# Map Projection Distortion

No map projection transformation can maintain scale everywhere. Angles, areas, distances and directions will be altered in the planar representation of the ellipsoidal earth. The distortions created during the map projection transformation may be analyzed using a measure of distortion.

Nicolas Auguste Tissot was one of many contributers during the 19th century of the application of sound mathematical pricipals to map projection science. During this period there was more than an eight fold increase in the number of publications relating to map projections.

Tissot's Indicatrix was develped during this era when new mathematics were being applied to map projections. In 1881 he published his: Memoire sur la representation des surfaces et les projections des cartes geographiques. In it Tissot "proposed a analy sis of distortion that has had a major impact on the work of many 20th century writeres on map projections."(Snyder, 1993, p 147) For over a 100 years the distortion characteristics inherent in map projection transformation have been revealed using the Indicatrix.

### The Indicatrix

A primary concept of Tissots's theory of deformation of map projections is the geometric deformation indicator: the Indicatrix. An infinately small circle on the surface of the Earth projects as an infinately small ellipse on the map projection plane. This ellipse describes characteristics locally at and near the infinately small ellipse.

The area described as an infinitesimally small on the surface of the Earth's ellipsoid can be dealt with as if it were a on a plane and remains infinitesimally small on the projection surface. The infinitesimally small circle and the projected ellipse are related to one another by a 2-dimensional afine transformation and hence the rules of projective Euclidean geometry apply.

The semiaxes a and b of the distortion ellipse, both in size and direction, are determined by the equations of the map projection and the geoemtric properties of the Earth's ellipsoidal surface at the point being evaluated. The local properties being of the transformation being evaluated by the Indicatrix include distortions in lengths, angles, and areas.

According to Snyder "the orientation (of the axis of the ellipse) is of much less interest than the size of the deformation." (Snyder, Handbook 21) Scale distortion is "most often calculated as the ratio of the scale along the meridian or along the parallel at a given point to the scale at a standard point or along a stndard line, which is made true to scale."(Synder, 1987, p21)

Laskowski, Piotr H. 1989. "The traditional and modern look at Tissot's indicatrix." Chapter 14 in Accuracy of Spatial Databases. Eds. Michael Goodchild and Sucharita Gopal. Taylor and Francis. Bristol, PA. p. 155-174.

Maling, D.H. 1992. Coordinate Systems and Map Projections, 2nd Ed. Pergamon Press. Oxford.

Robinson, Arthur H., Randall D. Sale, Joel L. Morrison, and Phillip C. Muehrcke. 1984. Elements of Cartography, 5th Ed. John Wiley & Sons. New York.

Snyder, John P. 1993. Flattening the Earth: Two Thousand Years of Map Projections. University of Chicago Press. Chicago, IL.

------. 1987. Map Projections--A Working Manual. U.S. Geological Survey Professional Paper 1395. U.S. Government Printing Office. Washington, D.C.