Nationwide transformation of coordinates in the Bonne projection used for historical map series of the Netherlands
Jochem Lesparre, Nederlandse Samenwerking Geodetische Infrastructuur (NSGI)
The historical map series of the Netherlands the Topografische en Militaire Kaart (TMK) and the Chromotopografische Kaart des Rijks (Bonnebladen) use geographic coordinates from Krayenhoff’s triangulation and coordinates in the Bonne projection. The older Choro-topographische kaart der Noordelijke Provinciën van het Koningrijk der Nederlanden by Krayenhoff is based on the same triangulation, but used a different ellipsoid and a different projection. While a locally optimised transformation (e.g. rubbersheeting) will give the best results for georeferencing the Dutch historical map series based on the Bonne projection, a nationwide coordinate transformation can be useful. There are 2 options for this transformation:
- Original polynomial transformation from Bonne coordinates to RD (EPSG:28992)
- Approximate similarity transformation from Bonne coordinates to ETRS89
The original polynomial transformation uses a third degree polynomial published in 1925 by Van Riel.[TKL 1925, p.19] It is based on a subset of 41 points from the 64 triangulation points that Krayenhoff's triangulation and the Rijksdriehoeksmeting (RD) had incommon.
The approximate similarity transformation introduced in this document, is based on a least-squares estimation using 102 common points with coordinates in the Bonne projection as well as RDNAP coordinates. The points are a subset, evenly distributed over the Netherlands, obtained from the original polynomial transformation (Tafel van Van Riel). To reduce the size of the rotation angles of the similarity transformation, the longitude of the prime meridian is set to the value 4° 53′ 1.978″ E of Greenwich from RD.[Haasbroek 1972] For geographic Krayenhoff coordinates, the prime meridian of Amsterdam should be set to zero.
The residuals of the approximate similarity transformation reach up to 6 m and show a whirly pattern that is typical for the inaccuracies of a triangulation Figure. For the original polynomial transformation, the residuals are smaller Figure. However, the pattern and magnitude of the residuals are only representative for high-order triangulated points. The deviations of topographic features in the historical maps can be much larger (e.g. RMSE of 74 m for a sample of 32 points in the municipality Dieren)[TKL 1925, p.55] for both transformations.
The parameters of the Dutch Bonne projection are listed in several historical sources.[e.g. Van der Schraaf 1979] Some sources also provide lists of points with coordinates in Bonne projection as well as geographic Krayenhoff coordinates,[Topografisch Bureau 1864] with errors up to 5 m due to the numeric inaccuracy of historical computation methods like slide rulers.
Commands for the cct program of PROJ are given. A PROJ4-style string is provided too.
PROJ pipeline for the original third degree polynomial (Horner) transformation from coordinates in Dutch Bonne projection to RD (EPSG:28992), e.g. for use in cct of PROJ:
cct -d 4 -z 0 +proj=pipeline
+step +proj=horner +deg=3 +fwd_origin=0,0 +fwd_u=+120025.8705,+0.999986846,-0.0250515e-8,+0.1471965e-14,+0.006918172,-0.0860518e-8,-0.0398590e-14,+0.0011249e-8,+0.4556545e-14,+0.1991105e-14 +fwd_v=+390181.7849,+0.999993024,-0.0338879e-8,+0.1184845e-14,-0.006907254,-0.0492600e-8,-0.1945916e-14,+0.0222878e-8,+0.9244571e-14,-0.1091684e-14 +inv_tolerance=0.0001
For transformation to ETRS89 or WGS 84, additional steps of the transformation RDNAPTRANS™ should be added.[NSGI]
PROJ pipeline for an approximate 7-parameter similarity (Helmert) transformation from coordinates in Dutch Bonne projection to ETRS89, e.g. for use in cct of PROJ:
cct -d 9 -z 0 +proj=pipeline
+step +proj=bonne +lat_1=51.5 +lon_0=0 +a=6376950.4 +rf=309.65 +pm=4.883882778 +inv
+step +proj=cart +a=6376950.4 +rf=309.65
+step +proj=helmert +convention=coordinate_frame +exact +x=932.9862 +y=86.2986 +z=-197.9356 +rx=2.276813 +ry=1.478043 +rz=4.673555 +s=50.09450
+step +proj=cart +ellps=GRS80 +inv
For transformation to RD, additional steps of the transformation RDNAPTRANS™ should be added.[NSGI] For transformation to WGS 84, no additional steps are needed if a null transformation is used, as recommended for most applications.[Geonovum]
PROJ4-style string for an an approximate 7-parameter similarity (Helmert) transformation from coordinates in Dutch Bonne projection to ETRS89, e.g. for defining a custom CRS in QGIS:
+proj=bonne +lat_1=51.5 +lon_0=0 +a=6376950.4 +rf=309.65 +pm=4.883882778 +towgs84=932.9862,86.2986,-197.9356,-2.276813,-1.478043,-4.673555,50.09450 +type=crs
NB: A null transformation between ETRS89 and WGS84 is used, as recommended for most applications.[Geonovum]
For transformation to RD, an additional PROJ4 string is needed for RDNAPTRANS™.[NSGI]