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Concept
Equal-area projection
Summary
In cartography, an equivalent, authalic, or equal-area projection is a map projection that preserves relative area measure between any and all map regions. Equivalent projections are widely used for thematic maps showing scenario distribution such as population, farmland distribution, forested areas, and so forth, because an equal-area map does not change apparent density of the phenomenon being mapped.
By Gauss's Theorema Egregium, an equal-area projection cannot be conformal. This implies that an equal-area projection inevitably distorts shapes. Even though a point or points or a path or paths on a map might have no distortion, the greater the area of the region being mapped, the greater and more obvious the distortion of shapes inevitably becomes.
In order for a map projection of the sphere to be equal-area, its generating formulae must meet this Cauchy-Riemann-like condition:
where is constant throughout the map. Here, represents latitude; represents longitude; and and are the projected (planar) coordinates for a given coordinate pair.
For example, the sinusoidal projection is a very simple equal-area projections. Its generating formulæ are:
where is the radius of the globe. Computing the partial derivatives,
and so
with taking the value of the constant .
For an equal-area map of the ellipsoid, the corresponding differential condition that must be met is:
where is the eccentricity of the ellipsoid of revolution.
The term "statistical grid" refers to a discrete grid (global or local) of an equal-area surface representation, used for data visualization, geocode and statistical spatial analysis.
These are some projections that preserve area:
Azimuthal
Lambert azimuthal equal-area
Wiechel (pseudoazimuthal)
Conic
Albers
Lambert equal-area conic projection
Pseudoconical
Bonne
Bottomley
Werner
Cylindrical (with latitude of no distortion)
Lambert cylindrical equal-area (0°)
Behrmann (30°)
Hobo–Dyer (37°30′)
Gall–Peters (45°)
Pseudocylindrical
Boggs eumorphic
Collignon
Eckert II, IV and VI
Equal Earth
Goode's homolosine
Mollweide
Sinusoidal
Tobler hyperelliptical
Other
Eckert-Greifendorff
McBryde-Thomas Flat-Polar Quartic Projection
Hammer
Strebe 1995
Snyder equal-area projection, used for geodesic grids.
Official source
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Bonne projection
The Bonne projection is a pseudoconical equal-area map projection, sometimes called a dépôt de la guerre, modified Flamsteed, or a Sylvanus projection. Although named after Rigobert Bonne (1727–1795), the projection was in use prior to his birth, in 1511 by Sylvanus, Honter in 1561, De l'Isle before 1700 and Coronelli in 1696. Both Sylvanus and Honter's usages were approximate, however, and it is not clear they intended to be the same projection.
Tissot's indicatrix
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.
Geodesics on an ellipsoid
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry .