Tissot's Indicatrix

Tissot's indicatrix is a graphical means for the depiction of the amount and direction of distortion inherent in a map projection transformation. Invented by French mathematician Nicolas August Tissot, the elliptical figure is the result of a theoretical pointsized circle of unit radius on the globe or spheroid having gone through the map projection, and so having been distorted in size and shape.

Indicatrixes are normally shown as open circles or ellipses on a projected map and are repeated at regular intervals of latitude and longitude so that their collective pattern communicates to the map user the overall pattern of projection distortion. The ellipses show orientation, since their axes are in the directions of, and in proportion to, the maximum and minimum map scales at the chosen point. For example, if each circle simply expands or contracts but remains a circle, then directions at a point are preserved by the projection and the map is conformal. Similarly, if the circles all become ellipses of various orientations but are all of the same area, then the projection is of equal area or equivalent.

The indicatrix can thus show three dimensions of projection distortion. The orientation shows the direction of maximum scale distortion; the size of the indicatrix shows the amount of scale distortion; and the flattening of the ellipse shows differential directional scale distortion.

The indicatrix has been shown to be a highly efficient means of communicating projection distortion and blends both visual and mathematical methods. Means for its depiction are often included in map projection and geographic information systems software. Figure 1 shows two projections: left, a sinusoidal projection and, right, a Lambert Conformal Conic with Tissot's indicatrixes. The sinusoidal projection (equivalent, i.e., equal area) shows circles at the equator, becoming increasingly elliptical and [Page 479]angularly distorted as latitude increases and longitude diverges from the central meridian, but remaining the same size. The Lambert Conformal (shape-preserving) Conic projection shows only circles, but these get larger at high and low latitudes and do not change with longitude.

Figure 1 Tissot's Indicatrixes for Sinusoidal and Lambert Conformal Projections