# Does Web Mercator imply erroneous geospatial positioning?

Web Mercator nowadays is probably the most frequently used map projection. Don’t worry if you have never heard about it though. This new projection is used by Google Maps, Bing Maps by Microsoft, ArcGIS Online by Esri and the OpenLayers community. It became the standard of displaying geographic data on the web.

There are rumours that this standard is bad because it implies tremendous geospatial positioning errors. The Office of Geomatics at the U.S. National Geospatial-Intelligence Agency (NGA) puts the cost of using Web Mercator as follows:

up to 40.000 meters of erroneous geospatial positioning

So, what *is* wrong with Web Mercator?

Let’s take a step back: A map projection solves the problem of representing the 3D earth on a 2D plane, i.e. on a paper map or on a computer screen. The transformation from 3D to 2D involves distortions in one way or another. Of course, simply superimposing the outlines of Great Britain in two different projections (as the NGA did in the publication Implementation Practice Web Mercator Map Projection) results in shifts, see NGA’s figure below. And yes, the outlines do not match.

Distortions are what map projections are about! We therefore cannot talk about positioning errors when comparing different map projections such as the siblings Mercator and Web Mercator.

Given the fact that we have to live with distortions on maps we have to choose a map projection that suits a given purpose well. A map projection is therefore more or less appropriate for depicting a particular spatial fact or for accomplishing a specific task in space. Thus, a projection is neither good nor bad. Rather, its suitability has to be judged with the purpose in mind.

Alice had no idea what Latitude was, nor Longitude either, but thought they were nice grand words to say.

These are nice grand words indeed! However, other than Lewis Caroll’s *Alice in the Wonderland* we must have an idea at least what latitude means when talking about map projections. There are many latitudes to choose from:

On the sphere we know the **spherical latitude**. It is the angle between the normal to the sphere and the equatorial plane. On the ellipsoid with the two semi axis *a* and *b* (see figure below) we distinguish the **geodetic latitude** Φ or φ. This is the angle between the normal to the ellipsoid and the equatorial plane. The **geocentric latitude** γ is the angle between the radius (from the centre to the point on the ellipsoid) and the equatorial plane. Finally, there is the **reduced latitude** β that is used when transforming latitudes from sphere to ellipsoid.

When talking about latitude or **geographic latitude** we generally refer to the *geodetic* latitude. What the above figure also shows is that the definition of the latitude must be accompanied with the definition of an ellipsoid (e.g., semi-major axis and eccentricity).

On top of knowing the reference ellipsoid we must also know the **geodetic datum** that defines the position and orientation of the ellipsoid in relation to the geoid. When somebody uses **WGS84 coordinates** this most probably means

- The data is in geographic coordinates: geodetic latitude φ and longitude λ
- The coordinates refer to the WGS84 ellipsoid
- The ellipsoid is positioned and orientated relative to the geoid according the WGS84 datum

We now know the inputs, geodetic latitude φ and geodetic longitude λ, to our equation. Let’s examine the equation itself, the (Web) Mercator projection. There is a difference in how the vertical coordinate value (northing) is computed:

where:

*a = semi-major axis of ellipsoid*

*e = ellipsoid eccentricity*

*φ = geodetic latitude a.k.a. geographic latitude*

As the formulas show, Web Mercator simplifies the computation of northing considerably. In fact, it uses the equation for the sphere in combination with a sphere radius corresponding to the semi-major axis of the WGS84 ellipsoid. This simplification speeds up the projection computation.

Thus, when you require a projection that covers the entire globe seamlessly, Web Mercator is appropriate when you are interested in fast coordinate transformations. On the other hand, when your map needs to be conformal Mercator is your choice. I think it was a smart idea to simplify Mercator for web mapping purposes into Web Mercator.

Perhaps you noted that we didn’t talk about elevations. That’s another tricky story. Would you believe that Lake Constance has three different mean sea levels? I will post about that another time.

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Finally, some math on our blog! 🙂 Besides ease of calculation, Web Mercator has another advantage for some applications. For example, take our project walkalytics.com: We chose Mercator because the projection is conformal: a small part of the map (let’s say 2km, within a pedestrian’s reach) has_locally_ only small distortions. That’s everywhere in the world – even in Scotland or Iceland. So for Walkalytics, we can use one single projection for the whole world, which is a huge benefit in terms of processing and data storage. Additionally, directions are preserved (North is always upright, since WebMercator is cylindrical), which is nice for routing purposes.