Equal-area projection
Incartography,anequivalent,authalic,orequal-area projectionis amap projectionthat preserves relativeareameasure 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.
ByGauss'sTheorema Egregium,an equal-area projection cannot beconformal.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.
Description
[edit]In order for a map projection of the sphere to be equal-area, its generating formulae must meet thisCauchy-Riemann-like condition:[1]
whereis constant throughout the map. Here,represents latitude;represents longitude; andandare the projected (planar) coordinates for a givencoordinate pair.
For example, thesinusoidal projectionis a very simple equal-area projection. Its generating formulae are:
whereis the radius of the globe. Computing the partial derivatives,
and so
withtaking the value of the constant.
For an equal-area map of theellipsoid,the corresponding differential condition that must be met is:[1]
whereis theeccentricityof the ellipsoid of revolution.
Statistical grid
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The term "statistical grid" refers to adiscrete grid(global or local) of an equal-area surface representation, used fordata visualization,geocodeandstatistical spatial analysis.[2][3][4][5][6]
List of equal-area projections
[edit]These are some projections that preserve area:
- Azimuthal
- Lambert azimuthal equal-area
- Wiechel(pseudoazimuthal)
- Cylindrical(with latitude of no distortion)
- Lambert cylindrical equal-area(0°)
- Behrmann(30°)
- Hobo–Dyer(37°30′)
- Gall–Peters(45°)
- Pseudocylindrical
- Other
- Eckert-Greifendorff
- McBryde-Thomas Flat-Polar Quartic Projection[7]
- Hammer
- Strebe 1995
- Snyder equal-area projection,used forgeodesic grids.
See also
[edit]- Authalic latitude
- Authalic radius
- Equiareal map (mathematics)
- Measure-preserving dynamical system
- Geodesic polygon area
References
[edit]- ^abSnyder, John P. (1987).Map projections — A working manual.USGS Professional Paper. Vol. 1395. Washington: United States Government Printing Office. p. 28.doi:10.3133/pp1395.
- ^"INSPIRE helpdesk | INSPIRE".Archived fromthe originalon 22 January 2021.Retrieved1 December2019.
- ^http://scorus.org/wp-content/uploads/2012/10/2010JurmalaP4.5.pdf[dead link]
- ^IBGE (2016), "Grade Estatística". Arquivo
grade_estatistica.pdf
em FTP ou HTTP,Censo 2010Archived2 December 2019 at theWayback Machine - ^Tsoulos, Lysandros (2003)."An Equal Area Projection for Statistical Mapping in the EU".In Annoni, Alessandro; Luzet, Claude; Gubler, Erich (eds.).Map projections for Europe.Joint Research Centre,European Commission. pp. 50–55.
- ^Brodzik, Mary J.; Billingsley, Brendan; Haran, Terry; Raup, Bruce; Savoie, Matthew H. (13 March 2012)."EASE-Grid 2.0: Incremental but Significant Improvements for Earth-Gridded Data Sets".ISPRS International Journal of Geo-Information.1(1). MDPI AG: 32–45.doi:10.3390/ijgi1010032.ISSN2220-9964.
- ^"McBryde-Thomas Flat-Polar Quartic Projection - MATLAB".mathworks.Retrieved3 January2024.