# Rainy Lake Experiment: WGS84 Calculator

All calculations on this page are based on the reference ellipsoid of the WGS84 system. WGS84 uses 2 coordinate systems to specifiy a position on earth:

• Latitude/Longitude/Height (LLH) Geodetic coordinates (angles in degree)
• X/Y/Z in Earth Centered Earth Fixed (ECEF) coordinates in meters. The X-axes points to latitude = 0 and longitude = 0, the Y-Axes points to latitude = 0 and longitude = 90 and the Z-axes points to the North Pole.

In the panels you can enter a location either Geodetic LLH or in ECEF coordinates. The other coordinates are automatically computed from the input.

All calculations use the earth parameters as defined in [1].

## Vector Origin-Target Input

The following panels can be used to convert between ECEF (earth centered, earth fixed) coordinates and LLH (latitude, longitude, height) too.

Use the following panels to compute data from the Origin Refrence Point to a Target Point. The Results are displayed at Vector Origin-Target Calculations. Some values are expressed in a local coordinate system positioned at Origin Point. The Local Coord Orientation Reference Point defines the orientation of the local coordinate system in such a way, that the y axes (North) is pointing from Origin to the Orientation Reference Point, the x axes is poining to the right of y (East) and the z axes is pointing away from the surface (Up).

Don't change the Reference Point or set lat = 90 if you want the local coordinate system to be a East/North/Up system. If you want to get Target Point local coordinates with respect to another target point Ref2, enter the coordinates of Ref2 above.

## Vector Origin-Target Calculations

The values shown in the following panel are the result of a vector from Origin Point to Target Point, both in ECEF and local coordinates.

All data is calculated using the WGS84 ellipsoid model. ECEF = Earth Centered, Earth Fixed coordinates. Local coordinates are coordinates with respect to the local coordinate system at the Origin point. This is calculated in such a way, that Z is the perpendicular to the reference ellipsoid of the Origin point, Y is in direction to the Reference point and X points to the right wrt. Y.

O: Origin point data.

TS: Target Surface point data. The Target Surface point is calculated from the Target point by calculating the ellipsoid point with the same lat/long plus the elevation of the Origin point.

Length: Length of the Vector.

Azimuth: Direction on the surface from Origin to Target or Target Surface with respect to the Y axes of the local coordinate system at Origin. Angles from 0° to 360°.

Elevation: Angle from Origin to Target or Target Surface with respect to the surface at Origin. Negative angles point down through the surface. The Zenith angle is 90° - Elevation.

Distance: Distance along the ellipsoid surface from Origin point to Target Surface point. To Compute the distance the length of the vector Origin to Target Surface and the radius of the ellipsoid at the Origin is used.

OT: Data of the vector Origin to Target point.

OTS: Data of the vector Origin to Target Surface point.

O Ellipsoid Rad: Radius of the reference ellipsoid at the Origin. This is the distance of the ellipsoid point at origin location from the center of the earth.

OT ΔHeight: Height of Target minus height of Origin.

OT Angle: Angle between the vectors perpendicular to the ellipsoid surface at Origin and Target. This corresponds to the leaning of a target tower wrt. a origin tower.

ViewDist: Distance of Target point in the local Y direction. Y direction is from Origin to Reference point.

SideDist: Distance of Target point from the View Direction in the X direction, which points to the right of Y.

## Create Computer Model Data

The following form was used to convert the ECEF coordinates to the data format used in the Computer Model . Insert the ECEF Data from CurveAppData.csv into the field and click Create Curve App Data. The Computer Model needs the distance of the targets from the Origin and the Side Position from the center line.

The first line must specify the Origin point and the second line must specify the Reference Point. The center line is the line from the Origin to the Refrence Point.

The Data must have the following format for each line: ID, X, Y, Z, Description
where X, Y, Z are in ECEF (Earth Centered, Earth Fixed) coordinates.

## WGS84 Coordinate System

The WGS 84 Coordinate System is a Conventional Terrestrial Reference System (CTRS). The definition of this coordinate system follows the criteria outlined in the International Earth Rotation Service (IERS) Technical Note 21 [2]. These criteria are repeated below:

• It is geocentric, the center of mass being defined for the whole Earth including oceans and atmosphere
• Its scale is that of the local Earth frame, in the meaning of a relativistic theory of gravitation
• Its orientation was initially given by the Bureau International de l’Heure (BIH) orientation of 1984.0
• Its time evolution in orientation will create no residual global rotation with regards to the crust

The WGS 84 Coordinate System is a right-handed, Earth-fixed orthogonal coordinate system and is graphically depicted in Figure 2.1.

Figure 2.1 The WGS 84 Coordinate System Definition

The origin and axes are defined as follows:

Origin
Earth’s center of mass
Z-Axis
The direction of the IERS Reference Pole (IRP). This direction corresponds to the direction of the BIH Conventional Terrestrial Pole (CTP) (epoch 1984.0) with an uncertainty of 0.005"
X-Axis
Intersection of the IERS Reference Meridian (IRM) and the plane passing through the origin and normal to the Z-axis. The IRM is coincident with the BIH Zero Meridian (epoch 1984.0) with an uncertainty of 0.005"
Y-Axis
Completes a right-handed, Earth-Centered Earth-Fixed (ECEF) orthogonal coordinate system

The WGS 84 Coordinate System origin also serves as the geometric center of the WGS 84 Ellipsoid and the Z-axis serves as the rotational axis of this ellipsoid of revolution. [1]

## Converting ellipsoidal to ECEF coordinates

To convert from ellipsoidal latitude, longitude, height to ECEF cartesian coordinates, the formulas published in Wikipedia are used [3].

(1)
(2)
(3)
with
and
where'
 $X, Y, Z$ ' =' 'ECEF cartesian coordinates $\varphi$ ' =' 'latitude in radian $\lambda$ ' =' 'longitude in radian $h$ ' =' 'height above reference ellipsoid $a$ ' =' '6,378,137 m = semi-major axis of reference ellipsoid $b$ ' =' '6,356,752.314245 m = semi-minor axis of reference ellipsoid $N$ ' =' 'prime vertical radius of curvature = distance from the surface to the Z-axis along the ellipsoid normal $e^2$ ' =' 'square of the first numerical eccentricity of the ellipsoid

To convert latitude and longitude from degrees into radian, multiply them by π/180°.

## Converting ECEF to ellipsoidal coordinates

To convert from ECEF cartesian coordinates to ellipsoidal latitude, longitude, height, the formulas published in Wikipedia are used [3]. This transformation can only be calculated iteratively. The calculator on this page uses the Newton method.

The longitude can be calculated directly. The calculator uses vector geometry to accomplish this. If the X and Y components are 0 then the longitude is undefined and set to 0. Otherwise longitude is calculated as folows:

(4)
where'
 $X, Y$ ' =' 'components of the ECEF coordinates $\lambda$ ' =' 'longitude in radian $\mathrm{sign}(Y)$ ' =' '1 if Y ≥ 0, else -1

To convert latitude and longitude from radian into degrees, multiply them by 180°/π.

If the X and Y components are 0 then latitude is +90° for Z > 0 or −90° for Z < 0. If Z is 0 then latitude is 0°. In all other cases the latitude has to be calculated iteratively:

(5)
with
and
where'
 $\varphi$ ' =' 'latitude in radian $X, Y$ ' =' 'components of the ECEF coordinates $\mathrm{solve}()$ ' =' 'Newton solver algorithmus $f(k)$ ' =' 'function to find the root for with the Newton solver $k_0$ ' =' 'start value (guess) for the Newton solver

The function to find the root for is:

 (6)

If latitude $\varphi$ is +90° or −90° then the ellipsoid height is:

 (7)

If latitude $-90° \lt \varphi \lt 90°$ then the ellipsoid height is:

(8)
with
and
where'
 $h$ ' =' 'orthometric height above ellipsoid $N$ ' =' 'prime vertical radius of curvature = distance from the surface to the Z-axis along the ellipsoid normal $a$ ' =' '6,378,137 m = semi-major axis of reference ellipsoid $b$ ' =' '6,356,752.314245 m = semi-minor axis of reference ellipsoid $e^2$ ' =' 'square of the first numerical eccentricity of the ellipsoid

## References

[1]
DEPARTMENT OF DEFENSE WORLD GEODETIC SYSTEM 1984
Its Definition and Relationships with Local Geodetic Systems
https://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf
IERS Technical Note 21, IERS Conventions (1996), D. McCarthy, editor, Observatoire de Paris; l July 1996.
[3]
Geographic coordinate conversion
https://en.wikipedia.org/wiki/Geographic%5Fcoordinate%5Fconversion

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