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 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. Because the infinitesimal circles represented by the ellipses on the map all have the same area on the underlying curved geometric model, the distortion imposed by the map projection is evident. There is a one-to-one correspondence between the Tissot indicatrix and the metric tensor of the map projection coordinate conversion.
Description 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:
A map distorts distances (linear distortion) wherever the quotient between the lengths of an infinitesimally short line as projected onto the projection surface, and as it originally is on the Earth model, deviates from 1. The quotient is called the scale factor. Unless the projection is conformal at the point being considered, the scale factor varies by direction around the point. A map distorts angles wherever the angles measured on the model of the Earth are not conserved in the projection. This is expressed by an ellipse of distortion which is not a circle. A map distorts areas wherever areas measured in the model of the Earth are not conserved in the projection. This is expressed by ellipses of distortion whose areas vary across the map. 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.
Mathematics In the diagram below, the circle A B C D {\displaystyle ABCD} has unit area as defined on the surface of a sphere. The ellipse A ′ B ′ C ′ D ′ {\displaystyle {A'B'C'D'}} is the Tissot's indicatrix that results from some projection of A B C D {\displaystyle ABCD} onto a plane. Linear scale has not been preserved in this projection, as O A ′ ≆ O A {\displaystyle {OA'\ncong OA}} and O B ′ ≆ O B {\displaystyle OB'\ncong OB} . Because ∠ M ′ O A ′ ≆ ∠ M O A {\displaystyle {\angle M'OA'\ncong \angle MOA}} , we know that there is an angular distortion. Because Area ( A ′ B ′ C ′ D ′ ) ≠ Area ( A B C D ) {\displaystyle \operatorname {Area} (A'B'C'D')\neq \operatorname {Area} (ABCD)} , we know there is an areal distortion.
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