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Geographical position calculations – simple and exact solutions by means of n-vector 🌍

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# *n*-vector library in Matlab and Octave (plus many other languages)
Author: Kenneth Gade (PhD, Principal Scientist at [FFI](https://www.ffi.no/en))

This library provides tools for a wide range of different geographical position calculations, such as adding, subtracting, interpolating, and averaging positions. Exact results are returned for either ellipsoidal or spherical Earth model. This code is compatible with Matlab and the free [Octave](https://octave.org/).

All calculations are based on [*n*-vector](https://www.navlab.net/nvector) (which can replace e.g. latitude and longitude), giving several advantages:

* **Simple and intuitive** calculations.
  * Reason: *n*-vector is a 3D vector and hence the powerful vector algebra can be used to solve many position calculations intuitively and with few code lines.
* **Non-singular** calculations: work equally well at or near the North/South Pole as any other global position.
  * Reason: The *n*-vector representation is inherently non-singular for all Earth positions.
* **No discontinuities** in the calculations:  work equally well across the dateline (±180° longitude meridian) as any other global positions.
  * Reason: The *n*-vector representation has no discontinuities.

For more details and 10 examples of usage, see https://www.navlab.net/nvector

**References:** 
> * K. Gade (2010): A Non-singular Horizontal Position Representation, *The Journal of Navigation*, Volume 63, Issue 03, pp 395-417, July 2010. *<https://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf>*
> * K. Gade (2025): The *n*-vector page *<https://www.ffi.no/en/research/n-vector>*

## Also available in many other programming languages

There are several *n*-vector libraries (from other authors) available in other programming languages that are either based on this Matlab library, or based directly on the reference article ([Gade, 2010](https://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf)). Some of these are:

* **Python**: https://github.com/pbrod/Nvector
  * Alternative 2: https://github.com/mrJean1/PyGeodesy 
  * Alternative 3: https://github.com/mhogan-nwra/envector
  * Alternative 4: https://github.com/lxnt/ccnvector
* **JavaScript**: https://github.com/chrisveness/geodesy/blob/master/latlon-nvector-spherical.js
* **C++**: https://www.navlab.net/nvector/#download
  * Alternative 2: https://github.com/GPSBabel/gpsbabel/blob/master/src/core/nvector.cc
* **C# (C Sharp)**: https://www.navlab.net/nvector/#download
  * Alternative 2: https://github.com/johnjore/hajk/tree/master/Libraries/FFI.NVector
* **Go / Google's Go / Golang**: https://github.com/ezzatron/nvector-go
  * Alternative 2: https://github.com/fortyninemaps/nvector 
* **TypeScript**: https://github.com/ezzatron/nvector-js
* **Haskell**: https://github.com/ofmooseandmen/jord
* **R**: https://github.com/euctrl-pru/nvctr 
* **Java**: https://github.com/huiAlex/TRIAD/tree/main/dataset/dronology/unprocessed/code/src
* **Rust**: https://github.com/ofmooseandmen/jord-rs

## Contributing
If you have comments to the code or tips about additional libraries to be added to the list above, please contact the author at kenneth.gade@ffi.no.

## Using this code
The code works with all Matlab versions and Octave (no toolboxes are needed). A simple example of usage is Example 7 from https://www.navlab.net/nvector, where a horizontal midpoint is calculated:

```matlab
% Three positions A, B and C are given as lat/long in degrees.
% Convert all three to radians and then to n-vectors:
n_EA_E = lat_long2n_E(rad(90),rad(0));
n_EB_E = lat_long2n_E(rad(60),rad(10));
n_EC_E = lat_long2n_E(rad(50),rad(-20));
 
% Find the horizontal mean position, M:
n_EM_E = unit(n_EA_E+n_EB_E+n_EC_E);
```

As described below, the file `examples.m` includes the above solution, and it also gives the following plot:

![Earth figure from Example 7](https://raw.githubusercontent.com/FFI-no/n-vector/main/Example7.png)

## Matlab files included

Below is a list of the files that are included in this library (19 files in total). The file name syntax, mathematical symbols and coordinate frames are defined at https://www.navlab.net/nvector.

Convert between lat/long and *n*-vector:

* `lat_long2n_E.m` Converts latitude and longitude to *n*-vector
* `n_E2lat_long.m` Converts *n*-vector to latitude and longitude

Convert between delta (i.e. local position vector) and  *n*-vectors:

* `n_EA_E_and_n_EB_E2p_AB_E.m` From two positions *A* and *B*, finds the delta position
* `n_EA_E_and_p_AB_E2n_EB_E.m` From position *A* and delta, finds position *B*

Convert between *n*-vector and ECEF-vector (i.e. position vector from Earth center, in meters):

* `n_EB_E2p_EB_E.m` Converts *n*-vector to ECEF-vector
* `p_EB_E2n_EB_E.m` Converts ECEF-vector to *n*-vector

Convert between *n*-vector and rotation matrix (i.e. ***R****EN* or ***R****EL*):

* `R_EN2n_E.m` Finds *n*-vector from ***R****EN*
* `n_E2R_EN.m` Finds ***R****EN* from *n*-vector
* `R_EL2n_E.m` Finds *n*-vector from ***R****EL*
* `n_E_and_wa2R_EL.m` Finds ***R****EL* from *n*-vector and wander azimuth angle

Convert between Euler angles and rotation matrix:

* `xyz2R.m` Creates a rotation matrix from 3 angles about new axes in the xyz order
* `R2xyz.m` Three angles about new axes in the xyz order are found from a rotation matrix
* `zyx2R.m` Creates a rotation matrix from 3 angles about new axes in the zyx order (e.g. yaw-pitch-roll)
* `R2zyx.m` Three angles about new axes in the zyx order (e.g. yaw-pitch-roll) are found from a rotation matrix

Miscellaneous simple utilities:

* `unit.m` Makes input vector unit length (i.e. norm = 1)
* `rad.m` Converts angle from degrees to radians
* `deg.m` Converts angle from radians to degrees
* `R_Ee.m` Selects axes of the coordinate frame *E*

Examples of how to use the *n*-vector library for position calculations:

* `examples.m` Contains solutions to the 10 examples given at https://www.navlab.net/nvector

A note about the axes of coordinate frame *E*: This library uses the most common *E*-frame by default (i.e. the *z*-axis towards the North Pole). However, if a less common (but in many cases better) choice is preferred (with the *x*-axis towards the North Pole), this can be switched in the `R_Ee.m`-file. More details are available in the help text of this file and in Table 2 of [Gade (2010)](https://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf).

## License
This repository is available under the MIT License. See the license file for details.

## Citing this code
The *n*-vector library is based on the following article, and hence it should be cited in publications using the library:

> Kenneth Gade (2010): A Non-singular Horizontal Position Representation, *The Journal of Navigation*, Volume 63, Issue 03, pp 395-417, July 2010, [DOI: 10.1017/S0373463309990415](https://doi.org/10.1017/S0373463309990415). 
*<https://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf>*

Bibtex entry as follows:
```
@article{gade2010non,
  title={A Non-singular Horizontal Position Representation},
  author={Gade, Kenneth},
  journal={The Journal of Navigation},
  volume={63},
  number={3},
  pages={395--417},
  year={2010},
  url={https://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf}, 
  doi={10.1017/S0373463309990415},
  publisher={Cambridge University Press}
}
```

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