Python geolocation calculation for the Waldrain plot of land.

To begin with, I had six unconfirmed latitude and longitude coordinates for the six corner points for a plot of land.

I also had pretty precise edge length and area measurements in metres:

That leads to the following data:

pts = [
  [47.61240287934088,7.668455564143808],
  [47.61238603493116,7.66886803694362],
  [47.61227235282722,7.668805013356426],
  [47.612081232450755,7.668710772100395],
  [47.61209766306042,7.668317607008359],
  [47.612263038360155,7.668392271613928]]

tags = ['NW','NO','OM','SO','SW','WM']

edge_length = [ # in metres
  31.10, # Nord
  13.34, 22.51, # Ost
  29.63, # Sued
  19.26, 16.24 ] # West

area = 1043 # square metres

Calculate the expected edge lengths and area from the given latitude and longtitude coordinates.

I found various answers and articles describing how to calculate the distance in metres between two points given their latitude and longitude.

The simplest suggestion is to apply the Haversine formula, assuming that the Earth is a sphere with a circumference of 40075 km. In that case, the length in meters of 1 degree of latitude is always 111.32 km, and the length in meters of 1 degree of longitude equals 40075 km * cos( latitude ) / 360.

More precise calculations are suggested by the following three articles:

1. How to convert latitude or longitude to meters?
   → Calculate distance, bearing and more between Latitude/Longitude points
2. How to convert latitude or longitude to meters?
   → Solution suggested in Wikipedia on the geographic coordinate system
3. Understanding terms in Length of Degree formula

Here is the result of calculating the distances between along the edges between the six given points and comparing with the given edge lengths:

Edge    | Given | 1.            | 2.            | 3.
NW - NO | 31.10 | 31.01 (-0.09) | 30.96 (-0.14) | 31.07 (-0.03)
NO - OM | 13.34 | 13.38 (+0.04) | 13.36 (+0.02) | 13.37 (+0.03)
OM - SO | 22.51 | 22.55 (+0.04) | 22.52 (+0.01) | 22.53 (+0.02)
SO - SW | 29.63 | 29.56 (-0.07) | 29.51 (-0.12) | 29.62 (-0.01)
SW - WM | 19.26 | 19.24 (-0.02) | 19.22 (-0.04) | 19.23 (-0.03)
WM - NW | 16.24 | 16.28 (+0.04) | 16.25 (+0.01) | 16.26 (+0.02)

The third algorithm seems to return the most precise results, assuming the given points and edge distances are correct to start with.

Next, I used the metre-based X and Y coordinates produced by the third algorithm to also calculate the area and compare that with the expected result.

I tweaked the original latitude and longitude coordinates a bit to reduce the errors, even though I am not sure whether they stem from the coordinates or my processing.

The full report after adding that looks like this:

6 points:
  NW [47.61240288, 7.66845556]
  NO [47.6123859, 7.6688685]
  OM [47.61227361, 7.66880501]
  SO [47.6120811, 7.6687109]
  SW [47.6120972, 7.66831761]
  WM [47.612263, 7.66839227]
centre point:
  [47.612250614999994, 7.668591641666667]
edge lengths:
  NW - NO: 31.10 31.05 (-0.05) 30.99 (-0.11) 31.10 (+0.00)
  NO - OM: 13.34 13.38 (+0.04) 13.36 (+0.02) 13.37 (+0.03)
  OM - SO: 22.51 22.56 (+0.05) 22.54 (+0.03) 22.54 (+0.03)
  SO - SW: 29.63 29.57 (-0.06) 29.52 (-0.11) 29.62 (-0.01)
  SW - WM: 19.26 19.29 (+0.03) 19.26 (+0.00) 19.27 (+0.01)
  WM - NW: 16.24 16.28 (+0.04) 16.26 (+0.02) 16.26 (+0.02)
area calculated:
  1042.28 (error -0.72)