n-vector library in Matlab

Author: Kenneth Gade (PhD, principal scientist at FFI)

This library is for performing a range of different geographical position calculations, e.g. adding, subtracting, interpolating, and averaging positions. Exact results are returned for either ellipsoidal or spherical Earth model.

All calculations are based on n-vector (which can replace e.g. latitude and longitude), giving several advantages:

• The calculations are typically simple and intuitive
  • 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.
• The calculations are non-singular (i.e. work equally well at or near the poles as any other global position)
  • Reason: The n-vector representation is inherently non-singular for all Earth positions.
• The calculations have no discontinuities (i.e. 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

Reference:

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://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf

Also available in several other programming languages:

There are several n-vector libraries (from other authors) available in other programming languages that are either based on this library, or based directly on the reference article (Gade, 2010). Some of these are listed below. Note: the listed code is not verified by the author of this library.

• C# (C Sharp): https://www.navlab.net/nvector/#download
• C++: https://www.navlab.net/nvector/#download
• Python: https://github.com/pbrod/Nvector
  • Alternative 2: https://github.com/mrJean1/PyGeodesy
  • Alternative 3: https://github.com/lxnt/ccnvector
• JavaScript: https://github.com/chrisveness/geodesy
• Haskell: https://github.com/ofmooseandmen/jord
• Go / Google's Go / Golang: https://github.com/fortyninemaps/nvector
• R: https://github.com/euctrl-pru/nvctr
• Java: https://github.com/omeruluoglu/JGeodesy
• Rust: https://github.com/ofmooseandmen/jord-rs

Contributing

If you have suggestions or other comments to the code, please contact the author at kenneth.gade@ffi.no.

Using this code

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

% 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:

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).

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://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}
}