Editing Map projection (section)

===Distortion===
{{main|Tissot's indicatrix}}
[[Carl Friedrich Gauss]]'s ''[[Theorema Egregium]]'' <!-- M. Lapaine notes that the proof antedates Gauss; this material needs to be updated --> proved that a sphere's surface cannot be represented on a plane without distortion. The same applies to other reference surfaces used as models for the Earth, such as oblate [[spheroid]]s, [[ellipsoids]], and [[geoid]]s. Since any map projection is a representation of one of those surfaces on a plane, all map projections distort.<ref name="EGmap"/>

[[File:Tissot mercator.png|thumb|Tissot's indicatrices on the [[Mercator projection]]]]
The classical way of showing the distortion inherent in a projection is to use [[Tissot's indicatrix]]. For a given point, using the scale factor ''h'' along the meridian, the scale factor ''k'' along the parallel, and the angle ''θ''′ between them, [[Nicolas Auguste Tissot|Nicolas Tissot]] described how to construct an ellipse that illustrates the amount and orientation of the components of distortion.<ref name="SnyderFlattening"/>{{rp|147–149}}<ref name="WorkingManual" />{{rp|p=24}} By spacing the ellipses regularly along the meridians and parallels, the network of indicatrices shows how distortion varies across the map.

====Other distortion metrics====
Many other ways have been described of showing the distortion in projections.<ref>{{cite journal |last1=Mulcahy |first1=Karen A. |last2=Clarke |first2=Keith C. |title=Symbolization of Map Projection Distortion: A Review |journal=Cartography and Geographic Information Science |date=January 2001 |volume=28 |issue=3 |pages=167–182 |doi=10.1559/152304001782153044 |publisher=Cartography and Geographic Information Society |s2cid=26611469 |url=http://www.geog.ucsb.edu/~kclarke/Geography232/MulcahyClarke2001.pdf}}</ref><ref>{{cite book |last1=Canters |first1=Frank |title=Small-scale map projection design |date=2002 |publisher=Taylor & Francis |location=London |isbn=9780203472095 |page=[https://books.google.com/books?id=8cR7yG5ohHoC&pg=PA291 291] |series=Research monographs in geographic information systems}}</ref> Like Tissot's indicatrix, the '''Goldberg-Gott indicatrix''' is based on infinitesimals, and depicts ''flexion'' and ''skewness'' (bending and lopsidedness) distortions.<ref name="Goldberg-Gott">{{cite journal
  | url        = http://www.physics.drexel.edu/~goldberg/projections/goldberg_gott.pdf
  | title      = Flexion and Skewness in Map Projections of the Earth
  | year       = 2007
  | first1     = David M.
  | last1      = Goldberg
  | first2     = J. Richard
  | last2      = Gott III
  | journal    = Cartographica
  | volume     = 42
  | issue      = 4
  | pages      = 297–318
  | access-date = 2011-11-14
  | doi=10.3138/carto.42.4.297
  | arxiv= astro-ph/0608501
  | s2cid = 11359702
 }}</ref>

Rather than the original (enlarged) infinitesimal circle as in Tissot's indicatrix, some visual methods project finite shapes that span a part of the map.
For example, a [[small circle]] of fixed radius (e.g., 15 degrees [[angular radius]]).<ref name="tiss">{{cite conference |url=http://www.geolab.polimi.it/wp-content/uploads/GW12_FOSS4G-eu15.pdf |title=Real-time projection visualisation with Indicatrix Mapper QGIS Plugin |author1-first=Ervin |author1-last=Wirth |author2-first=Péter |author2-last=Kun |date=July 2015 |conference=FOSS4G Europe 2015 |editor1-first=Maria Antonia |editor1-last=Brovelli |editor2-first=Marco |editor2-last=Minghini |editor3-first=Marco |editor3-last=Negreti |series=Geomatics Workbooks |volume=12 |book-title=Open Innovation for Europe |publisher=Polytechnic University of Milan |archive-url=https://web.archive.org/web/20220723153559/http://www.geolab.polimi.it/wp-content/uploads/GW12_FOSS4G-eu15.pdf |archive-date=23 July 2022 |url-status=live |location=Como, Italy |pages=697–700 |issn=1591-092X}}</ref> Sometimes [[spherical triangle]]s are used.{{citation needed|date=August 2016}}
In the first half of the 20th century, projecting a human head onto different projections was common to show how distortion varies across one projection as compared to another.<ref name="rutgers">{{cite news
  | title      = This is your brain on maps
  | date       = 18 September 2013
  | url        = http://bigthink.com/strange-maps/624-this-is-your-brain-on-maps
  | first      = Frank
  | last       = Jacobs
  | department = Strange Maps
  | work       = Big Think
 }}</ref>
In dynamic media, shapes of familiar coastlines and boundaries can be dragged across an interactive map to show how the projection distorts sizes and shapes according to position on the map.<ref>{{cite web |url=https://bramus.github.io/mercator-puzzle-redux/ |title=Mercator Puzzle Redux |access-date=24 January 2018 |first=Bramus |last=Van Damme}}</ref>

Another way to visualize local distortion is through grayscale or color gradations whose shade represents the magnitude of the angular deformation or areal inflation. Sometimes both are shown simultaneously by blending two colors to create a [[bivariate map]].<ref name="cornucopia">{{cite web
  | title      = A cornucopia of map projections
  | url        = http://www.mapthematics.com/Downloads/Images/Cornucopia33.jpg
  | website    = Mapthematics
 }}</ref>

To measure distortion globally across areas instead of at just a single point necessarily involves choosing priorities to reach a compromise. Some schemes use distance distortion as a proxy for the combination of angular deformation and areal inflation; such methods arbitrarily choose what paths to measure and how to weight them in order to yield a single result. Many have been described.<ref name="Goldberg-Gott" /><ref name="AB Peters">
{{cite journal
| last1 = Peters
| first1 = A. B.
| title = Uber Weltkartenverzerrunngen und Weltkartenmittelpunkte
| journal = {{ill|Kartographische Nachrichten|de}}
| pages = 106–113
| year = 1978
}}</ref><ref name = "GMC">
{{cite arXiv
| last1 = Gott, III
| first1 = J. Richard
| last2 = Mugnolo
| first2 = Charles
| last3 = Colley
| first3 = Wesley N.
| title = Map projections for minimizing distance errors
| eprint = astro-ph/0608500v1
| year = 2006
}}</ref><ref name = "Laskowski">
{{cite journal
| last1 = Laskowski
| first1 = P.
| title = Distortion-spectrum fundamentals: A new tool for analyzing and visualizing map distortions
| journal = Cartographica
| volume = 34
| issue = 3
| year = 1997
| doi = 10.3138/Y51X-1590-PV21-136G
| doi-access = free
}}</ref><ref name = "Airy">
{{cite journal
| last1 = Airy
| first1 = G.B.
| title = Explanation of a projection by balance of errors for maps applying to a very large extent of the Earth's surface; and comparison of this projection with other projections
| journal = London, Edinburgh, and Dublin Philosophical Magazine
| year = 1861
| series = 4
| volume = 22
| issue = 149
| pages = 409–421
| doi = 10.1080/14786446108643179
}}</ref>