Equal-area projection

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.