A serious study of map projections usually involves a comparison of how they are affected by the three main kinds of distortion — area, shape and distance. Distortion may be visually estimated by inspecting graticule patterns and the general shape of coastlines; it can also be evaluated by measuring distances between selected sets of points. However, a systematic approach to quantitatively calculating distortion had to wait for Nicolas A. Tissot's extremely influential papers of 1878 and 1881 (some ideas were already introduced in a work of 1859) presenting his ellipse of distortion, universally known today as Tissot's indicatrix.
Tissot imagined an infinitesimally small circle centered on some point on the surface of the Earth, and considered its shape after transformed by a given map projection. He proved it to be a perfect ellipse, centered exactly on the corresponding point on the map. Also,
Equatorial Mercator map, clipped at 85°N and 85°S, with Tissot's indicatrices (again, the oversized circles here used for illustration are actually ovals which slightly violate conformality) |
The characteristic distortion pattern of a projection may be roughly visualized by means of an array of Tissot's indicatrices regularly spaced along the map. For some projections, the angle and major and minor radii of each ellipse can be calculated analytically. In practice, commonly the original circles are simply numerically projected and rendered on the map; in order to be visible — even after scaling — they must usually be much larger than infinitesimal and don't necessarily look like perfect ellipses.
All equal-area projections distort shape nearly everywhere. For instance, Hammer's elliptical projection is not conformal, except, in the normal aspect, at the intersection of the Equator and the central meridian. Mollweide's projection is only free of shape distortion at the intersection of the central meridian with the two standard parallels, at about 40° N and S. Eckert's second projection also has two standard parallels; if numerically projected, the sharp break of angular deformation at the Equator creates some peculiarly shaped indicatrices.
On the other hand, in Mercator's conformal projection all indicatrices remain circular in shape, parallels keep parallelism, meridians are straight lines and always perpendicular to every parallel. Areas are not preserved, but greatly increase towards the top and bottom of the map: circles at the poles would be infinitely large (this is to be expected, since meridians cross one another on a sphere but never touch in a Mercator map. Only infinite circles on different meridians could be all concentric as in the globe's poles).
Oblique Mercator map. Compared with the equatorial version, the circles moved around but their relative sizes still depend only on their distance from a horizontal line. |
Another conformal projection, the azimuthal stereographic preserves the shape of every circle, naturally including indicatrices.
In a projection neither conformal nor equal-area like the azimuthal orthographic, Tissot's indicatrices keep neither original shape nor area.
Given a circle on Earth, Tissot considered pairs of diameters; he proved there is always one pair which intersects orthogonally (i.e., at a right angle) both on the circle's center and on its projection in the map, where they comprise the ellipse's major and minor axes. In addition, there is another pair intersecting at a right angle on the circle, but maximally far from a right angle on the map. This deviation is the maximum angular deformation at that point and is, of course, zero if the projection is conformal there.
Tissot developed equations relating the scale distortion (either compression or stretching) at any point of a projection with the maximum angular deformation: for conformality, for every pair of orthogonal directions their scales must be the same; for areal equivalence, they must be reciprocal. With his formulas the deformation pattern may be calculated with any desired precision and, when plotted on a map, directly evinces zones of major areal or shape distortion.
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On the left, equatorial and oblique azimuthal orthographic maps; on the right, oblique azimuthal stereographic map. All clipped to one hemisphere. |
As shown by oblique orthographic and Mercator maps, Tissot's indicatrices display an overall deformation pattern, not affected by aspect changes: in an azimuthal map, distortion depends only on the radial distance from the conceptual center of the map; in a cylindrical map, only on the vertical distance from the conceptual "Equator". These are exactly the same patterns presented by the normal versions.
Equatorial cylindrical equidistant map |
Finally, indicatrices may help distinguishing between easily confused projections, for instance three designs bounded by a circle, ordinarily used in the equatorial aspect, with meridians curved away from the Equator and extreme areal exaggeration near the poles: