Sinusoidal projection: Difference between revisions

Browse history interactively

← Previous edit

Content deleted Content added

VisualWikitext

Inline

Latest revision as of 22:24, 3 December 2024

The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson–Flamsteed or the Mercator equal-area projection. Jean Cossin of Dieppe was one of the first mapmakers to use the sinusoidal, using it in a world map in 1570.^[1]

The projection represents the poles as points, as they are on the sphere, but the meridians and continents are distorted. The equator and the central meridian are the most accurate parts of the map, having no distortion at all, and the further away from those that one examines, the greater the distortion.^[2]

The projection is defined by:

{\begin{aligned}x&=\left(\lambda -\lambda _{0}\right)\cos \varphi \\y&=\varphi \,\end{aligned}}

where $\varphi$ is the latitude, λ is the longitude, and λ₀ is the longitude of the central meridian.^[3]

Scale is constant along the central meridian, and east–west scale is constant throughout the map. Therefore, the length of each parallel on the map is proportional to the cosine of the latitude, as it is on the globe. This makes the left and right bounding meridians of the map into half of a sine wave, each mirroring the other. Each meridian is half of a sine wave with only the amplitude differing, giving the projection its name. Each is shown on the map as longer than the central meridian, whereas on the globe all are the same length.

The true distance between two points on a meridian can be measured on the map as the vertical distance between the parallels that intersect the meridian at those points. With no distortion along the central meridian and the equator, distances along those lines are correct, as are the angles of intersection of other lines with those two lines. Distortion is lowest throughout the region of the map close to those lines.

Similar projections which wrap the east and west parts of the sinusoidal projection around the North Pole are the Werner and the intermediate Bonne and Bottomley projections.

The MODLAND Integerized Sinusoidal Grid, based on the sinusoidal projection, is a geodesic grid developed by the NASA's Moderate-Resolution Imaging Spectroradiometer (MODIS) science team.^[4]

References

^ Jean Cossin, Carte cosmographique ou Universelle description du monde avec le vrai traict des vents, 1570.
^ "Finally, a World Map That Doesn't Lie". D-brief. 2016-11-03. Archived from the original on 2019-10-09. Retrieved 2019-07-01.
^ Map Projections—A Working Manual, USGS Professional Paper 1395, John P. Snyder, 1987, pp. 243–248
^ NASA: "MODLAND Integerized Sinusoidal Grid"

External links

Media related to Sinusoidal projection at Wikimedia Commons
Pseudocylindrical Projections
Table of examples and properties of all common projections, from radicalcartography.net

[1] Jean Cossin, Carte cosmographique ou Universelle description du monde avec le vrai traict des vents, 1570.

[2] "Finally, a World Map That Doesn't Lie". D-brief. 2016-11-03. Archived from the original on 2019-10-09. Retrieved 2019-07-01.

[3] Map Projections—A Working Manual, USGS Professional Paper 1395, John P. Snyder, 1987, pp. 243–248

[4] NASA: "MODLAND Integerized Sinusoidal Grid"

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
+{{Short description|Pseudocylindrical equal-area map projection}}
-[[Image:Sinusoidal projection.gif|thumb|250px|right|Sinusoidal projection]]
+[[File:Sinusoidal projection SW.jpg|300px|thumb|Sinusoidal projection of the world.]]
-A '''sinusoidal projection''' is a pseudocylindrical equal-area [[map projection]], sometimes called the '''Sanson-Flamsteed''' or the '''Mercator equal-area projection'''.
+[[File:Sinusoidal with Tissot's Indicatrices of Distortion.svg|thumb|right|300px|The sinusoidal projection with [[Tissot's indicatrix]] of deformation]]
+[[File:Carte cosmographique ou Universelle description du monde, Jean Cossin, Dieppe, 1570.png|thumb|Jean Cossin, Carte cosmographique ou Universelle description du monde, Dieppe, 1570]]
+The '''sinusoidal projection''' is a pseudocylindrical [[equal-area projection|equal-area]] [[map projection]], sometimes called the '''Sanson–Flamsteed''' or the '''Mercator equal-area projection'''. Jean Cossin of [[Dieppe maps|Dieppe]] was one of the first mapmakers to use the sinusoidal, using it in a world map in 1570.<ref>Jean Cossin, [http://gallica.bnf.fr/ark:/12148/btv1b5901087w ''Carte cosmographique ou Universelle description du monde avec le vrai traict des vents''], 1570.</ref>
-The north-south scale is the same everywhere at the central [[meridian (geography)|meridian]], and the east-west scale is throughout the map the same as that; correspondingly, on the map, as in reality, the length of each parallel is proportional to the cosine of the latitude; thus the shape of the map for the whole earth is the area between two symmetric rotated cosine curves. The true distance between two points on the same meridian corresponds to the distance on the map between the two parallels, which is smaller than the distance between the two points on the map.  There is no distortion on the central meridian or the [[equator]].
+The projection represents the poles as points, as they are on the sphere, but the meridians and continents are distorted. The equator and the central meridian are the most accurate parts of the map, having no distortion at all, and the further away from those that one examines, the greater the distortion.<ref>{{Cite web|url=http://blogs.discovermagazine.com/d-brief/2016/11/03/most-accurate-world-map/#.XZ2CH-zP30o|title=Finally, a World Map That Doesn't Lie|date=2016-11-03|website=D-brief|archive-url=https://web.archive.org/web/20191009064547/http://blogs.discovermagazine.com/d-brief/2016/11/03/most-accurate-world-map/#.XZ2CH-zP30o |access-date=2019-07-01|archive-date=2019-10-09 }}</ref>
-[[Image:Usgs_map_sinousidal_equal_area.PNG|400px|frame|center|A sinusoidal projection shows relative sizes accurately, but distorts shapes and directions. Distortion can be reduced by "interrupting" the map.]]
+The projection is defined by:
-Similar projections which wrap the east and west parts of the sinusoidal projection around the [[north pole]] are the [[Werner projection|Werner]] and the intermediate [[Bonne projection|Bonne]] and [[Bottomley projection]]s.
+:<math>\begin{align} x &= \left(\lambda - \lambda_0\right) \cos \varphi \\ y &= \varphi\,\end{align}</math>
-==External link==
-*[http://193.55.107.45/articles/241eng.pdf Cybergeo article]
+where <math>\varphi</math> is the latitude, ''λ'' is the longitude, and ''λ''{{sub|0}} is the longitude of the central meridian.<ref>[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 ''Map Projections—A Working Manual''], [[United States Geological Survey|USGS]] Professional Paper 1395, John P. Snyder, 1987, pp. 243–248</ref>
+Scale is constant along the central [[meridian (geography)|meridian]], and east–west scale is constant throughout the map. Therefore, the length of each parallel on the map is proportional to the cosine of the latitude, as it is on the globe. This makes the left and right bounding meridians of the map into half of a sine wave, each mirroring the other. Each meridian is half of a sine wave with only the amplitude differing, giving the projection its name. Each is shown on the map as longer than the central meridian, whereas on the globe all are the same length.
-{{Cartography-stub}}
-[[Category:Cartographic projections]]
+The true distance between two points on a meridian can be measured on the map as the vertical distance between the parallels that intersect the meridian at those points. With no distortion along the central meridian and the [[equator]], distances along those lines are correct, as are the angles of intersection of other lines with those two lines. Distortion is lowest throughout the region of the map close to those lines.
-[[ca:Projecció sinusoidal]]
-<!--[[en:Sinusoidal projection]]-->
+[[File:Usgs map sinousidal equal area.PNG|350px|thumb|right|A sinusoidal projection shows relative sizes accurately, but distorts shapes and directions. Distortion can be reduced by "interrupting" the map.]]
-[[fr:Projection de Sanson-Flamsteed]]
+Similar projections which wrap the east and west parts of the sinusoidal projection around the [[North Pole]] are the [[Werner projection|Werner]] and the intermediate [[Bonne projection|Bonne]] and [[Bottomley projection|Bottomley]] projections.
-[[ja:サンソン図法]]
+The MODLAND Integerized Sinusoidal Grid, based on the sinusoidal projection, is a [[geodesic grid]] developed by the NASA's [[Moderate-Resolution Imaging Spectroradiometer]] ([[MODIS]]) science team.<ref>NASA: [https://web.archive.org/web/20020625130235/http://modland.nascom.nasa.gov/developers/is_tiles/is_grid.html "MODLAND Integerized Sinusoidal Grid"]</ref>
+==See also==
+* [[List of map projections]]
+* [[Gerardus Mercator]], [[Nicolas Sanson]], and [[John Flamsteed]] – mathematicians who developed the technique.
+==References==
+{{Reflist}}
+==External links==
+*{{Commons category-inline}}
+*[http://www.progonos.com/furuti/MapProj/Normal/ProjPCyl/projPCyl.html#SansonFlamsteed Pseudocylindrical Projections]
+*[http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net
+{{Map projections}}
+{{Authority control}}
+[[Category:Equal-area projections]]