View source for Cylindrical equal-area projection

{{Short description|Family of map projections}}[[File:Lambert_cylindrical_equal-area_projection_SW.jpg|400px|thumb|Lambert cylindrical equal-area projection of the world; standard parallel at 0°]]
[[File:Tissot indicatrix world map Lambert cyl equal-area proj.svg|400px|thumb|The Lambert (standard parallel at 0°, normal) cylindrical equal-area projection with [[Tissot's indicatrix]] of deformation]]
In [[cartography]], the '''normal cylindrical equal-area projection''' is a family of [[Map projection#Normal cylindrical projection|normal cylindrical]], [[equal-area projection|equal-area]] [[map projection]]s.

== History ==
The invention of the Lambert cylindrical equal-area projection is attributed to the [[Switzerland|Swiss]] mathematician [[Johann Heinrich Lambert]] in 1772.<ref name="mulcahy_lambert">{{cite web
  | last =Mulcahy
  | first =Karen 
  | title =Cylindrical Projections
  | publisher = [[City University of New York]]
  | url =http://www.geo.hunter.cuny.edu/mp/cylind.html
  | access-date = 2007-03-30 }}</ref> Variations of it appeared over the years by inventors who stretched the height of the Lambert and compressed the width commensurately in various ratios.

== Description ==
[[File:Cilinderprojectie-constructie.jpg|thumb|How the Earth is projected onto a cylinder]]
The projection:
* is [[Map_projection#Cylindrical|cylindrical]], that means it has a cylindrical projection surface<ref>{{cite web | url=https://www.britannica.com/science/cylindrical-projection | title=Cylindrical projection {{pipe}} cartography {{pipe}} Britannica }}</ref>
* is normal, that means it has a normal [[Map projection#Aspect|aspect]]
* is an equal-area projection, that means any two areas in the map have the same relative size compared to their size on the sphere.

The term "normal cylindrical projection" is used to refer to any projection in which [[Meridian (geography)|meridians]] are mapped to equally spaced vertical lines and [[circles of latitude]] are mapped to horizontal lines (or, ''[[mutatis mutandis]]'', more generally, radial lines from a fixed point are mapped to equally spaced parallel lines and concentric circles around it are mapped to perpendicular lines).

The mapping of meridians to vertical lines can be visualized by imagining a cylinder whose axis coincides with the Earth's axis of rotation, then projecting onto the cylinder, and subsequently unfolding the cylinder.

By the geometry of their construction, cylindrical projections stretch distances east-west. The amount of stretch is the same at any chosen latitude on all cylindrical projections, and is given by the [[Secant (trigonometry)|secant]] of the [[latitude]] as a multiple of the equator's scale. The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude is given by ''φ''):

The only normal cylindrical projections that preserve area have a north-south compression precisely the reciprocal of east-west stretching ([[cosine|cos]]&nbsp;''φ''). This divides north-south distances by a factor equal to the secant of the latitude, preserving area but distorting shapes.

=== East–west scale matching the north–south scale ===
Depending on the stretch factor S, any particular cylindrical equal-area projection either has zero, one or two latitudes for which the east–west scale matches the north–south scale.
* S>1 : zero
* S=1 : one, that latitude is the equator
* S<1 : a pair of identical latitudes of opposite sign

== Formulae ==
The formulae presume a [[figure of the Earth|spherical model]] and use these definitions:<ref>[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 ''Map Projections – A Working Manual''] {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}, [[United States Geological Survey|USGS]] Professional Paper 1395, John P. Snyder, 1987, pp.76–85</ref>
* ''λ'' is the longitude
* ''λ''{{sub|0}} is the central meridian
* ''φ'' is the latitude
* ''φ''{{sub|0}} is the standard latitude
* ''S'' is the stretch factor
* ''x'' is the horizontal coordinate of the projected location on the map
* ''y'' is the vertical coordinate of the projected location on the map

{| class="wikitable"
! 
! using standard latitude ''φ''{{sub|0}}
! using stretch factor ''S''
! S=1, ''φ''{{sub|0}}=0
|-
! using radians 
| <math>\begin{align}x &= \cos \varphi_0 ( \lambda - \lambda_0 ) \\
  y &= \frac{\sin \varphi}{\cos \varphi_0}\end{align}</math>
| <math>\begin{align}x &= \sqrt{S} (\lambda - \lambda_0) \\ 
  y &= \frac{\sin\varphi}{\sqrt{S}} \end{align} </math>
| <math>\begin{align}x &= \lambda - \lambda_0\\
  y &= \sin \varphi\end{align}</math>
|-
! using degrees
| <math>\begin{align}x &= \frac{\cos\varphi_0 \pi ( \lambda - \lambda_0 )}{180^\circ}\\
  y &= \frac{\sin \varphi}{\cos \varphi_0}\end{align}</math>
|<math>\begin{align}x &= \frac{\sqrt{S} \pi ( \lambda - \lambda_0 )}{180^\circ} \\
  y &= \frac{\sin \varphi}{\sqrt{S}} \end{align}</math>
|<math>\begin{align}x &= \frac{\pi (\lambda - \lambda_0)}{180^\circ}\\
  y &= \sin \varphi \end{align}</math>
|}

Relationship between <math>S</math> and <math>\varphi_0</math>:
* <math>S = \cos^2 \varphi_0</math>
* <math>\varphi_0 = \arccos \sqrt{S}</math>

== Specializations ==
The specializations differ only in the ratio of the vertical to horizontal axis. Some specializations have been described, promoted, or otherwise named.<ref name="Snyder2">Snyder, John P. (1989). ''An Album of Map Projections'' p. 19. Washington, D.C.: U.S. Geological Survey Professional Paper 1453. (Mathematical properties of the Gall–Peters and related projections.)</ref><ref name="Monmonier">Monmonier, Mark (2004). ''Rhumb Lines and Map Wars: A Social History of the Mercator Projection'' p. 152. Chicago: The University of Chicago Press. (Thorough treatment of the social history of the Mercator projection and Gall–Peters projections.)</ref><ref name="Smyth">Smyth, C. Piazzi. (1870). ''On an Equal-Surface Projection and its Anthropological Applications''. Edinburgh: Edmonton & Douglas. (Monograph describing an equal-area cylindric projection and its virtues, specifically disparaging Mercator's projection.)</ref><ref>
Weisstein, Eric W. "Cylindrical Equal-Area Projection." From MathWorld—A Wolfram Web Resource. https://mathworld.wolfram.com/CylindricalEqual-AreaProjection.html
</ref><ref name="Tobler and Chen">Tobler, Waldo and Chen, Zi-tan(1986). ''A Quadtree for Global Information Storage''. http://www.geog.ucsb.edu/~kclarke/Geography232/Tobler1986.pdf</ref>

{| class="wikitable sortable"
|+ Specializations of the normal cylindrical equal-area projection, images showing projection centered on the [[Greenwich meridian]]
|-
! Stretch factor<br />''S''
! Aspect ratio<br /><small>(width-to-height)</small><br />{{pi}}''S''
! Standard parallel(s)<br />''φ''<sub>0</sub>
! Image (Tissot's indicatrix)
! Image (Blue Marble)
! Name
! Publisher
! Year of publication
|-
| 1
| {{pi}} ≈ 3.142
| data-sort-value="0"| 0°
| style="text-align:center"| [[File:Tissot indicatrix world map Lambert cyl equal-area proj.svg|314px]]
| style="text-align:center"| [[File:Lambert_cylindrical_equal-area_projection_SW.jpg|314px]]
| [[Lambert cylindrical equal-area projection|Lambert cylindrical equal-area]]
| [[Johann Heinrich Lambert]] 
| 1772
|-
| data-sort-value="0.75"| {{sfrac|3|4}} <br />= 0.75
| {{sfrac|3{{pi}}|4}} ≈ 2.356
| data-sort-value="30"| 30°
| style="text-align:center"| [[File:Tissot indicatrix world map Behrmann equal-area proj.svg|272px]]
| style="text-align:center"| [[File:Behrmann_projection_SW.jpg|272px]]
| [[Behrmann projection|Behrmann]]
| [[Walter Behrmann]]
| 1910
|-
| data-sort-value="0.63661977236"| {{sfrac|2|{{pi}}}} <br />≈ 0.6366
| 2
| data-sort-value="37.071435"| <math>\arccos \sqrt{\tfrac{2}{\pi}}</math> <br />≈ 37°04{{prime}}17{{pprime}} <br />≈ 37.0714°
| style="text-align:center"| [[File:Tissot indicatrix world map Smyth-Craster equal-area proj.svg|251px]]
| style="text-align:center"| [[File:Smyth-Craster_equal-area_projection_SW.jpg|251px]]
| [[Smyth equal-surface]]<br />= Craster rectangular 
| [[Charles Piazzi Smyth]] 
| 1870
|-
| data-sort-value="0.63109458932"| cos<sup>2</sup>(37.4°) <br />≈ 0.6311
| {{pi}}·cos<sup>2</sup>(37.4°) <br />≈ 1.983
| data-sort-value="37.4"| 37°24{{prime}} <br />= 37.4°
| style="text-align:center"| [[File:Tissot indicatrix world map Trystan Edwards equal-area proj.svg|251px]]
| style="text-align:center"| [[File:Trystan_Edwards_equal-area_projection_SW.jpg|251px]]
| Trystan Edwards
| [[Trystan Edwards]]
| 1953
|-
| data-sort-value="0.62940952255"| cos<sup>2</sup>(37.5°) <br />≈ 0.6294
| {{pi}}·cos<sup>2</sup>(37.5°) <br />≈ 1.977
| data-sort-value="37.5"| 37°30{{prime}} <br />= 37.5°
| style="text-align:center"| [[File:Tissot indicatrix world map Hobo-Dyer equal-area proj.svg|249px]]
| style="text-align:center"| [[File:Hobo–Dyer_projection_SW.jpg|249px]]
| [[Hobo–Dyer projection|Hobo–Dyer]]
| Mick Dyer 
| 2002
|-
| data-sort-value="0.58682408883"| cos<sup>2</sup>(40°) <br />≈ 0.5868
| {{pi}}·cos<sup>2</sup>(40°) <br />≈ 1.844
| data-sort-value="40"| 40°
| style="text-align:center"| [[File:Tissot indicatrix world map unnamed 40 deg equal-area proj.svg|241px]]
| style="text-align:center"| [[File:Cylindrical_equal-area_projection_SW.jpg|241px]]
| (unnamed)
| 
|
|-
| data-sort-value="0.5"| {{sfrac|1|2}} <br />=0.5
| {{sfrac|{{pi}}|2}} ≈ 1.571
| data-sort-value="45"| 45°
| style="text-align:center"| [[File:Tissot indicatrix world map Gall-Peters equal-area proj.svg|222px]]
| style="text-align:center"| [[File:Gall–Peters_projection_SW.jpg|222px]]
| [[Gall–Peters projection|Gall–Peters]]<br />= Gall orthographic<br />= Peters
| [[James Gall]],<br />Promoted by [[Arno Peters]] as his own invention
| 1855 (Gall),<br />1967 (Peters)
|-
| data-sort-value="0.41317591116"| cos<sup>2</sup>(50°) <br />≈ 0.4132
| {{pi}}·cos<sup>2</sup>(50°) <br />≈ 1.298
| data-sort-value="50"| 50°
| style="text-align:center"| [[File:Tissot indicatrix world map Balthasart equal-area proj.svg|200px]]
| style="text-align:center"| [[File:Balthasart_equal-area_projection_SW.jpg|200px]]
| Balthasart 
| M. Balthasart 
| 1935 
|-
| data-sort-value="0.31830988618"| {{sfrac|1|{{pi}}}} <br />≈ 0.3183
| 1 
| data-sort-value="55.6539665"| <math>\arccos \sqrt{\tfrac{1}{\pi}}</math> <br />≈ 55°39{{prime}}14{{pprime}} <br />≈ 55.6540°
| style="text-align:center"| [[File:Tissot indicatrix world map Tobler equal-area proj.svg|177px]]
| style="text-align:center"| [[File:Tobler_equal-area_projection_SW.jpg|177px]]
| Tobler's world in a square 
| [[Waldo Tobler]] 
| 1986
|}

== Derivatives ==

The [[Tobler hyperelliptical projection]], first described by Tobler in 1973, is a further generalization of the cylindrical equal-area family.

The [[HEALPix]] projection is an equal-area hybrid combination of: the Lambert cylindrical equal-area projection, for the equatorial regions of the sphere; and an interrupted [[Collignon projection]], for the polar regions.

==References==
{{reflist}}

==External links==
{{Commons cat}}
* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net

{{Map projections}}

[[Category:Cylindrical equal-area projections| ]]
[[Category:Equal-area projections]]
[[Category:Cylindrical projections]]