n-vector

The n-vector representation (also called geodetic normal or ellipsoid normal vector) is a three-parameter

in mathematical calculations and computer algorithms.

Geometrically, the n-vector for a given position on an

property.

More in general, the concept can be applied to representing positions on the boundary of a strictly

. In this general case, the n-vector consists of k parameters.

General properties

A

used as a position representation. [1]

For most applications the surface is the

, as shown in the figure.

A surface position has two

.

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

coordinate system, called e, it is clear that going from latitude/longitude to n-vector, is achieved by:

${\displaystyle \mathbf {n} ^{e}=\left[{\begin{matrix}\cos(\mathrm {latitude} )\cos(\mathrm {longitude} )\\\cos(\mathrm {latitude} )\sin(\mathrm {longitude} )\\\sin(\mathrm {latitude} )\\\end{matrix}}\right]}$

The superscript e means that n-vector is

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

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

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

Longitude is found using:

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

In these expressions ${\displaystyle \arctan(y/x)}$ should be implemented using a call to

(0,0) is undefined. Note that the equations are exact both for spherical and ellipsoidal Earth model.

Example: Great circle distance

Finding the

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]

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

where ${\displaystyle \mathbf {n} _{a}}$ and ${\displaystyle \mathbf {n} _{b}}$ are the n-vectors representing the two positions a and b. ${\displaystyle \Delta \sigma }$ 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.

See also

• Earth section paths
• Horizontal position representation
• Latitude
• Longitude
• Universal Transverse Mercator coordinate system
• Quaternion

References

External links

• Solving 10 problems by means of the n-vector