4.1. n-vector explained

4.1.1. What is n-vector?

n-vector is just the 3D normal vector to the Earth ellipsoid (or sphere, if spherical Earth model is used). It can replace latitude and longitude (and other alternatives) to represent horizontal position, giving several advantages. This page gives a brief introduction, see Gade (2010). for a more detailed explanation.

../_images/EarthEllipsoid_with_nvector.png — Fig. 4.1 The n-vector is the normal vector to the Earth model (reference ellipsoid).

4.1.2. Why is n-vector needed?

Performing global position calculations often involves one or more of the following concerns:

Complex implementations (many, and often complex lines of program code needed)

Approximations, e.g.

distortion in map projections

assuming spherical Earth when ellipsoidal should be used Errors often increase with increasing distances

Equation/code not valid/accurate for all Earth positions, e.g.

Latitude/longitude:

Singularities at Poles

Discontinuity at International Date Line ( $\pm 180^\circ$ meridian)

Map projections: usually valid for a limited area, e.g. UTM

Iterations (need to iterate to achieve needed accuracy)

By using the solutions provided here, these concerns can be avoided for a majority of position calculations.

4.1.2.1. What can go wrong if using latitude and longitude?

Even the simple discontinuity of longitude at the International Date Line ( $\pm 180^\circ$ meridian) can give serious consequences if not handled correctly in all the code. An example illustrating this was the computer crash experienced by 12 fighter aircrafts (F-22 Raptors) when they crossed this meridian in 2007.

../_images/F22_PacificReducedSize.jpg — Fig. 4.2 The discontinuity of longitude caused serious problems for the F-22 Raptors when the longitude instantly changed from $-180^{\circ}$ to $+180^{\circ}$ (when crossing the International Date Line).

The discontinuity can be quite easily handled by adding specific code, while the singularities at the Poles are much more challenging to deal with. Examples of how the singularities can affect calculations with latitude and longitude are found in Section 6 of the reference Gade (2010).

4.1.3. n-vector

As discussed above, using latitude and longitude in calculations may lead to several problems due to the singularities at the Poles and complex behavior near the Poles, and due to the discontinuity at the Date Line ( $\pm 180^\circ$ meridian).

Instead of latitude and longitude, we represent position with an “n-vector”, which is the normal vector to the Earth model (the same reference ellipsoid that is used for latitude and longitude). When using n-vector, all Earth-positions are treated equally, and there is no need to worry about singularities or discontinuities. An additional benefit with using n-vector is that many position calculations can be solved with simple vector algebra (e.g. dot product and cross product).

In documents we use $\mathbf{n}^{\mathbf{E}}$ , while in program code we use n_E. E denotes an Earth-fixed coordinate frame, and it indicates that the three components of n-vector are along the three axes of E. More details about the notation used are found here. The position of the North Pole given as n-vector is

$\mathbf{n}^{E} = \begin{bmatrix} 0 \\ 0 \\ 1 \\ \end{bmatrix}$

(assuming the coordinate frame E has its z-axis pointing to the North Pole). For all the details about n-vector, see the reference Gade (2010).

4.1.4. Converting between n-vector and lat/lon

Converting between n-vector and latitude/longitude is done using

$\mathbf{n}^{E} = \begin{bmatrix} cos(lat) \, cos(lon) \\ cos(lat) \, sin(lon)\\ sin(lat)\\ \end{bmatrix}$

This can be seen from the geometry and the axes of E. The relation is true both for ellipsoidal or spherical Earth model.

To find latitude from n-vector we have

$lat = asin(n_{z}^{E}) = atan2(n_{z}^{E}, \sqrt{ (n_{x}^{E})^2 + (n_{y}^{E})^2 })$

where the x, y and z subscripts indicate elements 1, 2 and 3 of the n-vector. Note that the atan2-variant has better numerical properties, and is thus recommended for implementation.

Longitude is found from

$lon = atan2(n_{y}^{E}, n_{x}^{E})$

In the nvector library, two functions lat_lon2n_E and n_E2lat_lon are used for these conversions.

4.1.5. Many position calculations can be solved with only two functions

It turns out that very often, the problem to be solved is one of these:

A and B => delta: Given two positions, A and B (e.g. as latitudes/longitudes/heights), find the (delta) vector from A to B in meters. See also Example 1.

A and delta => B: Given position A (e.g. as lat/long/height) and a vector in meters from A to B, find position B. See also Example 2.

Two functions performing these calculations are available here, using n-vectors for position A and B as input/output. The n-vectors for position A and B are written $\mathbf{n}_{EA}^{E}$ and $\mathbf{n}_{EB}^{E}$ while in program code we use n_EA_E and n_EB_E. The (delta) position vector from A to B is written $\mathbf{p}_{AB}^{E}$ (p_AB_E). More info about the notation is found in the next section.

Based on the above variable names, the two above functions are called:

n_EA_E_and_n_EB_E2p_AB_E

n_EA_E_and_p_AB_E2n_EB_E

4.2. Vector symbols explained

The notation system used for the n-vector library is presented in Chapter 2 of the following thesis: Gade (2018): “Inertial Navigation - Theory and Applications”. A simplified presentation is given here.

4.2.1. Coordinate frame

A coordinate frame has a position (origin), and three axes (basis vectors) x, y and z (orthonormal). Thus a coordinate frame can represent both position and orientation, i.e. 6 degrees of freedom. It can be used to represent a rigid body, such as a vehicle or the Earth, and it can also be used to represent a “virtual” coordinate frame such as North-East-Down.

Coordinate frames are designated with capital letters, e.g. the three generic coordinate frames A, B, and C.

We also have specific names for some common coordinate frames:

Coordinate frame	Description
E	Earth
N	North-East-Down
B	Body, i.e. the vehicle

4.2.2. General vector

A 3D vector given with numbers is written e.g. $\begin{bmatrix} 2 \\ 4 \\ 6 \\ \end{bmatrix}$ . The three numbers are the vector components along the x, y and z axes of a coordinate frame. If the name of the vector is k, and the coordinate frame is A, we will use bold k and A as trailing superscript, i.e.:

$\mathbf{k}^{\mathbf{A}} = \begin{bmatrix} 2 \\ 4 \\ 6 \\ \end{bmatrix}$

Thus $\mathbf{k}^{\mathbf{A}}$ is the 3D vector that is constructed by going 2 units along the x-axis of coordinate frame A, 4 units along the y-axis, and 6 along the z-axis. We say that the vector k is decomposed in A.

4.2.3. Position vector

Instead of the general vector k, we can have a specific vector that goes from A to B. This vector can be decomposed in C. A, B, and C are three arbitrary coordinate frames. We would write this vector:

$\mathbf{p}_{\mathbf{AB}}^{\mathbf{C}}$

In program code: p_AB_C

The letter p is used since this is a position vector (the position of B relative to A, decomosed/resolved in the axes of C).

Example a):

$\mathbf{p}_{\mathbf{EB}}^{\mathbf{E}} = \begin{bmatrix} 0 \\ 0 \\ 6371 \\ \end{bmatrix} km$

From the subscript we see that this is the vector that goes from E (center of the Earth) to B (the vehicle). The superscript tells us that it is decomposed in E, which we now assume has its z-axis pointing towards the North Pole. From the values we see that the vector goes 6371 km towards the North Pole, and zero in the x and y directions. If we assume that the Earth is a sphere with radius 6371 km, we see that B is at the North Pole.

Example b):

$\mathbf{p}_{\mathbf{BC}}^{\mathbf{N}} = \begin{bmatrix} 50 \\ 60 \\ -5 \\ \end{bmatrix} m$

The vector goes from B, e.g. an aircraft, to C, e.g. an object. The vector is decomposed in N (which has North-East-Down axes), i.e. C is 50 m north of B and 60 m east, and C is also 5 m above B.

4.2.4. Properties of the position vector

For the general position vector , we have the property:

$\mathbf{p}_{\mathbf{AB}}^{\mathbf{C}} = - \mathbf{p}_{\mathbf{BA}}^{\mathbf{C}}$

I.e. swapping the coordinate frames in the subscript gives a vector that goes in the opposite direction. We also have:

$\mathbf{p}_{\mathbf{AD}}^{\mathbf{C}} = \mathbf{p}_{\mathbf{AB}}^{\mathbf{C}} + \mathbf{p}_{\mathbf{BD}}^{\mathbf{C}}$

I.e., going from A to D is the same as first going from A to B, then from B to D. From the equation, we see that B is cancelled out. A, B, C, and D are arbitrary coordinate frames.

4.2.5. Rotation matrix

If we return to the general vector $\mathbf{k}^{\mathbf{A}}$ , we could also have a coordinate frame B, with different orientation than A. The same vector k could be expressed by components along the x, y and z-axes of B instead A, i.e. it can also be decomposed in B, written $\mathbf{k}^{\mathbf{B}}$ . Note that the length of $\mathbf{k}^{\mathbf{B}}$ equals the length of $\mathbf{k}^{\mathbf{A}}$ . We will now have the relation:

$\mathbf{k}^{\mathbf{A}} = \mathbf{R}_{\mathbf{AB}} \mathbf{k}^{\mathbf{B}}$

$\mathbf{R}_{\mathbf{AB}}$ is the 9 element (3x3) rotation matrix (also called direction cosine matrix) that transforms vectors decomposed in B to vectors decomposed in A. Note that the B in $\mathbf{R}_{\mathbf{AB}}$ . should be closest to the vector decomposed in B (following the “the rule of closest frames”, see Section 2.5.3 in “Inertial Navigation - Theory and Applications” for details). If we need to go in the other direction we have:

$\mathbf{k}^{\mathbf{B}} = \mathbf{R}_{\mathbf{BA}}\mathbf{k}^{\mathbf{A}}$

Now we see that A is closest to A.

4.2.6. Properties of the rotation matrix

We have that

$\mathbf{R}_{\mathbf{AB}} = (\mathbf{R}_{\mathbf{BA}})^{T}$

where the T means matrix transpose. We also have the following property (closest frames are cancelled):

$\mathbf{R}_{\mathbf{AC}} = \mathbf{R}_{\mathbf{AB}} \mathbf{R}_{\mathbf{BC}}$

If we compare these properties with the position vector, we see that they are very similar: minus is replaced by transpose, and plus is replaced by matrix multiplication. A, B, and C are three arbitrary coordinate frames.

4.2.7. n-vector

The n-vector is in almost all cases decomposed in E, and in the simplest form we will write it

$\mathbf{n}^{E}$

This simple form can be used in cases where it is no doubt about what n-vector expresses the position of. In such cases we can also express the position using the variables lat and lon, without further specification.

However, if we are interested in the position of multiple objects, e.g. A and B, we must specify which of the two, both for n-vector and for latitude/longitude. In this case we will write:

$\mathbf{n}_{EA}^{E}$ and $\mathbf{n}_{EB}^{E}$ (program code: n_EA_E and n_EB_E)

And

$lat_{EA}, lon_{EA}$ and $lat_{EB}, lon_{EB}$

The subscript E might seem redundant here, it could be sufficient to use only A or B. However, we have chosen to also include the E, since both n-vector and latitude/longitude are depending on the reference ellipsoid that is associated with E (see Section 4.1. in Gade (2010) for more about this). Note however, that the subscript rules (swapping and cancelling) we had for $\mathbf{p}_{\mathbf{AB}}^{\mathbf{C}}$ and $\mathbf{R}_{\mathbf{AB}}$ cannot be used for n-vector or lat/lon.

For spherical Earth, we have a simple relation between $\mathbf{p}_{\mathbf{EB}}^{\mathbf{E}}$ and $\mathbf{n}_{\mathbf{EB}}^{\mathbf{E}}$ :

$\mathbf{p}_{EB}^{E} = \mathbf{n}_{EB}^{E} \dot (r_{earth}+h_{EB}) = \mathbf{n}_{EB}^{E} \dot (r_{earth}-z_{EB})$

where $r_{earth}$ is the radius of the Earth, $h_{EB}$ is the height of B and $z_{EB}$ is the depth. For more information about how to use n-vector in various calculations, see the 10 examples and Gade (2010).