[PROJ] How does proj deal with ellipsoid with respect to reprojection

[Charles Karney]

Jochem,

Here's the equivalent table for the ellipsoidal gnomonic projection as
it applies (roughly) to the Netherlands.  I use the ellipsoidal gnomonic
projection centered at lat = 52 deg.  I considered geodesic line
segments of length S whose centers are 500 km from the center of the
projection.  I then computed D, the maximum deviation of the straight
line in the projection from the geodesic.

       S     max-deviation
     500 km   99 mm
     200 km   16 mm
     100 km    4 mm
      50 km  985 um
      20 km  158 um
      10 km   40 um
       5 km   10 um
       2 km    2 um
       1 km  394 nm
      500 m   99 nm
      200 m   18 nm

On 3/29/20 2:28 AM, Lesparre, Jochem wrote:
> Doing point in polygon in a projection will result in occasional wrong 
> conclusions. A point near de edge can seem to be inside the polygon 
> while it's outside, or the other way around, since a straight line in 
> the projection deviates from the geodesic.
> 
> I analysed this problem for the Netherlands (51 - 55 degrees north) in 
> the azimuthal projection (+proj=sterea) of the national coordinate 
> reference system RD (epsg:28992) and in plate-caree projection 
> (+proj=lonlat). The deviation depends on the length, orientation and 
> location of a polygon segment. I computed the maximum possible deviation 
> in the Netherlands for both projections to advise the Dutch government 
> not to allow any segments longer than 200 m in a new digital storage 
> system for policy and zoning borders.
> 
> Table for the Netherlands:
> Segment length, RD deviation, lonlat deviation;
> 
> 500 km, 160 m, 6 km;
> 200 km, 25 m, 1 km;
> 100 km, 6.4 m, 0.2 km;
> 50 km, 1.6 m, 60 m;
> 20 km, 26 cm, 9.7 m;
> 10 km, 8 cm, 2.4 m;
> 5 km, 3 cm, 60 cm;
> 2 km, 5 mm, 9.7 cm;
> 1 km, 1.3 mm, 2.4 cm;
> 500 m, 0.3 mm, 6 mm;
> 200 m, <0.1 mm, 1 mm;
> 100 m, <0.1 mm, 0.2 mm;
> 50 m, <0.1 mm, <0.1 mm
> 
> This means that when you first split long segments of polygons by adding 
> enough points along the geodesic, you could do a point in polygon even 
> in lonlat. The required distance between the added points depends on the 
> location (latitude). Near the poles it will get a bit tricky, as the 
> maximum deviation increases to 50% of the size of the segment length.
> 
> Regards, Jochem
>