Editing Map projection (section)

==Projections by surface==

The three developable surfaces (plane, cylinder, cone) provide useful models for understanding, describing, and developing map projections. However, these models are limited in two fundamental ways. For one thing, most world projections in use do not fall into any of those categories. For another thing, even most projections that do fall into those categories are not naturally attainable through physical projection. As [[L. P. Lee]] notes,

{{quote|No reference has been made in the above definitions to cylinders, cones or planes. The projections are termed cylindric or conic because they can be regarded as developed on a cylinder or a cone, as the case may be, but it is as well to dispense with picturing cylinders and cones, since they have given rise to much misunderstanding. Particularly is this so with regard to the conic projections with two standard parallels: they may be regarded as developed on cones, but they are cones which bear no simple relationship to the sphere. In reality, cylinders and cones provide us with convenient descriptive terms, but little else.<ref name=Lee>
{{cite journal
  | first   = L.P.
  | last    = Lee
  | author-link = Laurence Patrick Lee
  | year    = 1944
  | title   = The nomenclature and classification of map projections
  | journal = Empire Survey Review
  | volume  = VII
  | issue   = 51
  | pages   = 190–200
  | doi     = 10.1179/sre.1944.7.51.190
  }} p. 193</ref>}}

Lee's objection refers to the way the terms ''cylindrical'', ''conic'', and ''planar'' (azimuthal) have been abstracted in the field of map projections. If maps were projected as in light shining through a globe onto a developable surface, then the spacing of parallels would follow a very limited set of possibilities. Such a cylindrical projection (for example) is one which:
# Is rectangular;
# Has straight vertical meridians, spaced evenly;
# Has straight parallels symmetrically placed about the equator;
# Has parallels constrained to where they fall when light shines through the globe onto the cylinder, with the light source someplace along the line formed by the intersection of the prime meridian with the equator, and the center of the sphere.

(If you rotate the globe before projecting then the parallels and meridians will not necessarily still be straight lines. Rotations are normally ignored for the purpose of classification.)

Where the light source emanates along the line described in this last constraint is what yields the differences between the various "natural" cylindrical projections. But the term ''cylindrical'' as used in the field of map projections relaxes the last constraint entirely. Instead the parallels can be placed according to any algorithm the designer has decided suits the needs of the map. The famous Mercator projection is one in which the placement of parallels does not arise by projection; instead parallels are placed how they need to be in order to satisfy the property that a course of constant bearing is always plotted as a straight line.

===Cylindrical===
{{See also|List of map projections#Cylindrical}}
<!-- use this somewhere else later: [[File:Usgs map space oblique mercator.PNG|frame|right|The [[space-oblique Mercator projection]] was developed by the [[USGS]] for use in [[Landsat]] images.]] -->

====Normal cylindrical====
[[File:Usgs map mercator.svg|frame|right|The Mercator projection shows [[rhumbs]] as straight lines. A rhumb is a course of constant bearing. Bearing is the compass direction of movement.]]
A normal cylindrical projection is any projection in which [[Meridian (geography)|meridians]] are mapped to equally spaced vertical lines and [[circles of latitude]] (parallels) 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. This cylinder is wrapped around the Earth, projected onto, and then unrolled.

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 [[Trigonometric function|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):
* North-south stretching equals east-west stretching ([[Secant (trigonometry)|sec]] ''φ''): The east-west scale matches the north-south scale: conformal cylindrical or [[Mercator projection|Mercator]]; this distorts areas excessively in high latitudes.
* North-south stretching grows with latitude  faster than east-west stretching (sec{{sup|2}} ''φ''): The cylindric perspective (or [[central cylindrical projection|central cylindrical]]) projection; unsuitable because distortion is even worse than in the Mercator projection.
* North-south stretching grows with latitude, but less quickly than the east-west stretching: such as the [[Miller cylindrical projection]] (sec {{sfrac|4|5}}''φ'').
* North-south distances neither stretched nor compressed (1): [[equirectangular projection]] or "plate carrée".
* North-south compression equals the cosine of the latitude (the reciprocal of east-west stretching): [[cylindrical equal-area projection|equal-area cylindrical]].  This projection has many named specializations differing only in the scaling constant, such as the [[Gall–Peters projection|Gall–Peters]] or Gall orthographic (undistorted at the 45° parallels), [[Behrmann projection|Behrmann]] (undistorted at the 30° parallels), and [[Lambert cylindrical equal-area projection|Lambert cylindrical equal-area]] (undistorted at the equator). Since this projection scales north-south distances by the reciprocal of east-west stretching, it preserves area at the expense of shapes.

In the first case (Mercator), the east-west scale always equals the north-south scale. In the second case (central cylindrical), the north-south scale exceeds the east-west scale everywhere away from the equator. Each remaining case has a pair of [[secant line]]s—a pair of identical latitudes of opposite sign (or else the equator) at which the east-west scale matches the north-south-scale.

Normal cylindrical projections map the whole Earth as a finite rectangle, except in the first two cases, where the rectangle stretches infinitely tall while retaining constant width.

====Transverse cylindrical====
A transverse cylindrical projection is a cylindrical projection that in the tangent case uses a great circle along a meridian as contact line for the cylinder.

See: [[Transverse Mercator projection|transverse Mercator]].

====Oblique cylindrical====
[[File:Cylindrical Equal-Area Projection Oblique Case Map of the World.png|400px|thumb|Cylindrical equal-area projection with oblique orientation]]
An oblique cylindrical projection aligns with a great circle, but not the equator and not a meridian.

===Pseudocylindrical===
{{See also|List of map projections#pseudocylindrical}}
[[File:Usgs map sinousidal equal area.PNG|thumb|upright=2|A sinusoidal projection shows relative sizes accurately, but grossly distorts shapes. Distortion can be reduced by "[[Interrupted projection|interrupting]]" the map.]]

Pseudocylindrical projections represent the ''central'' [[meridian (geography)|meridian]] as a straight line segment. Other meridians are longer than the central meridian and bow outward, away from the central meridian. Pseudocylindrical projections map [[circle of latitude|parallels]] as straight lines. Along parallels, each point from the surface is mapped at a distance from the central meridian that is proportional to its difference in longitude from the central meridian. Therefore, meridians are equally spaced along a given parallel. On a pseudocylindrical map, any point further from the equator than some other point has a higher latitude than the other point, preserving north-south relationships. This trait is useful when illustrating phenomena that depend on latitude, such as climate. Examples of pseudocylindrical projections include:
* [[Sinusoidal projection|Sinusoidal]], which was the first pseudocylindrical projection developed. On the map, as in reality, the length of each parallel is proportional to the cosine of the latitude.<ref>{{MathWorld | urlname= SinusoidalProjection | title= Sinusoidal Projection}}</ref> The area of any region is true.
* [[Collignon projection]], which in its most common forms represents each meridian as two straight line segments, one from each pole to the equator.
{|
|
* [[Tobler hyperelliptical projection|Tobler hyperelliptical]]
: [[File:Tobler hyperelliptical projection SW.jpg|200px]]
||
* [[Mollweide projection|Mollweide]]
: [[File:Mollweide projection SW.jpg|200px]]
||
* [[Goode homolosine projection|Goode homolosine]]
: [[File:Goode homolosine projection SW.jpg|200px]]
|-
|
* [[Eckert IV projection|Eckert IV]]
: [[File:Ecker IV projection SW.jpg|200px]]
||
* [[Eckert VI projection|Eckert VI]]
: [[File:Ecker VI projection SW.jpg|200px]]
||
* [[Kavrayskiy VII projection|Kavrayskiy VII]]
: [[File:Kavraiskiy VII projection SW.jpg|200px]]
|}

===Hybrid===
The [[HEALPix]] projection combines an equal-area cylindrical projection in equatorial regions with the [[Collignon projection]] in polar areas.

===Conic===
[[File:Usgs map albers equal area conic.PNG|thumb|upright=2|Albers conic]]

The term "conic projection" is used to refer to any projection in which [[Meridian (geography)|meridians]] are mapped to equally spaced lines radiating out from the apex and [[circles of latitude]] (parallels) are mapped to circular arcs centered on the apex.<ref>{{cite web |last1=Furuti |first1=Carlos A. |title=Conic Projections |url=https://www.progonos.com/furuti/MapProj/Normal/ProjCon/projCon.html |website=Prógonos |archive-url=https://web.archive.org/web/20161212000553/progonos.com/furuti/MapProj/Normal/ProjCon/projCon.html |archive-date=12 December 2016 |date=11 April 2016 |url-status=usurped}}</ref>
<!-- (or, [[mutatis mutandis]], more generally, radial lines from a fixed point are mapped to equally spaced radial lines and concentric circles around it are mapped to circular arcs). -->

When making a conic map, the map maker arbitrarily picks two standard parallels.  Those standard parallels may be visualized as [[secant line]]s where the cone intersects the globe—or, if the map maker chooses the same parallel twice, as the tangent line where the cone is tangent to the globe.  The resulting conic map has low distortion in scale, shape, and area near those standard parallels.  Distances along the parallels to the north of both standard parallels or to the south of both standard parallels are stretched; distances along parallels between the standard parallels are compressed.  When a single standard parallel is used, distances along all other parallels are stretched.

Conic projections that are commonly used are:
* [[Equidistant conic projection|Equidistant conic]], which keeps parallels evenly spaced along the meridians to preserve a constant distance scale along each meridian, typically the same or similar scale as along the standard parallels.
* [[Albers conic projection|Albers conic]], which adjusts the north-south distance between non-standard parallels to compensate for the east-west stretching or compression, giving an equal-area map.
* [[Lambert conformal conic projection|Lambert conformal conic]], which adjusts the north-south distance between non-standard parallels to equal the east-west stretching, giving a conformal map.

===Pseudoconic===
* [[Bonne projection|Bonne]], an equal-area projection on which most meridians and parallels appear as curved lines. It has a configurable standard parallel along which there is no distortion.
* [[Werner projection|Werner cordiform]], upon which distances are correct from one pole, as well as along all parallels.
* [[Polyconic projection|American polyconic]] and other projections in the [[polyconic projection class]].

===Azimuthal (projections onto a plane){{anchor|Azimuthal}}===
{{See also|List of map projections#azimuthal}}
[[File:Usgs map azimuthal equidistant.PNG|thumb|upright=2|An azimuthal equidistant projection shows distances and directions accurately from the center point, but distorts shapes and sizes elsewhere.]]

[[Azimuth]]al projections have the property that directions from a central point are preserved and therefore [[great circle]]s through the central point are represented by straight lines on the map. These projections also have radial symmetry in the scales and hence in the distortions: map distances from the central point are computed by a function ''r''(''d'') of the true distance ''d'', independent of the angle; correspondingly, circles with the central point as center are mapped into circles which have as center the central point on the map.

The mapping of radial lines can be visualized by imagining a [[Plane (geometry)|plane]] tangent to the Earth, with the central point as [[tangent]] point.

The radial scale is ''r′''(''d'') and the transverse scale ''r''(''d'')/(''R''&nbsp;sin&nbsp;{{sfrac|''d''|''R''}}) where ''R'' is the radius of the Earth.

Some azimuthal projections are true [[perspective projection]]s; that is, they can be constructed mechanically, projecting the surface of the Earth by extending lines from a [[point of perspective]] (along an infinite line through the tangent point and the tangent point's [[antipodal point|antipode]]) onto the plane:
* The [[gnomonic projection]] displays [[great circle]]s as straight lines. Can be constructed by using a point of perspective at the center of the Earth. ''r''(''d'') = ''c''&nbsp;tan&nbsp;{{sfrac|''d''|''R''}}; so that even just a hemisphere is already infinite in extent.<ref>{{MathWorld | urlname= GnomonicProjection | title= Gnomonic Projection}}</ref><ref>{{cite web
  | title        = The Gnomonic Projection
  | url          = http://members.shaw.ca/quadibloc/maps/maz0201.htm
  | access-date  = November 18, 2005
  | archive-url  = https://web.archive.org/web/20160430145233/members.shaw.ca/quadibloc/maps/maz0201.htm
  | archive-date = 30 April 2016
  | first        = John
  | last         = Savard
  }}</ref>
* The [[Orthographic projection (cartography)|orthographic projection]] maps each point on the Earth to the closest point on the plane. Can be constructed from a point of perspective an infinite distance from the tangent point; ''r''(''d'') = ''c''&nbsp;sin&nbsp;{{sfrac|''d''|''R''}}.<ref>{{MathWorld | urlname= OrthographicProjection | title= Orthographic Projection}}</ref> Can display up to a hemisphere on a finite circle. Photographs of Earth from far enough away, such as the [[Moon]], approximate this perspective.
* Near-sided perspective projection, which simulates the view from space at a finite distance and therefore shows less than a full hemisphere, such as used in ''[[The Blue Marble 2012]]'').<ref name="PROJ 7.1.1 documentation 2020">{{cite web | title=Near-sided perspective | website=PROJ 7.1.1 documentation | date=2020-09-17 | url=https://proj.org/operations/projections/nsper.html | access-date=2020-10-05}}</ref>
* The [[General Perspective projection]] can be constructed by using a point of perspective outside the Earth. Photographs of Earth (such as those from the [[International Space Station]]) give this perspective. It is a generalization of near-sided perspective projection, allowing tilt.
* The [[Stereographic projection in cartography|stereographic projection]], which is conformal, can be constructed by using the tangent point's [[antipodal point|antipode]] as the point of perspective.  ''r''(''d'') = ''c''&nbsp;tan&nbsp;{{sfrac|''d''|2''R''}}; the scale is ''c''/(2''R''&nbsp;cos{{sup|2}}&nbsp;{{sfrac|''d''|2''R''}}).<ref>{{MathWorld | urlname= StereographicProjection | title= Stereographic Projection}}</ref> Can display nearly the entire sphere's surface on a finite circle. The sphere's full surface requires an infinite map.

Other azimuthal projections are not true [[Perspective (graphical)|perspective]] projections:
* [[Azimuthal equidistant projection|Azimuthal equidistant]]: ''r''(''d'') = ''cd''; it is used by [[amateur radio]] operators to know the direction to point their antennas toward a point and see the distance to it. Distance from the tangent point on the map is proportional to surface distance on the Earth (;<ref>{{MathWorld | urlname= AzimuthalEquidistantProjection | title= Azimuthal Equidistant Projection}}</ref> for the case where the tangent point is the North Pole, see the [[flag of the United Nations]])
* [[Lambert azimuthal equal-area projection|Lambert azimuthal equal-area]]. Distance from the tangent point on the map is proportional to straight-line distance through the Earth: ''r''(''d'') = ''c''&nbsp;sin&nbsp;{{sfrac|''d''|2''R''}}<ref>{{MathWorld | urlname= LambertAzimuthalEqual-AreaProjection | title= Lambert Azimuthal Equal-Area Projection}}</ref>
* [[Logarithmic azimuthal projection|Logarithmic azimuthal]] is constructed so that each point's distance from the center of the map is the logarithm of its distance from the tangent point on the Earth. ''r''(''d'') = ''c''&nbsp;ln&nbsp;{{sfrac|''d''|''d''<sub>0</sub>}}); locations closer than at a distance equal to the constant ''d''<sub>0</sub> are not shown.<ref name="Enlarging">{{cite book |chapter=Enlarging the Heart of a Map |last=Snyder |first=John P. |chapter-url=http://www.gis.psu.edu/projection/chapter6.html |archive-url=https://web.archive.org/web/20100702083430/http://www.gis.psu.edu/projection/chapter6.html |title=Matching the Map Projection to the Need |archive-date=2 July 2010 |url-status=dead |access-date=14 April 2016 |editor1-first=Arthur H. |editor1-last=Robinson |editor2-first=John P. |editor2-last=Snyder |date=1997 |publisher=Cartography and Geographic Information Society}}<br />Reprinted in: {{cite book |last1=Snyder |first1=John P. |title=Choosing a Map Projection |chapter=Matching the Map Projection to the Need |series=Lecture Notes in Geoinformation and Cartography |editor1-last=Lapaine |editor1-first=Miljenko |editor2-last=Usery |editor2-first=E. Lynn |date=2017 |publisher=International Cartographic Association |location=Cham, Switzerland |isbn=978-3-319-51835-0 |pages=78–83 |doi=10.1007/978-3-319-51835-0_3}}</ref>
{{comparison_azimuthal_projections.svg|600px|}}

===Polyhedral===
{{See also|List of map projections#Polyhedral}}
[[File:Dymaxion projection.png|thumb|right|Buckminster Fuller's Dymaxion map]]
[[Polyhedral map projection]]s use a [[polyhedron]] to subdivide the globe into faces, and then projects each face to the globe. The most well-known polyhedral map projection is Buckminster Fuller's [[Dymaxion map]].