equal-area projection

In order for a map projection of the sphere to be equal-area, its generating formulae must meet this Cauchy-Riemann-like condition:{{cite book | title = Map projections — A working manual | last1 = Snyder | first1 = John P. | year = 1987 | series = USGS Professional Paper | volume = 1395 | page = 28| publisher = United States Government Printing Office | location = Washington | doi = 10.3133/pp1395 | url = https://pubs.er.usgs.gov/publication/pp1395}}

:$\backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash varphi\}\; \backslash cdot\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash lambda\}\; -\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash lambda\}\; \backslash cdot\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash varphi\}\; =\; s\; \backslash cdot\; \backslash cos\; \backslash varphi$

where $s$ is constant throughout the map. Here, $\backslash varphi$ represents latitude; $\backslash lambda$ represents longitude; and $x$ and $y$ are the projected (planar) coordinates for a given $(\backslash varphi,\; \backslash lambda)$ coordinate pair.

For example, the sinusoidal projection is a very simple equal-area projection. Its generating formulae are:

:$\backslash begin\{align\}$

x &= R \cdot \lambda \cos \varphi \\

y &= R \cdot \varphi

\end{align}

where $R$ is the radius of the globe. Computing the partial derivatives,

:$\backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash varphi\}\; =\; -R\; \backslash cdot\; \backslash lambda\; \backslash cdot\; \backslash sin\; \backslash varphi,\backslash quad\; R\; \backslash cdot\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash lambda\}\; =\; R\; \backslash cdot\; \backslash cos\; \backslash varphi,\backslash quad\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash varphi\}\; =\; R,\backslash quad\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash lambda\}\; =\; 0$

and so

:$\backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash varphi\}\; \backslash cdot\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash lambda\}\; -\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash lambda\}\; \backslash cdot\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash varphi\}\; =\; R\; \backslash cdot\; R\; \backslash cdot\; \backslash cos\; \backslash varphi\; -\; 0\; \backslash cdot\; (-R\; \backslash cdot\; \backslash lambda\; \backslash cdot\; \backslash sin\; \backslash varphi)\; =\; R^2\; \backslash cdot\; \backslash cos\; \backslash varphi\; =\; s\; \backslash cdot\; \backslash cos\; \backslash varphi$

with $s$ taking the value of the constant $R^2$.

For an equal-area map of the ellipsoid, the corresponding differential condition that must be met is:

:$\backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash varphi\}\; \backslash cdot\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash lambda\}\; -\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash lambda\}\; \backslash cdot\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash varphi\}\; =\; s\; \backslash cdot\; \backslash cos\; \backslash varphi\; \backslash cdot\; \backslash frac\{(1-e^2)\}\{(1-e^2\; \backslash sin^2\; \backslash varphi)^2\}$

where $e$ is the eccentricity of the ellipsoid of revolution.

{{stub section|date=April 2020}}

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.{{Cite web|url=https://inspire.ec.europa.eu/forum/discussion/view/10928/use-of-the-equal-area-grid-grid-etrs89-laea|title=INSPIRE helpdesk | INSPIRE|access-date=1 December 2019|archive-date=22 January 2021|archive-url=https://web.archive.org/web/20210122065047/https://inspire.ec.europa.eu/forum/discussion/view/10928/use-of-the-equal-area-grid-grid-etrs89-laea|url-status=dead}}http://scorus.org/wp-content/uploads/2012/10/2010JurmalaP4.5.pdf IBGE (2016), "Grade Estatística". Arquivo grade_estatistica.pdf em FTP ou HTTP, [http://geoftp.ibge.gov.br/recortes_para_fins_estatisticos/grade_estatistica/censo_2010 Censo 2010] {{Webarchive|url=https://web.archive.org/web/20191202014124/http://geoftp.ibge.gov.br/recortes_para_fins_estatisticos/grade_estatistica/censo_2010/ |date=2 December 2019 }}{{cite book |first=Lysandros |last=Tsoulos |pages=50–55 |chapter=An Equal Area Projection for Statistical Mapping in the EU |chapter-url=https://www.researchgate.net/publication/236852866 |title=Map projections for Europe |publisher=Joint Research Centre, European Commission |date=2003 |editor-last1=Annoni |editor-first1=Alessandro |editor-last2=Luzet |editor-first2=Claude |editor-last3=Gubler |editor-first3=Erich}}{{cite journal | last1=Brodzik | first1=Mary J. | last2=Billingsley | first2=Brendan | last3=Haran | first3=Terry | last4=Raup | first4=Bruce | last5=Savoie | first5=Matthew H. | title=EASE-Grid 2.0: Incremental but Significant Improvements for Earth-Gridded Data Sets | journal=ISPRS International Journal of Geo-Information | publisher=MDPI AG | volume=1 | issue=1 | date=2012-03-13 | issn=2220-9964 | doi=10.3390/ijgi1010032 | pages=32–45| doi-access=free }}

These are some projections that preserve area:

• Azimuthal
• Lambert azimuthal equal-area
• Wiechel (pseudoazimuthal)

File:Albers projection SW.jpg

• Conic
• Albers
• Lambert equal-area conic projection

File:Bottomley projection SW.JPG

• Pseudoconical
• Bonne
• Bottomley
• Werner

File:Lambert cylindrical equal-area projection SW.jpg

• Cylindrical (with latitude of no distortion)
• Lambert cylindrical equal-area (0°)
• Behrmann (30°)
• Hobo–Dyer (37°30′)
• Gall–Peters (45°)

File:Equal Earth projection SW.jpg

• 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{{Cite web|url=https://www.mathworks.com/help/map/flatplrq.html|title=McBryde-Thomas Flat-Polar Quartic Projection - MATLAB|website=www.mathworks.com|accessdate=3 January 2024}}
• Hammer
• Strebe 1995
• Snyder equal-area projection, used for geodesic grids.

• Authalic latitude
• Authalic radius
• Equiareal map (mathematics)
• Measure-preserving dynamical system
• Geodesic polygon area

{{Map projections}}

Category:Map projections