Tissot's indicatrices

In cartography, a Tissot's indicatrix (Tissot indicatrix, Tissot's ellipse, Tissot ellipse, ellipse of distortion) (plural: "Tissot's indicatrices") is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

A single indicatrix describes the distortion at a single point. Because distortion varies across a map, generally Tissot's indicatrices are placed across a map to illustrate the spatial change in distortion. A common scheme places them at each intersection of displayed meridians and parallels. These schematics are important in the study of map projections, both to illustrate distortion and to provide the basis for the calculations that represent the magnitude of distortion precisely at each point.

There is a one-to-one correspondence between the Tissot indicatrix and the metric tensor of the map projection coordinate conversion.

Tissot's theory was developed in the context of cartographic analysis. Generally the geometric model represents the Earth, and comes in the form of a sphere or ellipsoid.

Tissot's indicatrices illustrate linear, angular, and areal distortions of maps:

In conformal maps, where each point preserves angles projected from the geometric model, the Tissot's indicatrices are all circles of size varying by location, possibly also with varying orientation (given the four circle quadrants split by meridians and parallels). In equal-area projections, where area proportions between objects are conserved, the Tissot's indicatrices all have the same area, though their shapes and orientations vary with location. In arbitrary projections, both area and shape vary across the map.

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