What's the accuracy of Web Mercator EPSG:3857?

Question

Are there any references on the accuracy or (min/max/avg) error of Web Mercator when it comes to e.g. distance calculations?

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For the sake of completeness, I haven't found any reference to the accuracy Web Mercator. In contrast, spatial computations in WGS-84 projection are, due to Charles Karney, highly accurate as described in http://arxiv.org/abs/1109.4448 This, however, requires Gnomonic projection of coordinates for point to line projections which is very costly in GIS software. An alternative is the use of de-facto standard Web Mercator which must be questioned if accuracy is sufficient.

Answer 1

As a first step, you could look at the distortions of the Mercator projection, which is a conformal projection. Distance with this projection is only correct along the equator, then the error increase with the latitude. Indeed, as you can see on a global view, the parallels keep the same legnth on the maps. For example, the horizontal scale factor, which is 1 at the equator, is equal to 1.15 at a latitude of 30° (15% error), 2 at a latitude of 60° and 11.5 at a latitude of 85° (scale factor 1/cos(latitude) )

Web Mercator uses WGS 84 Lat/Long as if they were angles on a sphere, presumably to reduce the processing time. This leads to more distortions, especially when you move away from the equator, that make the Web Mercator non-conformal. However,the impact of the "pseudo-sphericity" on distance measurements at high latitude is a lot smaller than the design of the Mercator projection.

The same reasoning holds for area distortion.

Answer 2

You can study the error by making SQL queries with Spatialite-gui which has a function ST_Length with a description at https://www.gaia-gis.it/gaia-sins/spatialite-sql-latest.html

return the length of c (measured in meters). If the use_ellipsoid argument is set to TRUE the precise (but slower) length will be computed on the Ellipsoid, otherwise will be computed on the Great Cicle (approximative, but faster).

Examples:

• Example one, east-west oriented linestring, length one degree, located at latitude 20°N

Great circle length:

select ST_Length(ST_GeomFromText('LINESTRING (0 20,1 20)',4326),0)
104489.040752

Length along ellipsoid:

select ST_Length(ST_GeomFromText('LINESTRING (0 20,1 20)',4326),1)
104646.930934

Then the length when the linestring is projected first into EPSG:3857:

select ST_Length(ST_Transform(ST_GeomFromText('LINESTRING (0 20,1 20)',4326),3857))
111319.490793

Difference to length along ellipsoid 6673 m.

• Example two, east-west oriented linestring, length one degree, located at latitude 60°N

Great circle length:

select ST_Length(ST_GeomFromText('LINESTRING (0 60,1 60)',4326),0)
55597.010615

Length along ellipsoid:

select ST_Length(ST_GeomFromText('LINESTRING (0 60,1 60)',4326),1)
55799.470393

Then the length when the linestring is projected first into EPSG:3857:

select ST_Length(ST_Transform(ST_GeomFromText('LINESTRING (0 60,1 60)',4326),3857))
111319.490793

Difference to length along ellipsoid 55520 m.

You can do similar queries with PostGIS but you must use "geography" http://postgis.net/docs/ST_Length.html.