The distortion of the surface geometry is a necessary consequence of changing the media's shape, but often one or several geometric properties can be preserved to a great extent. Specific map projections are chosen to minimize the statistical properties of a distortion of particular interest; this can be improved if there are specific areas of interest on a map.

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 developed 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 an analysis of distortion that has had a major impact on the work of many 20th century writeres on map projections."(Snyder, 1993) 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 geometric 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 standard line, which is made true to scale."(Synder)

Transformation of Angles

Conformal or orthomorphic projections are those that maintain angular relations; cardinal directions remain 90° apart at any point (directions and angles are correct and the immediate surruounding area, not over regions of any significant size). To do this, the scale factor must be constant in every direction at every point, but then the scale factor varies from 1; therefore lose equal-area. If a projection is conformal, parallels and meridians will always meet at right angles (but not nec reverse).

Transformation of Areas

Equal-area or equivalent projections are those in which regions are portrayed in their correct relative sizes. This is done by designing a transformation such that the product of the scale factors in the principal directions always equals 1. This means that at most points, the scale factors are not equal in different directions, therefore conformality is lost.

Transformation of Distances

Accurate measurements of distance require constant scale along a line, and this scale must be consistent with the principal scale on the reference globe. There are two ways to maintain this on a map:

1. A scale factor of 1.0 is maintained along one or more parallel lines (can only be done on these lines), which are then called standard lines or standard parallels.
2. A scale factor of 1.0 is maintained in all directions from specific points on the map, which are called standard points, and the resultant map projection is called equidistant.

Thus, it is impossible to maintain all distances with a consistent map scale everywhere.

Transformation of Directions

It is also impossible to represent all true directions on a map with straight lines. For example, it is impossible to have a map show all great circle routes as straight lines, and have the same angular relationships to the map graticule as the globe graticule. The Mercator projection is often cited as one that shows "true direction", but recall that the straight lines on these maps are actually rhumbs, lines of constant bearing, not great circle routes.

"Correct" direction on a map can be thought of as a straight line on a map which actually represents a great circle route. At the starting point, this line will have the same azimuth with respect to the reference globe and the map projection's meridians. There are two categories of approaches to attempt taking this constraint beyond the starting point:

1. Great circle arcs between all points on a map may be shown as straight lines for a very limited area. Proper azimuths with respect to meridians can not be maintained for a large area (i.e. you cannot extend this across a hemisphere).
2. Great circle arcs with correct azimuths can be shown as straight lines for all directions from one or possibly two points. These are called azimuthal projections.

Analysis of Distortion

1. Visual Analysis

   Don't forget that when using the graticule and the cardinal directions as a method of inspecting projections, the projection is often positioned so that standard points or lines coincide with prominent points on the graticule, for convenience. Changing this coincidence can drastically change the appearance of the graticule, however the distortion in the projection is still the same. The following list of characteristics, as presented by Robinson et al. 1995, is convenient for visual analyses of projections:

   1. Parallels are parallel,
   2. Parallels, when shown at a constant interval, are equally spaced along meridians,
   3. Meridians and great circles on a globe appear as straight lines when viewed orthogonally,
   4. Meridians converge towards the poles and diverge towards the equator,
   5. Meridians, when shown at a constant interval, are equally spaced on the parallels, but their spacing decreases from the equator to the pole,
   6. When meridians and parallels are both shown at the same intervals, they are equally spaced at or near the equator,
   7. When meridians and parallels are both shown at the same intervals, meridians at 60° latitude are half as far apart as parallels, and
   8. The surface area bounded by any two parallels and two meridians (a given distance apart) is the same anywhere between the same parallels.
   
   
2. Quantitative Analysis

   Can use the changes is scale factor in cardinal directions as an analysis of the distortion. From the scale factor in two orthogonal directions, we can calculate the angular distortion (2Ω), and the product of a and b (S). These are commonly used to compare the distortions of various map projections.

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.