Editing Map projection (section)

===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|}}