Tissot's Indicatrix

Tissot’s indicatrix, or ellipse of distortion, is a circle or ellipse that measures and illustrates distortion caused by map projection. It is a concept developed by French mathematician Nicolas Auguste Tissot. In papers published in both 1859 and 1871, Tissot illustrated that distortions occur when a true spherical model of the earth is converted to a flattened map projection. When an infinitesimally small circle is projected on a geometric model of the Earth (a sphere or an ellipsoid), the resulting figure is an ellipse. The length of the axes of an ellipse of distortion demonstrates the extent to which the landforms represented on that area of the map will be distorted.

Tissot proved that this distortion figure is normally an ellipse, whose axes indicate the two principal directions of the projection at a certain point. These are the directions along which its scale is a maximum or a minimum. Tissot’s indicatrices are repeated at regular intervals across a map in order to illustrate the spatial pattern of distortion which varies across a map. Indicatrices are commonly placed at the intersections of parallels and meridians. In conformal projections, like the Mercator projection, where angles are preserved, the Tissot’s indicatrix is composed of circles with varying sizes. In equal-area projections, such as the Albers equal-area conic projection, where area proportions between objects are conserved, the Tissot’s indicatrix displays figures that all have the same area, while their respective shapes and orientations vary by location.

The Four General Instances of the Indicatrix
(Refer to Figure 1)

1. There are no changes in the semi-major axis a and semi-minor axis b, indicating no distortion of area nor angle.

2. The lengths of major axis a and minor axis b are unequal, creating an ellipse The area of the ellipse, however, is equal to the area of the unit circle. There is an increase in the length of the semi-major axis, but a proportionate decrease in the length of the semi-minor axis. In this case, there is angular distortion, but no areal distortion.

3. The length of both major axis a and minor axis b change the same. (e.g. a = 2.0 and b = 2.0). In this instance the indicatrix is still a circle, but it is larger or smaller than the unit circle. In this case there is no angular distortion, but there is areal distortion.

4. The lengths of semi-major axis a and semi-minor axis b change unequally, creating an ellipse, but the area of the ellipse is not equal to the area of the unit circle. There is both areal and angular distortion.  



Tissot’s indicatrix are used to graphically illustrate linear, angular and area distortions of maps:


 * A linear distortion occurs when the quotient between corresponding lengths in the projection surface and in the Earth model is different from the principal scale of the map. In the ellipse of distortion this is expressed by a radius length in the direction considered.
 * An angular distortion occurs when, in a particular location, the angles measured on the model of the Earth are not conserved in the projection. This is expressed by an ellipse of distortion that is not circular
 * An area distortion occurs when areas measured in the model of the Earth are not conserved in the projection. As a consequence, the corresponding ellipses of distortion have areas different measurements.

Consider Figure 1: ABCD is a circle with unit area defined in a spherical or ellipsoidal model of the Earth, and A’B’C’D’ is the Tissot’s indicatrix that results from its projection on the plane. Segment OA is transformed in OA’, and segment OB is transformed in OB’. Linear scale is not conserved along these two directions, since OA’ is not equal to OA and OB’ is not equal to OB. Angle MOA, in the unit are circle, is transformed in angle M’OA' in the distortion ellipse. Because M'OA'<MOA, there is an angular distortion. The area of circle ABCD is, by definition, equal to 1. Because the area of ellipse A’B’ is less than 1, an area distortion occurs.

Applications to GIS and Cartography
The principles illustrated by Tissot's Indicatrices are very important in GIS and Cartography as individuals seek to represent the world using a projection that is both functional and aesthetically pleasing. These two goals often contradict one another because maps that minimize distortion may not be especially pretty to look at and may even confuse viewers who are not especially familiar with the mathematical principles of projections and their distortions.

When choosing a map projection, it is best to have detailed information about the type, amount, and distribution of distortion for that projection. This allows the geographer to determine which projection will best fit his or her data. Because this information is so easily communicated using figures such as Tissot's ellipses of distortion, most GIS have a method by which one can calculate these indicatrices.

For methods of creating Tissot's indicatrices in GIS, see:
 * Making Tissot's Indicatrices in ArcGIS
 * Creating a Tissot's Indicatrix in QGIS