# Globe and Flat Earth Transformations and Mappings

Calculator and Equations to convert between WGS84 ellipsoid and Flat Earth coordinates, in geographic and cartesian coordines.

This page explains why using the same geographic coordinates on the Globe and Flat Earth result in different vectors (different cartesian coordinates, lenghts and directions) and hence different distances between positions. Using the same values for latitude and longitude on this models is not the same at all, as flat earther often claim. GPS latitude and longitude only work on the Globe. This is due to the fact that the geographic coordinate systems of Globe and Flat Earth have completely different geometries.

• Vector
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## Transformation from Globe to Flat Earth

Calculating Speed and Acceleration from GPS data can be done using simple vector geometry, if we have cartesian (X,Y,Z) corrdinates for every location instead of geodetic (Lat/Long/Alt) coordinates. GPS does all calculations internally in Earth Centered Earth Fixed (ECEF) cartesian (X,Y,Z) coordinates and only transforms this coordinates into (Lat,Long,Elev) as the last step for our convenience. The App on this page does all calculations in Globe and Flat Earth cartesian coordinates.

We can use Coordinate System Transformations to convert between cartesian and geodetic coordinates in the same domain. Such Coordinate System Transformations have the property that they retain vectors. The values of the components of a vector are different in the different coordinate systems of the same domain, but the vectors have the same length and direction in each such coordinate system. That means that derived values like the distances, velocities and accelerations between vectors are the same in all such coordinate systems of the same domain.

To make calculations in cartesian coordinates on the Flat Earth from vectors given in cartesian Globe coordinates, we have to make a projection transformation from the Globe domain into the Flat Earth domain. The whole transformation from Globe (X,Y,Z) coordinates to Flat Earth (X,Y,Z) coordinates is a sequence of the following transformations:

1. Coordinate system transformation from Globe (X,Y,Z) into geodetic Globe (Lat,Long,Alt) coordinates: retains vectors
2. Mapping/Projection into the Flat Earth domain = changing the geometry of the underlying coordinate system: does not retain vectors!
3. Coordinate system transformation from geodetic Flat Earth (Lat,Long,Alt) into cartesian Flat Earth (X,Y,Z) coordinates: retains vectors

Because the projection does change the geometry, the combined transformations do not retain vectors.

Example above: Even though the geodetic (Lat,Long,Alt) coordinates of the magenta vectors are the same on the Globe and Flat Earth domain, the corresponding vectors are not the same. They have different directions and lengths and hence different cartesian coordinates. The lengths of the magenta vectors are shown at |Vglobe| and |Vfe| respectively.

Note: Coordinate System Transformations (CS Transf) do not change the length and direction of vectors. They only change the values of the vector components to get the same vector in the corresponding coordinate system.

Projection however do change the length and direction of vectors, even if their components have the same values in the different domains. This is because projections are a change in the geometry of the underlying coordinate system. The following figure illustrates this. The (red,blue,green) components have the same value on the Globe and Flat Earth coordinate systems. But because the geometries of the coordinate systems are different, the resulting vectors are also different.

Projection = Change of the geometry of the coordinate system → change of all vectors

This is the reason why length and direction of vectors, distances between vectors and all derived values like velocities and accelerations are not retained by the Globe to Flat Earth projection. That's the reason why navigation and measurements are not the same on Globe and Flat Earth, even if they use the same geodetic coordinates. They may use the same geodetic coordinates for vectors, but the underlying geodetic coordinate systems have a different geometry. One is based on a sphere, the other on a circular plane.

## 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 [1]. 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. [2]

## Transforming Geographic Globe Coordinates into ECEF Coordinates

To convert from ellipsoidal latitude, longitude, height to ECEF cartesian coordinates, we can use the formulas published in Wikipedia [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°.

## Transforming ECEF into Geographic Globe Coordinates

To convert from ECEF cartesian coordinates to ellipsoidal latitude, longitude, height, we can use the formulas published in Wikipedia [3]. This transformation can only be calculated iteratively.

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

## Mapping Geographic Globe Coordinates onto a Flat Earth

Transformations between coordinate systems of the same domain do not change vectors. Length and direction of each vector and the relations between vectors like angles and distances are retained. However, changing the domain from Globe to Flat Earth or vice versa, by changing the geometry of the underlying coordinate system, changes all vectors with it.

Using the same geographic Globe coordinates (φ,λ,h) in the geographic Flat Earth coordinate system does not retain the vectors, because the underlying coordinate systems have different geometries. Other relations derived from vectors like velocities and accelerations do also change accordingly.

So reusing geographic Globe coordinates as geographic Flat Earth coordinates is not a coordinate system transformation that retains vectors, but is a mapping projection into a coordinate system with a different geometry.

The mapping projection of geographic Globe coordinates onto geographic Flate Earth coordinates is simply:

 (9)

The underlying coordinate systems globe and fe have different geometries!

## Transforming Geographic Flat Earth Coordinates into Cartesian Flat Earth Coordinates

For vector calculations on the Flat Earth we have to transform geographic Flat Earth coordinates into cartesian Flat Earth coordinates. The origin of this cartesian coordinate system is the north pole. The X axis defines a line through the 0-meridian. The Y axis is a 90° counter clockwise rotation of the X axis on the flat earth plane as seen from above. The Z axis points up.

Using the latitude and longitude angle and the radius of the flat earth π·R we can use trigonometry to calculate the vectors pointing from the north pole to the observer locations on the Flat Earth in cartesian coordinates.

(10)
where'
 $\mathbf{P}$ ' =' 'observer location in FE cartesian coordinates $h$ ' =' 'altitude above surface $R$ ' =' '6371 km = radius of the earth $\varphi$ ' =' 'latitude of observer in radian $\lambda$ ' =' 'longitude of observer in radian

## Transforming Cartesian Flat Earth Coordinates into Geographic Flat Earth Coordinates

(11)
(12)
(13)
with
where'
 $\varphi$ ' =' 'latitude in radian $\lambda$ ' =' 'longitude in radian $h$ ' =' 'altitude above surface $X,Y,Z$ ' =' 'cartesian coordinates of a position on the flat earth $R$ ' =' '6371 km = radius of the Globe Earth

If r = 0 (point on the North Pole), then the longitude $\lambda$ is undefined and can be set to any angle.

## References

IERS Technical Note 21, IERS Conventions (1996), D. McCarthy, editor, Observatoire de Paris; l July 1996.
DEPARTMENT OF DEFENSE WORLD GEODETIC SYSTEM 1984
Its Definition and Relationships with Local Geodetic Systems
wgs84fin.pdf  local copy

https://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf
[3]
Geographic coordinate conversion
https://en.wikipedia.org/wiki/Geographic%5Fcoordinate%5Fconversion

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