# Cartogram

A cartogram is a map in which some thematic mapping variable – such as travel time, population, or GNP – is substituted for land area or distance. The geometry or space of the map is distorted, sometimes extremely, in order to convey the information of this alternate variable. They are primarily used to display emphasis and for analysis as nomographs.[1]

Two common types of cartograms are area and distance cartograms. Cartograms have a fairly long history, with examples from the mid-1800s.[2]

## Area cartograms

An area cartogram is sometimes referred to as a value-by-area map or an isodemographic map, the latter particularly for a population cartogram, which illustrates the relative sizes of the populations of the countries of the world by scaling the area of each country in proportion to its population; the shape and relative location of each country is retained to as large an extent as possible, but inevitably a large amount of distortion results. Other synonyms in use are anamorphic map, density-equalizing map and Gastner map.[4][5][6]

Area cartograms may be contiguous or noncontiguous. The area cartograms shown on this page are all contiguous, while a good example of a noncontiguous cartogram was published in The New York Times.[7][8] This method of cartogram creation is sometimes referred to as the projector method or scaled-down regions.

Cartograms may be classified also by the properties of shape and topology preservation. Classical area cartograms (shown on this page) are typically distorting the shape of spatial units to some degree, but they are strict at preserving correct neighborhood relationships between them. Scaled-down cartograms (from the NY Times example) are strictly shape-preserving. Another branch of cartograms introduced by Dorling, replaces actual shapes with circles scaled according to the mapped feature. Circles are distributed to resemble the original topology. Demers cartogram is a variation of Dorling cartogram, but it uses rectangles instead of circles, and attempts to retain visual cues at the expense of minimum distance. Schematic maps based on quad trees can be seen as non shape-preserving cartograms with some degree of neighborhood preservation.

A collection of contiguous area cartograms is available at Worldmapper,[9] which was started by a collaborative team of researchers at the Universities of Sheffield and Michigan.

## Production

One of the first cartographers to generate cartograms with the aid of computer visualization was Waldo Tobler of UC Santa Barbara in the 1960s. Prior to Tobler's work, cartograms were created by hand (as they occasionally still are). The National Center for Geographic Information and Analysis located on the UCSB campus maintains an online Cartogram Central with resources regarding cartograms.

A number of software packages generate cartograms. Most of the available cartogram generation tools work in conjunction with other GIS software tools as add-ons or independently produce cartographic outputs from GIS data formatted to work with commonly used GIS products. Examples of cartogram software include ScapeToad,[10][11] Cart,[12] and the Cartogram Processing Tool (an ArcScript for ESRI's ArcGIS), which all use the Gastner-Newman algorithm.[13][14] An alternative algorithm, Carto3F,[15] is also implemented as an independent program for non-commercial use on Windows platforms.[16] This program also provides an optimization to the original Dougenik rubber-sheet algorithm.[17] [18] The CRAN package recmap provides an implementation of a rectangular cartogram algorithm.[19]

Cartograms can also be constructed manually, either by hand or in a computer-assisted environment. Block cartograms are constructed by arranging geometrically regular equal-sized blocks, with the number of blocks allocated to each district proportional to the population variable. Several examples of block cartograms were published during the 2016 U.S. presidential election season by The Washington Post [20], the FiveThirtyEight blog [21], and the Wall Street Journal [22], among others.

### Algorithms

YearAuthorAlgorithmTypeShape preservationTopology preservation
1973ToblerRubber map methodarea contiguouswith distortionYes, but not guaranteed
1976OlsonProjector methodarea noncontiguousyesNo
1984Selvin et al.DEMP (Radial Expansion) methodarea contiguouswith distortionUnknown
1985Dougenik et al.Rubber Sheet Distortion method [18]area contiguouswith distortionYes, but not guaranteed
1986ToblerPseudo-Cartogram methodarea contiguouswith distortionYes
1987SnyderMagnifying glass azimuthal map projectionsdistance radialUnknownUnknown
1989Cauvin et al.Piezopleth mapsarea contiguouswith distortionUnknown
1990TorgusonInteractive polygon zipping methodarea contiguouswith distortionUnknown
1990DorlingCellular Automata Machine methodarea contiguouswith distortionYes
1993Gusein-Zade, TikunovLine Integral methodarea contiguouswith distortionYes
1996DorlingCircular cartogramarea noncontiguousno (circles)No
1997Edelsbrunner, WaupotitschCombinatorial-based approacharea contiguouswith distortionUnknown
1998Kocmoud, HouseConstraint-based approacharea contiguouswith distortionYes
2001Keim, North, PanseCartoDraw[23]area contiguouswith distortionYes, algorithmically guaranteed
2004Gastner, NewmanDiffusion-based method[4]area contiguouswith distortionYes, algorithmically guaranteed
2004SlugaLastna tehnika za izdelavo anamorfozarea contiguouswith distortionUnknown
2004van Kreveld, SpeckmannRectangular Cartogram[24]area contiguousno (rectangles)No
2004Heilmann, Keim et al.RecMap[19]area noncontiguousno (rectangles)No
2005Keim, North, PanseMedial-axis-based cartograms[25]area contiguouswith distortionYes, algorithmically guaranteed
2009Heriques, Bação, LoboCarto-SOMarea contiguouswith distortionYes
2013Shipeng SunOpti-DCN[17] and Carto3F[15]area contiguouswith distortionYes, algorithmically guaranteed
2014B. S. Daya SagarMathematical Morphology-Based Cartogramsarea contiguouswith local distortion, but no global distortionNo
2018Gastner, Seguy, MoreFast Flow-Based Method[26]area contiguouswith distortionYes, algorithmically guaranteed

## References

1. Tobler, Waldo (March 2004). "Thirty-Five Years of Computer Cartograms". Annals of the Association of American Geographers. 94 (1): 58–73. doi:10.1111/j.1467-8306.2004.09401004.x. JSTOR 3694068.
2. Johnson (2008-12-08). "Early cartograms". indiemaps.com/blog. Retrieved 2012-08-17.
3. Paull, John & Hennig, Benjamin (2016) Atlas of Organics: Four Maps of the World of Organic Agriculture Journal of Organics. 3(1): 25-32.
4. Michael T. Gastner; Mark E. J. Newman (2004). "Diffusion-based method for producing density equalizing maps". Proceedings of the National Academy of Sciences. 101 (20): 7499–7504. arXiv:. Bibcode:2004PNAS..101.7499G. doi:10.1073/pnas.0400280101. PMC . PMID 15136719.
5. Gallery of Data Visualization - Bright Ideas
6. UNEP GRID Ardenal: Anamorphic Maps
7. Johnson (2011-02-22). "Noncontiguous cartograms in OpenLayers and Polymaps". indiemaps.com/blog. Retrieved 2012-08-17.
8. Cowan, Sarah; Doyle, Stephen; Heffron, Drew (2008-11-02), "Op-Chart: How Much Is Your Vote Worth?", New York Times, retrieved 2012-08-17
9. Worldmapper: Rediscover the world as you've never seen it before
11. The Art of Software: Cartogram Crash Course
12. Cart: Computer software for making cartograms
13. Cartogram Geoprocessing Tool
14. Hennig, Benjamin D.; Pritchard, John; Ramsden, Mark; Dorling, Danny, "Remapping the World's Population: Visualizing data using cartograms", ArcUser (Winter 2010): 66–69
15. Sun, Shipeng (2013), "A Fast, Free-Form Rubber-Sheet Algorithm for Contiguous Area Cartograms", International Journal of Geographic Information Science, 27 (3): 567–93, doi:10.1080/13658816.2012.709247
16. Personal Website of Shipeng Sun
17. Sun, Shipeng (2013), "An Optimized Rubber-Sheet Algorithm for Continuous Area Cartograms", The Professional Geographer, 16 (1): 16–30, doi:10.1080/00330124.2011.639613
18. Dougenik, James A.; Chrisman, Nicholas R.; Niemeyer, Duane R. (1985), "An Algorithm to Construct Continuous Area Cartograms", The Professional Geographer, 37 (1): 75–81, doi:10.1080/00330124.2011.639613
19. Heilmann, Roland; Keim, Daniel; Panse, Christian; Sips, Mike (2004). "RecMap : Rectangular Map Approximations". Proceedings of the 10th IEEE Symposium on Information Visualization: 33–40. doi:10.1109/INFVIS.2004.57.
20. "Poll: Redrawing the Electoral Map". Washington Post. Retrieved 4 February 2018.
21. "2016 Election Forecast". FiveThirtyEight blog. Retrieved 4 February 2018.
22. "Draw the 2016 Electoral College Map". Wall Street Journal. Retrieved 4 February 2018.
23. Keim, Daniel; North, Stephen; Panse, Christian (2004). "CartoDraw: a fast algorithm for generating contiguous cartograms". IEEE Trans Vis Comput Graph. 10 (1): 95–110. doi:10.1109/TVCG.2004.1260761.
24. van Kreveld, Marc; Speckmann, Bettina (2004). "On Rectangular Cartograms". In: Albers S., Radzik T. (eds) Algorithms – ESA 2004. ESA 2004. Lecture Notes in Computer Science. Springer, Berlin, Heidelberg. 3221. doi:10.1007/978-3-540-30140-0_64.
25. Keim, Daniel; Panse, Christian; North, Stephen (2005). "Medial-axis-based cartograms". IEEE Computer Graphics and Applications. 25 (3): 60–68. doi:10.1109/MCG.2005.64.
26. Michael T. Gastner; Vivien Seguy; Pratyush More (2018). "Fast flow-based algorithm for creating density-equalizing map projections". Proceedings of the National Academy of Sciences. 115 (10): E2156–E2164. arXiv:. doi:10.1073/pnas.1712674115.