be used with care near the ±180° meridian and the Poles.

Solving the same problem using n-vector is simpler due to the possibility of using vector algebra. The arccos expression is achieved from the dot product, while the magnitude of the cross product gives the arcsin expression. Combining the two gives the arctan expression:^[1]

{\begin{aligned}&\Delta \sigma =\arccos \left(\mathbf {n} _{a}\cdot \mathbf {n} _{b}\right)\\&\Delta \sigma =\arcsin \left(\left|\mathbf {n} _{a}\times \mathbf {n} _{b}\right|\right)\\&\Delta \sigma =\arctan \left({\frac {\left|\mathbf {n} _{a}\times \mathbf {n} _{b}\right|}{\mathbf {n} _{a}\cdot \mathbf {n} _{b}}}\right)\\\end{aligned}}

where $\mathbf {n} _{a}$ and $\mathbf {n} _{b}$ are the n-vectors representing the two positions a and b. $\Delta \sigma$ is the angular difference, and thus the great-circle distance is achieved by multiplying with the Earth radius. This expression also works at the poles and at the ±180° meridian.

There are several other examples where the use of vector algebra simplifies standard problems.^[1] For a general comparison of the various representations, see the horizontal position representations page.

Related Research Articles

In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes that are used in special applications.

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

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A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

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In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse.

In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

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Geographical distance Distance measured along the surface of the earth

Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem.

Solution of triangles is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.

Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude (north/south) $ϕ$ , longitude (east/west) $λ$ , and ellipsoidal height $h$ . The triad is also known as Earth ellipsoidal coordinates.

A position representation is the parameters used to express a position relative to a reference. When representing positions relative to the Earth, it is often most convenient to represent vertical position separately, and to use some other parameters to represent horizontal position. There are also several applications where only the horizontal position is of interest, this might e.g. be the case for ships and ground vehicles/cars. It is a type of geographic coordinate system.

The position of the Sun in the sky is a function of both the time and the geographic location of observation on Earth's surface. As Earth orbits the Sun over the course of a year, the Sun appears to move with respect to the fixed stars on the celestial sphere, along a circular path called the ecliptic.

References

1 2 3 4 Gade, Kenneth (2010). "A non-singular horizontal position representation" (PDF). The Journal of Navigation. Cambridge University Press. 63 (3): 395–417. doi:10.1017/S0373463309990415.
↑ Stuelpnagel, John (1964). "On the Parametrization of the Three-Dimensional Rotation Group". SIAM Review. Society for Industrial and Applied Mathematics. 6 (4): 422–430. doi:10.1137/1006093. JSTOR 2027966.

External links

Solving 10 problems by means of the n-vector

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[Gade-1] 1 2 3 4 Gade, Kenneth (2010). "A non-singular horizontal position representation" (PDF). The Journal of Navigation. Cambridge University Press. 63 (3): 395–417. doi:10.1017/S0373463309990415.

[2] Stuelpnagel, John (1964). "On the Parametrization of the Three-Dimensional Rotation Group". SIAM Review. Society for Industrial and Applied Mathematics. 6 (4): 422–430. doi:10.1137/1006093. JSTOR 2027966.

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