Tissot's Indicatrix

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Figure 1 - Tissot's indicatrix
Figure 1 - Tissot's indicatrix

Tissot’s indicatrix, or ellipse of distortion, is a concept developed by French mathematician Nicolas Auguste Tissot, in 1859 and 1871, to measure and illustrate map distortions. It is the theoretical figure that results from the orthogonal projection of an infinitesimal circle with unit radio, defined in a geometric model of the Earth (a sphere or an ellipsoid), on the projection plane. Tissot proved that this figure is normally an ellipse, whose axes indicate the two principal directions of the projection at a certain point, i.e., the directions along which its scale is maximum and minimum. When the Tissot’s indicatrix reduces to a circle it means that, at that particular point, the scale is independent of direction. In conformal projections, where angles are preserved around every location, the Tissot’s indicatrix are all circles, with varying sizes. In equal-area projections, where area proportions between objects are conserved, the Tissot’s indicatrix have all unit area, although their shapes and orientations vary with location.

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 different from unity, 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 which is not a circle.
  • 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 from unity.

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.

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