imported>RFracture: /* Anamorphic Projection */ Added links

2025-11-17T01:45:44Z

Anamorphic Projection: Added links

← Previous revision		Revision as of 01:45, 17 November 2025
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	{{Broader\|Thematic map}}		{{Broader\|Thematic map}}
	{{Distinguish\|Cartography}}		{{Distinguish\|Cartography}}
	[[File:Global population cartogram.png\|thumb\|upright=1.35\|Mosaic cartogram showing the distribution of the global population.		[[File:Global population cartogram.png\|thumb\|upright=1.5\|Mosaic cartogram showing the distribution of the global population.
	Each of the 15,266 pixels represents the home country of 500,000 people – cartogram by [[Max Roser]] for [[Our World in Data]]]]		Each of the 15,266 pixels represents the home country of 500,000 people – cartogram by [[Max Roser]] for [[Our World in Data]]]]

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	== Area cartograms ==		== Area cartograms ==
	[[File:Germany-population-cartogram.png\|thumb\|Cartogram of [[Germany]], with the states and districts resized according to population]]		[[File:Germany-population-cartogram.png\|thumb\|Cartogram of [[Germany]], with the states and districts resized according to population]]
	The area cartogram is by far the most common form; it scales a set of region features, usually administrative districts such as counties or countries, such that the [[area]] of each district is [[Proportionality (mathematics)\|directly proportional]] to a given variable. Usually this variable represents the total count or amount of something, such as total [[~~Population~~]], [[~~Gross~~ domestic product]], or the number of retail outlets of a given brand or type. Other strictly positive [[Level of measurement\|ratio]] variables can also be used, such as [[List of countries by GDP (nominal) per capita\|GDP per capita]] or [[~~Birth~~ rate]], but these can sometimes produce misleading results because of the natural tendency to interpret size as total amount.<ref name="bertin" /> Of these, total population is probably the most common variable, sometimes called an ''isodemographic map''.		The area cartogram is by far the most common form; it scales a set of region features, usually administrative districts such as counties or countries, such that the [[area]] of each district is [[Proportionality (mathematics)\|directly proportional]] to a given variable. Usually this variable represents the total count or amount of something, such as total [[population]], [[gross domestic product]], or the number of retail outlets of a given brand or type. Other strictly positive [[Level of measurement\|ratio]] variables can also be used, such as [[List of countries by GDP (nominal) per capita\|GDP per capita]] or [[birth rate]], but these can sometimes produce misleading results because of the natural tendency to interpret size as total amount.<ref name="bertin" /> Of these, total population is probably the most common variable, sometimes called an ''isodemographic map''.

	The various strategies and algorithms have been classified a number of ways, generally according to their strategies with respect to preserving shape and topology. Those that preserve shape are sometimes called ''equiform'', although ''isomorphic'' (same-shape) or ''homomorphic'' (similar-shape) may be better terms. Three broad categories are widely accepted: contiguous (preserve topology, distort shape), non-contiguous (preserve shape, distort topology), and diagrammatic (distort both). Recently, more thorough taxonomies by Nusrat and Kobourov, Markowska, and others have built on this basic framework in an attempt to capture the variety in approaches that have been proposed and in the appearances of the results.<ref name="nusrat2016">{{cite journal \|last1=Nusrat \|first1=Sabrina \|last2=Kobourov \|first2=Stephen \|title=The State of the Art in Cartograms \|journal=Computer Graphics Forum \|date=2016 \|volume=35 \|issue=3 \|pages=619–642 \|doi=10.1111/cgf.12932\|arxiv=1605.08485 \|hdl=10150/621282 \|s2cid=12180113 \|hdl-access=free }} Special issue: 18th Eurographics Conference on Visualization (EuroVis), State of the Art Report</ref><ref name="markowska2019">{{cite journal \|last1=Markowska \|first1=Anna \|title=Cartograms - classification and terminology \|journal=Polish Cartographical Review \|date=2019 \|volume=51 \|issue=2 \|pages=51–65 \|doi=10.2478/pcr-2019-0005\|bibcode=2019PCRv...51...51M \|doi-access=free }}</ref> The various taxonomies tend to agree on the following general types of area cartograms.		The various strategies and algorithms have been classified a number of ways, generally according to their strategies with respect to preserving shape and topology. Those that preserve shape are sometimes called ''equiform'', although ''isomorphic'' (same-shape) or ''homomorphic'' (similar-shape) may be better terms. Three broad categories are widely accepted: contiguous (preserve topology, distort shape), non-contiguous (preserve shape, distort topology), and diagrammatic (distort both). Recently, more thorough taxonomies by Nusrat and Kobourov, Markowska, and others have built on this basic framework in an attempt to capture the variety in approaches that have been proposed and in the appearances of the results.<ref name="nusrat2016">{{cite journal \|last1=Nusrat \|first1=Sabrina \|last2=Kobourov \|first2=Stephen \|title=The State of the Art in Cartograms \|journal=Computer Graphics Forum \|date=2016 \|volume=35 \|issue=3 \|pages=619–642 \|doi=10.1111/cgf.12932\|arxiv=1605.08485 \|hdl=10150/621282 \|s2cid=12180113 \|hdl-access=free }} Special issue: 18th Eurographics Conference on Visualization (EuroVis), State of the Art Report</ref><ref name="markowska2019">{{cite journal \|last1=Markowska \|first1=Anna \|title=Cartograms - classification and terminology \|journal=Polish Cartographical Review \|date=2019 \|volume=51 \|issue=2 \|pages=51–65 \|doi=10.2478/pcr-2019-0005\|bibcode=2019PCRv...51...51M \|doi-access=free }}</ref> The various taxonomies tend to agree on the following general types of area cartograms.
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	===Anamorphic Projection===		===Anamorphic Projection===
	{{see also \| Anamorphosis}}		{{see also \| Anamorphosis}}
	This is a type of contiguous cartogram that uses a single parametric mathematical formula (such as a [[Polynomial \| polynomial curved surface]]) to distort space itself to equalize the spatial distribution of the chosen variable, rather than distorting the individual features. Because of this distinction, some have preferred to call the result a ''pseudo-cartogram''.<ref name="bortins-demers">{{cite web \|last1=Bortins \|first1=Ian \|last2=Demers \|first2=Steve \|title=Cartogram Types \|url=http://www.ncgia.ucsb.edu/projects/Cartogram_Central/types.html \|website=Cartogram Central \|publisher=National Center for Geographic Information Analysis, UC Santa Barbara \|access-date=15 November 2020 \|archive-date=29 January 2021 \|archive-url=https://web.archive.org/web/20210129215218/http://www.ncgia.ucsb.edu/projects/Cartogram_Central/types.html \|url-status=dead }}</ref> [[Waldo R. Tobler\|Tobler's]] first computer cartogram algorithm was based on this strategy,<ref name="tobler1963" /><ref name="tobler1973">{{cite journal \|last1=Tobler \|first1=Waldo R. \|title=A Continuous Transformation Useful for Districting \|journal=Annals of the New York Academy of Sciences \|date=1973 \|volume=219 \|issue=1 \|pages=215–220 \|doi=10.1111/j.1749-6632.1973.tb41401.x\|pmid=4518429 \|bibcode=1973NYASA.219..215T \|hdl=2027.42/71945 \|s2cid=35585206 \|hdl-access=free }}</ref> for which he developed the general mathematical construct on which his and subsequent algorithms are based.<ref name="tobler1963" /> This approach first models the distribution of the chosen variable as a continuous density function (usually using a [[Curve fitting\|least squares fitting]]), then uses the inverse of that function to adjust the space such that the density is equalized. The Gastner-Newman algorithm, one of the most popular tools used today, is a more advanced version of this approach.<ref name="GSM-Fast-Flow-Based">{{cite journal \|author = Michael T. Gastner \|author2=Vivien Seguy \|author3=Pratyush More \|year = 2018 \|title = Fast flow-based algorithm for creating density-equalizing map projections \|journal = Proceedings of the National Academy of Sciences \|volume = 115 \|pages = E2156–E2164 \|doi = 10.1073/pnas.1712674115 \|pmid=29463721 \|pmc=5877977 \|issue = 10 \|arxiv = 1802.07625 \|bibcode=2018PNAS..115E2156G \|doi-access=free }}</ref><ref name="gastner-newman">{{cite journal \|last1=Gastner \|first1=Michael T. \|last2=Newman \|first2=M.E.J. \|title=Diffusion-based Method for Producing Density-Equalizing Maps \|journal=Proceedings of the National Academy of Sciences of the United States of America \|date=May 18, 2004 \|volume=101 \|issue=20 \|pages=7499–7504 \|doi=10.1073/pnas.0400280101 \|jstor=3372222 \|pmid=15136719 \|pmc=419634 \|url=\|arxiv=physics/0401102 \|s2cid=2487634 \|doi-access=free }}</ref> Because they do not directly scale the districts, there is no guarantee that the area of each district is exactly equal to its value.		This is a type of contiguous cartogram that uses a single parametric mathematical formula (such as a [[Polynomial \| polynomial curved surface]]) to distort space itself to equalize the spatial distribution of the chosen variable, rather than distorting the individual features. Because of this distinction, some have preferred to call the result a ''pseudo-cartogram''.<ref name="bortins-demers">{{cite web \|last1=Bortins \|first1=Ian \|last2=Demers \|first2=Steve \|title=Cartogram Types \|url=http://www.ncgia.ucsb.edu/projects/Cartogram_Central/types.html \|website=Cartogram Central \|publisher=National Center for Geographic Information Analysis, UC Santa Barbara \|access-date=15 November 2020 \|archive-date=29 January 2021 \|archive-url=https://web.archive.org/web/20210129215218/http://www.ncgia.ucsb.edu/projects/Cartogram_Central/types.html \|url-status=dead }}</ref> [[Waldo R. Tobler\|Tobler's]] first computer cartogram algorithm was based on this strategy,<ref name="tobler1963" /><ref name="tobler1973">{{cite journal \|last1=Tobler \|first1=Waldo R. \|title=A Continuous Transformation Useful for Districting \|journal=Annals of the New York Academy of Sciences \|date=1973 \|volume=219 \|issue=1 \|pages=215–220 \|doi=10.1111/j.1749-6632.1973.tb41401.x\|pmid=4518429 \|bibcode=1973NYASA.219..215T \|hdl=2027.42/71945 \|s2cid=35585206 \|hdl-access=free }}</ref> for which he developed the general mathematical construct on which his and subsequent algorithms are based.<ref name="tobler1963" /> This approach first models the distribution of the chosen variable as a continuous density function (usually using a [[Curve fitting\|least squares fitting]]), then uses the inverse of that function to adjust the space such that the density is equalized. The Gastner-[[Mark Newman\|Newman]] algorithm, one of the most popular tools used today, is a more advanced version of this approach.<ref name="GSM-Fast-Flow-Based">{{cite journal \|author = Michael T. Gastner \|author2=Vivien Seguy \|author3=Pratyush More \|year = 2018 \|title = Fast flow-based algorithm for creating density-equalizing map projections \|journal = Proceedings of the National Academy of Sciences \|volume = 115 \|pages = E2156–E2164 \|doi = 10.1073/pnas.1712674115 \|pmid=29463721 \|pmc=5877977 \|issue = 10 \|arxiv = 1802.07625 \|bibcode=2018PNAS..115E2156G \|doi-access=free }}</ref><ref name="gastner-newman">{{cite journal \|last1=Gastner \|first1=Michael T. \|last2=Newman \|first2=M.E.J. \|title=Diffusion-based Method for Producing Density-Equalizing Maps \|journal=Proceedings of the National Academy of Sciences of the United States of America \|date=May 18, 2004 \|volume=101 \|issue=20 \|pages=7499–7504 \|doi=10.1073/pnas.0400280101 \|jstor=3372222 \|pmid=15136719 \|pmc=419634 \|url=\|arxiv=physics/0401102 \|s2cid=2487634 \|doi-access=free }}</ref> Because they do not directly scale the districts, there is no guarantee that the area of each district is exactly equal to its value.

	===Shape-warping contiguous cartograms===		===Shape-warping contiguous cartograms===
	[[File:PaullHennig2016WorldMap.OAha.CC-BY-4.0.jpg\|thumb\|upright=1.35\|Contiguous cartogram (Gastner-Newman) of the world with each country rescaled in proportion to the hectares of certified [[organic farming]]<ref name=Atlas>Paull, John & Hennig, Benjamin (2016) [https://www.academia.edu/25648267/Atlas_of_Organics_Four_maps_of_the_world_of_organic_agriculture Atlas of Organics: Four Maps of the World of Organic Agriculture] Journal of Organics. 3(1): 25–32.</ref>]]		[[File:PaullHennig2016WorldMap.OAha.CC-BY-4.0.jpg\|thumb\|upright=1.35\|Contiguous cartogram (Gastner-Newman) of the world with each country rescaled in proportion to the hectares of certified [[organic farming]]<ref name=Atlas>Paull, John & Hennig, Benjamin (2016) [https://www.academia.edu/25648267/Atlas_of_Organics_Four_maps_of_the_world_of_organic_agriculture Atlas of Organics: Four Maps of the World of Organic Agriculture] Journal of Organics. 3(1): 25–32.</ref>]]
	Also called ''irregular cartograms'' or ''deformation cartograms'',<ref name="markowska2019" /> This is a family of very different algorithms that scale and deform the shape of each district while maintaining adjacent edges. This approach has its roots in the early 20th Century cartograms of Haack and Weichel and others, although these were rarely as mathematically precise as current computerized versions. The variety of approaches that have been proposed include [[Cellular automaton\|cellular automata]], [[Quadtree \| quadtree partitions]], [[cartographic generalization]], [[Medial axis\|medial axes]], spring-like forces, and simulations of inflation and deflation.<ref name="nusrat2016" /> Some attempt to preserve some semblance of the original shape (and may thus be termed ''homomorphic''),<ref name="kocmoud1998">{{cite book \|last1=House \|first1=Donald H. \|last2=Kocmoud \|first2=Christopher J. \|title=Proceedings Visualization '98 (Cat. No.98CB36276) \|chapter=Continuous cartogram construction \|date=October 1998 \|pages=197–204 \|doi=10.1109/VISUAL.1998.745303 \|isbn=0-8186-9176-X \|s2cid=14023382 \|chapter-url=https://www.researchgate.net/publication/3788051}}</ref> but these are often more complex and slower algorithms than those that severely distort shape.		Also called ''irregular cartograms'' or ''deformation cartograms'',<ref name="markowska2019" /> This is a family of very different algorithms that scale and deform the shape of each district while maintaining adjacent edges. This approach has its roots in the early 20th Century cartograms of Haack and Weichel and others, although these were rarely as mathematically precise as current computerized versions. The variety of approaches that have been proposed include [[Cellular automaton\|cellular automata]], [[Quadtree \| quadtree partitions]], [[cartographic generalization]], [[Medial axis\|medial axes]], spring-like forces, and simulations of inflation and deflation.<ref name="nusrat2016" /> Some attempt to preserve some semblance of the original shape (and may thus be termed ''homomorphic''),<ref name="kocmoud1998">{{cite book \|last1=House \|first1=Donald H. \|last2=Kocmoud \|first2=Christopher J. \|title=Proceedings Visualization '98 (Cat. No. 98CB36276) \|chapter=Continuous cartogram construction \|date=October 1998 \|pages=197–204 \|doi=10.1109/VISUAL.1998.745303 \|isbn=0-8186-9176-X \|s2cid=14023382 \|chapter-url=https://www.researchgate.net/publication/3788051}}</ref> but these are often more complex and slower algorithms than those that severely distort shape.

	===Non-contiguous isomorphic cartograms===		===Non-contiguous isomorphic cartograms===
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	===Mosaic cartograms===		===Mosaic cartograms===
	[[File:Germany_population_states_hexagonal.svg\|thumb\|Population cartogram of the states of Germany. Each hexagon represents 250~~ ~~000 people. The cartogram is topologically correct in that any states that touch on the cartogram also touch in reality.]]		[[File:Germany_population_states_hexagonal.svg\|thumb\|Population cartogram of the states of Germany. Each hexagon represents 250,000 people. The cartogram is topologically correct in that any states that touch on the cartogram also touch in reality.]]
	In this approach (also called ''block'' or ''regular cartograms''), each shape is not just scaled or warped, but is reconstructed from a discrete [[tessellation]] of space, usually into squares or hexagons. Each cell of the tessellation represents a constant value of the variable (e.g., 5000 residents), so the number of whole cells to be occupied can be calculated (although rounding error often means that the final area is not exactly proportional to the variable). Then a shape is assembled from those cells, usually with some attempt to retain the original shape, including salient features such as panhandles that aid recognition (for example, [[Long Island]] and [[Cape Cod]] are often exaggerated). Thus, these cartograms are usually homomorphic and at least partially contiguous.		In this approach (also called ''block'' or ''regular cartograms''), each shape is not just scaled or warped, but is reconstructed from a discrete [[tessellation]] of space, usually into squares or hexagons. Each cell of the tessellation represents a constant value of the variable (e.g., 5000 residents), so the number of whole cells to be occupied can be calculated (although rounding error often means that the final area is not exactly proportional to the variable). Then a shape is assembled from those cells, usually with some attempt to retain the original shape, including salient features such as panhandles that aid recognition (for example, [[Long Island]] and [[Cape Cod]] are often exaggerated). Thus, these cartograms are usually homomorphic and at least partially contiguous.

imported>NyanarWelden: Typo

2025-07-01T02:58:34Z

Typo

← Previous revision		Revision as of 02:58, 1 July 2025
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	Each of the 15,266 pixels represents the home country of 500,000 people – cartogram by [[Max Roser]] for [[Our World in Data]]]]		Each of the 15,266 pixels represents the home country of 500,000 people – cartogram by [[Max Roser]] for [[Our World in Data]]]]

	A '''cartogram''' (also called a '''value-area map''' or an '''anamorphic map''', the latter common among German-speakers) is a [[thematic map]] of a set of features (countries, provinces, etc.), in which their geographic size is altered to be [[Proportionality (mathematics)\|directly proportional]] to a selected variable, such as travel time, [[population]], or [[gross national income]]. Geographic space itself is thus warped, sometimes extremely, in order to visualize the distribution of the variable. It is one of the most abstract types of [[map]]; in fact, some forms may more properly be called [[diagram]]s. They are primarily used to display emphasis and for analysis as [[Nomography\|nomographs]].<ref name="tobler2004">{{Cite journal \| last = Tobler \| first = Waldo \| title = Thirty-Five Years of Computer Cartograms \| journal = Annals of the Association of American Geographers \| volume = 94 \| issue = 1 \| pages = 58–73 \| date = March 2022 \| jstor = 3694068 \| doi=10.1111/j.1467-8306.2004.09401004.x\| citeseerx = 10.1.1.551.7290 \| s2cid = 129840496 }}</ref>		A '''cartogram''' (also called a '''value-area map''' or an '''anamorphic map''', the latter common among German speakers) is a [[thematic map]] of a set of features (countries, provinces, etc.), in which their geographic size is altered to be [[Proportionality (mathematics)\|directly proportional]] to a selected variable, such as travel time, [[population]], or [[gross national income]]. Geographic space itself is thus warped, sometimes extremely, in order to visualize the distribution of the variable. It is one of the most abstract types of [[map]]; in fact, some forms may more properly be called [[diagram]]s. They are primarily used to display emphasis and for analysis as [[Nomography\|nomographs]].<ref name="tobler2004">{{Cite journal \| last = Tobler \| first = Waldo \| title = Thirty-Five Years of Computer Cartograms \| journal = Annals of the Association of American Geographers \| volume = 94 \| issue = 1 \| pages = 58–73 \| date = March 2022 \| jstor = 3694068 \| doi=10.1111/j.1467-8306.2004.09401004.x\| citeseerx = 10.1.1.551.7290 \| s2cid = 129840496 }}</ref>

	Cartograms leverage the fact that size is the most intuitive [[visual variable]] for representing a total amount.<ref name="bertin">Jacque Bertin, ''Sémiologie Graphique. Les diagrammes, les réseaux, les cartes''. With Marc Barbut [et al.]. Paris : Gauthier-Villars. ''Semiology of Graphics'', English Edition, Translation by William J. Berg, University of Wisconsin Press, 1983.)</ref> In this, it is a strategy that is similar to [[proportional symbol map]]s, which scale point features, and many [[flow map]]s, which scale the weight of linear features. However, these two techniques only scale the [[map symbol]], not space itself; a map that stretches the length of linear features is considered a linear cartogram (although additional flow map techniques may be added). Once constructed, cartograms are often used as a base for other thematic mapping techniques to visualize additional variables, such as [[choropleth map]]ping.		Cartograms leverage the fact that size is the most intuitive [[visual variable]] for representing a total amount.<ref name="bertin">Jacque Bertin, ''Sémiologie Graphique. Les diagrammes, les réseaux, les cartes''. With Marc Barbut [et al.]. Paris : Gauthier-Villars. ''Semiology of Graphics'', English Edition, Translation by William J. Berg, University of Wisconsin Press, 1983.)</ref> In this, it is a strategy that is similar to [[proportional symbol map]]s, which scale point features, and many [[flow map]]s, which scale the weight of linear features. However, these two techniques only scale the [[map symbol]], not space itself; a map that stretches the length of linear features is considered a linear cartogram (although additional flow map techniques may be added). Once constructed, cartograms are often used as a base for other thematic mapping techniques to visualize additional variables, such as [[choropleth map]]ping.

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