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.
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:
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 V_{globe} and V_{fe} 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.
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:
The WGS 84 Coordinate System is a righthanded, Earthfixed orthogonal coordinate system and is graphically depicted in Figure 2.1.
The origin and axes are defined as follows:
The WGS 84 Coordinate System origin also serves as the geometric center of the WGS 84 Ellipsoid and the Zaxis serves as the rotational axis of this ellipsoid of revolution. [2]
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^{'} 

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

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^{'} 

The function to find the root for is:
(6) 
If latitude
(7) 
If latitude
(8)  
with  
and  
where^{'} 

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!
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 0meridian. 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^{'} 

(11)  
(12)  
(13)  
with  
where^{'} 

If r = 0 (point on the North Pole), then the longitude