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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.
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 unity. 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.
|World maps comparing Tissot's indicatrices on some common projections|
In the image to the right, 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 area circle, is transformed in angle M′OA′ in the distortion ellipse. Because M′OA′ ≠ MOA, we know that 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, a distortion of area has occurred.
In dealing with a Tissot indicatrix, different notions of radius come into play. The first is the infinitesimal radius of the original circle. The resulting ellipse of distortion will also have infinitesimal radius, but by the mathematics of differentials, the ratios of these infinitesimal values are finite. So, for example, if the resulting ellipse of distortion is the same size of infinitesimal as on the sphere, then its radius is considered to be 1. Lastly, the size that the indicatrix gets drawn for human inspection on the map is arbitrary. When a network of indicatrices is drawn on a map, they are all scaled by the same arbitrary amount so that their sizes are proportionally correct.
Other distortion metrics
Many ways have been described for characterizing distortion in projections. Some, like Tissot's indicatrix, give visual indication of the distortion, such as the flexion and skewness (bending and lopsidedness) model.
Other visual methods project shapes that span a part of the map, rather than originally infinitesimal. In the first half of the 20th century, projecting a human head onto different projections was common to show how distortion varies across one projection as compared to another. Sometimes spherical triangles are used; sometimes other shapes. In dynamic media, shapes of familiar coastlines and boundaries can be dragged across an interactive map to show how the projection distorts sizes and shapes according to position on the map.
Another way to visualize local distortion is through grayscale or color gradations whose shade represents the value of the angular deformation or areal inflation. Sometimes both are shown simultaneously by blending two colors to create a bivariate map.
The problem of characterizing distortion globally across areas instead of at just a single point necessarily involves choosing priorities. Some schemes use distance distortion as a proxy for the combination of angular deformation and areal inflation; such methods arbitrarily choose what paths to measure and how to weight them in order to yield a single result. Many have been described.
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