Map Projections

Map Projection Research

# 5707.2 - A Loxodrome in the Web-Mercator Projection

Wednesday, July 5
4:30 PM - 4:50 PM
Location: Maryland B

## Presenting Author(s)

• Miljenko Lapaine, PhD

Professor
University of Zagreb, Faculty of Geodesy

• ## Moderator(s)

• Bojan Šavrič, PhD

Software Development Engineer
Environmental Systems Research Institute

In 2005, Google made available maps of the world on the Internet which could be zoomed to the largest scales. To produce any map, points on the surface of a rotating ellipsoid, which in geodesy and cartography approximates the irregular surface of the Earth, must be portrayed in a plane, by applying one of many map projections. To this end, Google applied the following procedure. The geodetic latitude and longitude on the ellipsoid were mapped onto a plane according to formulae of the Mercator projection for a sphere. The large semi-axis of the WGS84 ellipsoid, which is widely applied throughout the world today, was considered the sphere’s radius. Google applied this procedure because the formulae for mapping a sphere are simpler and the procedure is five times quicker than mapping an ellipsoid (Zinn 2010).

The mapping procedure described resulted in a new map projection, which is similar to the Mercator projection, but it lacks some of its properties. This projection is known as the ‘Web Mercator projection’, or ‘Web Mercator’ for short. It is sometimes called ‘Pseudo Mercator’. Like the Mercator projection, the Web Mercator is a normal aspect cylindrical projection, i.e. meridians are drawn as parallel lines, as are parallels of latitude, perpendicularly to meridians. The essential difference between the two projections is that the Web Mercator projection is not conformal. This means the linear scale for a given point in the Web Mercator projection is not the same in all directions, which is the case in all conformal projections.

After Google, many others used the Web Mercator projection for producing maps on the web. Battersby et al. (2014) stress that the Web Mercator projection is applied in all of the following: Google Maps, Microsoft Bing Maps, Yahoo Maps, Esri's ArcGIS Online, OpenStreetMap and The National Map of the US Geological Survey. It has thus become the standard in web map production.

It is well-known that the image of a loxodrome on a Mercator chart is a straight line. A question which follows from that fact is: What can we say about the image of a loxodrome in the plane of the Web Mercator projection? The answer to this question is given in the present paper. The equation of the image of a loxodrome in the Web Mercator projection was derived. First of all, it shows the image of a loxodrome is not a straight line like on a Mercator chart. On the other hand, the image of a loxodrome in the Web Mercator projection is close to a straight line. The derived formula enables us to estimate deviations of the image of a loxodrome in the Web Mercator projection from a straight line. These results are important because they enable one to conclude whether the Web Mercator chart can be used for navigation purposes.

# Miljenko Lapaine

Professor
University of Zagreb, Faculty of Geodesy

Miljenko Lapaine graduated from the Faculty of Science, University of Zagreb, in the field of Theoretical Mathematics. He obtained his PhD from the Faculty of Geodesy, University of Zagreb with a dissertation entitled Mapping in the Theory of Map Projections. He has been a full professor since 2003. He has published more than 900 papers, several textbooks and monographs. Prof. Lapaine is the Chairman of the ICA Commission on Map Projections, a founder and President of the Croatian Cartographic Society and the Executive editor of the Cartography and Geoinformation journal.

# Bojan Šavrič

Software Development Engineer
Environmental Systems Research Institute

Bojan Šavrič is a Software Development Engineer for Projection Engine team at Esri, Inc. He holds a Ph.D. in geography and Diploma degree in geodetic engineering. His main research interests are map projections, mathematical techniques in cartography, and the development of tools for cartographers. Bojan is also a member of the International Cartographic Association Commission on Map Projections.

# Assets

## Attendees who have favorited this

• Fritz Kessler
Senior Research Associate / Associate Professor
Penn State
• Bojan Savric
Software Development Engineer
Environmental Systems Research Institute