Gall–Peters projection

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The Gall-Peters projection of the world map Gall-Peters projection SW.jpg
The Gall–Peters projection of the world map

The Gall–Peters projection is a rectangular map projection that maps all areas such that they have the correct sizes relative to each other. Like any equal-area projection, it achieves this goal by distorting most shapes. The projection is a particular example of the cylindrical equal-area projection with latitudes 45° north and south as the regions on the map that have no distortion.

Contents

The projection is named after James Gall and Arno Peters. Gall is credited with describing the projection in 1855 at a science convention. He published a paper on it in 1885. [1] Peters brought the projection to a wider audience beginning in the early 1970s by means of the "Peters World Map". The name "Gall–Peters projection" seems to have been used first by Arthur H. Robinson in a pamphlet put out by the American Cartographic Association in 1986. [2]

Maps based on the projection are promoted by UNESCO, and they are also widely used by British schools. [3] The U.S. state of Massachusetts and Boston Public Schools began phasing in these maps in March 2017, becoming the first public school district and state in the United States to adopt Gall–Peters maps as their standard. [4]

The Gall–Peters projection achieved notoriety in the late 20th century as the centerpiece of a controversy about the political implications of map design. [5]

Description

Formula

The projection is conventionally defined as:

where λ is the longitude from the central meridian in degrees, φ is the latitude, and R is the radius of the globe used as the model of the earth for projection. For longitude given in radians, remove the π/180° factors.

Simplified formula

Stripping out unit conversion and uniform scaling, the formulae may be written:

where λ is the longitude from the central meridian (in radians), φ is the latitude, and R is the radius of the globe used as the model of the earth for projection. Hence the sphere is mapped onto the vertical cylinder, and the cylinder is stretched to double its length. The stretch factor, 2 in this case, is what distinguishes the variations of cylindric equal-area projection.

Discussion

The Gall-Peters cylindrical equal-area projection with Tissot's indicatrices of deformation Tissot indicatrix world map Gall-Peters equal-area proj.svg
The Gall–Peters cylindrical equal-area projection with Tissot's indicatrices of deformation

The various specializations of the cylindric equal-area projection differ only in the ratio of the vertical to horizontal axis. This ratio determines the standard parallel of the projection, which is the parallel at which there is no distortion and along which distances match the stated scale. There are always two standard parallels on the cylindric equal-area projection, each at the same distance north and south of the equator. The standard parallels of the Gall–Peters are 45° N and 45° S. Several other specializations of the equal-area cylindric have been described, promoted, or otherwise named. [6] [7] [8]

Named specializations of the cylindric equal-area projection
SpecializationStandard parallels N/S
Lambert cylindric equal-area Equator
Behrmann cylindric equal-area 30°
Smyth equal-surface (= Craster rectangular) 37°04′
Trystan Edwards 37°24′
Hobo–Dyer 37°30′
Gall–Peters (= Gall orthographic = Peters) 45°
Balthasart 50°
Tobler's world in a square55°39′

Origins and naming

The Gall–Peters projection was first described in 1855 by clergyman James Gall, who presented it along with two other projections at the Glasgow meeting of the British Association for the Advancement of Science (the BA). He gave it the name "orthographic" and formally published his work in 1885 in the Scottish Geographical Magazine. [1] The projection is suggestive of the Orthographic projection in that distances between parallels of the Gall–Peters are a constant multiple of the distances between the parallels of the orthographic. That constant is 2.

The name "Gall–Peters projection" seems to have been used first by Arthur H. Robinson in a pamphlet put out by the American Cartographic Association in 1986. [2] Before 1973 it had been known, when referred to at all, as the "Gall orthographic" or "Gall's orthographic." Most Peters supporters refer to it only as the "Peters projection." During the years of controversy the cartographic literature tended to mention both attributions, settling on one or the other for the purposes of the article. In recent years "Gall–Peters" seems to dominate.

Peters world map

The right and left borders of the Peters map are in the Bering Strait, so all of Russia is displayed on the right side.
Greenwich great circle
Bering Strait great circle (traversing Florence in Italy, see Florence meridian) Peters projection, date line in Bering strait.svg
The right and left borders of the Peters map are in the Bering Strait, so all of Russia is displayed on the right side.
  Greenwich great circle
  Bering Strait great circle (traversing Florence in Italy, see Florence meridian)
Comparison of the Gall-Peters projection and some cylindrical equal-area map projections with Tissot indicatrix, standard parallels and aspect ratio Tissot indicatrix world map cyl equal-area proj comparison.svg
Comparison of the Gall–Peters projection and some cylindrical equal-area map projections with Tissot indicatrix, standard parallels and aspect ratio

In 1967, Arno Peters, a German filmmaker, devised a map projection identical to Gall's orthographic projection and presented it in 1973 as a "new invention". He promoted it as a superior alternative to the Mercator projection, which was suited to navigation but also used commonly in world maps. The Mercator projection increasingly inflates the sizes of regions according to their distance from the equator. This inflation results, for example, in a representation of Greenland that is larger than Africa, which has a geographic area 14 times greater than Greenland's. Since much of the technologically underdeveloped world lies near the equator, these countries appear smaller on a Mercator and therefore, according to Peters, seem less significant. [7] :155 On Peters's projection, by contrast, areas of equal size on the globe are also equally sized on the map. By using his "new" projection, Peters argued that poorer, less powerful nations could be restored to their rightful proportions. This reasoning has been picked up by many educational and religious bodies, leading to adoption of the Gall–Peters projection among some socially concerned groups, including Oxfam, [9] National Council of Churches, [10] New Internationalist magazine, [11] and the Mennonite Central Committee. [12] However, Peters's choice of 45° N/S for the standard parallels means that the regions displayed with highest accuracy include Europe and the US, and not the tropics.

Peters's original description of the projection for his map contained a geometric error that, taken literally, implies standard parallels of 46°02′ N/S. However the text accompanying the description made it clear that he had intended the standard parallels to be 45° N/S, making his projection identical to Gall's orthographic. [13] In any case, the difference is negligible in a world map.

Controversy

At first, the cartographic community largely ignored Peters's foray into cartography. The preceding century had already witnessed many campaigns for new projections with little visible result. Just twenty years earlier, for example, Trystan Edwards described and promoted his own eponymous projection, disparaging the Mercator, and recommending his projection as the solution. [14] Peters's projection differed from Edwards's only in height-to-width ratio. More problematic, Peters's projection was identical to one that was already over a century old, though he probably did not realize it. [7] That projection—Gall's orthographic—passed unnoticed when it was announced in 1855.

Beyond the lack of novelty in the projection itself, the claims Peters made about the projection were also familiar to cartographers. Just as in the case of Peters, earlier projections generally were promoted as alternatives to the Mercator. Inappropriate use of the Mercator projection in world maps and the size disparities figuring prominently in Peters's arguments against the Mercator projection had been remarked upon for centuries and quite commonly in the 20th century. [14] [15] [16] [17] [18] [19] [20] As early as 1943, Stewart notes this phenomenon and compares the quest for the perfect projection to "squaring the circle or making pi come out even" [21] because the mathematics that governs map projections just does not permit development of a map projection that is objectively significantly better than the hundreds already devised. Even Peters's politicized interpretation of the common use of Mercator was nothing new, with Kelloway's 1946 text mentioning a similar controversy. [17]

Cartographers had long despaired over publishers' inapt use of the Mercator. [22] [23] [24] [25] A 1943 New York Times editorial stated that "The time has come to discard [the Mercator] for something that represents the continents and directions less deceptively ... Although its usage ... has diminished ... it is still highly popular as a wall map apparently in part because, as a rectangular map, it fills a rectangular wall space with more map, and clearly because its familiarity breeds more popularity." [26] Because of the lack of novelty both in the projection Peters devised and in the rhetoric surrounding its promotion, the cartographic community had no reason to think Peters would succeed any more than Edwards or his predecessors had. [26] :165

Peters, however, launched his campaign in a different world from that of Edwards. He announced his map at a time when themes of social justice resonated strongly in academia and politics. Suggesting "cartographic imperialism", Peters found ready audiences. The campaign was bolstered by the claim that the Peters projection was the only "area-correct" map. Other claims included "absolute angle conformality", "no extreme distortions of form", and "totally distance-factual". [27]

All of those claims were erroneous. [28] Some of the oldest projections are equal-area (the sinusoidal projection is also known as the "Mercator equal-area projection"), and hundreds have been described, refuting any implication that Peters's map is special in that regard. In any case, Mercator was not the pervasive projection Peters made it out to be: a wide variety of projections has always been used in world maps. [29] Peters's chosen projection suffers extreme distortion in the polar regions, as any cylindrical projection must, and its distortion along the equator is considerable. Several scholars have remarked on the irony of the projection's undistorted presentation of the mid latitudes, including Peters's native Germany, at the expense of the low latitudes, which host more of the technologically underdeveloped nations. [30] [31] The claim of distance fidelity is particularly problematic: Peters's map lacks distance fidelity everywhere except along the 45th parallels north and south, and then only in the direction of those parallels. No world projection is good at preserving distances everywhere; Peters's and all other cylindric projections are especially bad in that regard because east-west distances inevitably balloon toward the poles. [28] [32]

The cartographic community met Peters's 1973 press conference with amusement and mild exasperation, but little activity beyond a few articles commenting on the technical aspects of Peters's claims. In the ensuing years, however, it became clear that Peters and his map were no flash in the pan. By 1980 many cartographers had turned overtly hostile to his claims. In particular, Peters writes in The New Cartography,

Philosophers, astronomers, historians, popes and mathematicians have all drawn global maps long before cartographers as such existed. Cartographers appeared in the "Age of Discovery", which developed into the Age of European Conquest and Exploitation and took over the task of making maps.

By the authority of their profession they have hindered its development. Since Mercator produced his global map over four hundred years ago for the age of Europeans world domination, cartographers have clung to it despite its having been long outdated by events. They have sought to render it topical by cosmetic corrections.

... The European world concept, as the last expression of a subjective global view of primitive peoples, must give way to an objective global concept.

The cartographic profession is, by its retention of old precepts based on the Eurocentric global concept, incapable of developing this egalitarian world map which alone can demonstrate the parity of all peoples of the earth. [33]

This attack galled the cartographic community. Their most emphatic refutation of Peters's assertions was the long list of cartographers who, over the preceding century, had formally expressed frustration at publishers' overuse of the Mercator, as noted above. [26] Many of those cartographers had already developed projections they explicitly promoted as alternatives to the Mercator, including the most influential American cartographers of the twentieth century: John Paul Goode (Goode homolosine projection), Erwin Raisz (Armadillo projection), and Arthur H. Robinson (Robinson projection). Hence the cartographic community viewed Peters's narrative as ahistorical and mean-spirited.

The two camps never made any real attempts toward reconciliation. The Peters camp largely ignored the protests of the cartographers. Peters maintained there should be "one map for one world" [34] —his—and did not acknowledge the prior art of Gall [28] until the controversy had largely run its course, late in his life. While Peters likely reinvented the projection independently, his unscholarly conduct and refusal to engage the cartographic community undoubtedly contributed to the polarization and impasse. [5]

Frustrated by some very visible successes and mounting publicity stirred up by the industry that had sprung up around the Peters map, the cartographic community began to plan more coordinated efforts to restore balance, as they saw it. The 1980s saw a flurry of literature directed against the Peters phenomenon. Though Peters's map was not singled out, the controversy motivated the American Cartographic Association (now Cartography and Geographic Information Society) to produce a series of booklets (including Which Map Is Best [2] ) designed to educate the public about map projections and distortion in maps. In 1989 and 1990, after some internal debate, seven North American geographic organizations adopted the following resolution, [35] [36] which rejected all rectangular world maps, a category that includes both the Mercator and the Gall–Peters projections:

WHEREAS, the earth is round with a coordinate system composed entirely of circles, and

WHEREAS, flat world maps are more useful than globe maps, but flattening the globe surface necessarily greatly changes the appearance of Earth's features and coordinate systems, and

WHEREAS, world maps have a powerful and lasting effect on people's impressions of the shapes and sizes of lands and seas, their arrangement, and the nature of the coordinate system, and

WHEREAS, frequently seeing a greatly distorted map tends to make it "look right",

THEREFORE, we strongly urge book and map publishers, the media and government agencies to cease using rectangular world maps for general purposes or artistic displays. Such maps promote serious, erroneous conceptions by severely distorting large sections of the world, by showing the round Earth as having straight edges and sharp corners, by representing most distances and direct routes incorrectly, and by portraying the circular coordinate system as a squared grid. The most widely displayed rectangular world map is the Mercator (in fact a navigational diagram devised for nautical charts), but other rectangular world maps proposed as replacements for the Mercator also display a greatly distorted image of the spherical Earth.

One map society, the North American Cartographic Information Society (NACIS), declined to endorse the 1989 resolution, although no reasons were given.[ citation needed ]

The geographic and cartographic communities did not unanimously disparage the Peters World Map. Some cartographers, including J. Brian Harley, have credited the Peters phenomenon with demonstrating the social implications of map projections, at the very least. [37] Crampton sees the condemnation from the cartographic community as reactionary and perhaps demonstrative of immaturity in the profession, given that all maps are political. [5] Denis Wood sees the map as one of many useful tools. [34] Lastly, Terry Hardaker of Oxford Cartographers Limited, sympathetic to Peters's mission, became the map's official cartographer when Peters, overwhelmed by the technical aspects of cartography, sought to pass on those responsibilities. [34]

See also

Related Research Articles

Mercator projection Map projection for navigational use that distorts areas far from the equator

The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator, but accelerates with latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

Map projection Systematic representation of the surface of a sphere or ellipsoid onto a plane

In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. Every distinct map projection distorts in a distinct way, by definition. The study of map projections is the characterization of these distortions. There is no limit to the number of possible map projections. Projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds. However, "map projection" refers specifically to a cartographic projection.

Robinson projection compromise map projection defined via a look-up table of precomputed values

The Robinson projection is a map projection of a world map which shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of readily showing the whole globe as a flat image.

Transverse Mercator projection The transverse Mercator projection is the transverse aspect of the standard (or Normal) Mercator projection

The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the UTM. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

Orthographic projection in cartography map projection of cartography

The use of orthographic projection in cartography dates back to antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.

World map map of the surface of the Earth

A world map is a map of most or all of the surface of Earth. World maps form a distinctive category of maps due to the problem of projection. Maps by necessity distort the presentation of the earth's surface. These distortions reach extremes in a world map. The many ways of projecting the earth reflect diverse technical and aesthetic goals for world maps.

Scale (map) Ratio of distance on a map to the corresponding distance on the ground

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

Equirectangular projection map projection that maps meridians and parallels to vertical and horizontal straight lines, respectively, producing a rectangular grid

The equirectangular projection is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.

Sinusoidal projection pseudocylindrical equal-area map projection

The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson–Flamsteed or the Mercator equal-area projection. Jean Cossin of Dieppe was one of the first mapmakers to use the sinusoidal, appearing in a world map of 1570.

Tissots indicatrix

In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

Lambert cylindrical equal-area projection

In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is a cylindrical equal-area projection. This projection is undistorted along the equator, which is its standard parallel, but distortion increases rapidly towards the poles. Like any cylindrical projection, it stretches parallels increasingly away from the equator. The poles accrue infinite distortion, becoming lines instead of points.

Space-oblique Mercator projection map projection

Space-oblique Mercator projection is a map projection devised in the 1970s for preparing maps from Earth-survey satellite data. It is a generalization of the oblique Mercator projection that incorporates the time evolution of a given satellite ground track to optimize its representation on the map. The oblique Mercator projection, on the other hand, optimizes for a given geodesic.

Winkel tripel projection compromise map projection defined as the arithmetic mean of the equirectangular projection and the Aitoff projection

The Winkel tripel projection, a modified azimuthal map projection of the world, is one of three projections proposed by German cartographer Oswald Winkel in 1921. The projection is the arithmetic mean of the equirectangular projection and the Aitoff projection: The name tripel refers to Winkel's goal of minimizing three kinds of distortion: area, direction, and distance.

Kavrayskiy VII projection

The Kavrayskiy VII projection is a map projection invented by Soviet cartographer Vladimir V. Kavrayskiy in 1939 for use as a general-purpose pseudocylindrical projection. Like the Robinson projection, it is a compromise intended to produce good-quality maps with low distortion overall. It scores well in that respect compared to other popular projections, such as the Winkel tripel, despite straight, evenly spaced parallels and a simple formulation. Regardless, it has not been widely used outside the former Soviet Union.

Cylindrical equal-area projection

In cartography, the cylindrical equal-area projection is a family of cylindrical, equal-area map projections.

Central cylindrical projection

The central cylindrical projection is a perspective cylindrical map projection. It corresponds to projecting the Earth's surface onto a cylinder tangent to the equator as if from a light source at Earth's center. The cylinder is then cut along one of the projected meridians and unrolled into a flat map.

Gall stereographic projection Cylindrical map projection

The Gall stereographic projection, presented by James Gall in 1855, is a cylindrical projection. It is neither equal-area nor conformal but instead tries to balance the distortion inherent in any projection.

Equal Earth projection pseudocylindrical, equal-area map projection

The Equal Earth map projection is an equal-area pseudocylindrical projection for world maps, invented by Bojan Šavrič, Bernhard Jenny, and Tom Patterson in 2018. It is inspired by the widely used Robinson projection, but unlike the Robinson projection, retains the relative size of areas. The projection equations are simple to implement and fast to evaluate.

Strebe 1995 projection

The Strebe 1995 projection, Strebe projection, Strebe lenticular equal-area projection, or Strebe equal-area polyconic projection is an equal-area map projection presented by Daniel "daan" Strebe in 1994. Strebe designed the projection to keep all areas proportionally correct in size; to push as much of the inevitable distortion as feasible away from the continental masses and into the Pacific Ocean; to keep a familiar equatorial orientation; and to do all this without slicing up the map.

Stereographic projection in cartography

The stereographic projection, also known as the planisphere projection or the azimuthal conformal projection, is a conformal map projection whose use dates back to antiquity. Like the orthographic projection and gnomonic projection, the stereographic projection is an azimuthal projection, and when on a sphere, also a perspective projection.

References

Notes

  1. 1 2 Gall, James (1885). "Use of cylindrical projections for geographical, astronomical, and scientific purposes" (PDF). Scottish Geographical Magazine. 1 (4): 119–123. doi:10.1080/14702548508553829.
  2. 1 2 3 American Cartographic Association's Committee on Map Projections, 1986. Which Map is Best p. 12. Falls Church: American Congress on Surveying and Mapping.
  3. Higgins, Hannah B. The Grid Book. Cambridge, Massachusetts: MIT Press, 2009. ISBN   9780262512404 p.94. "Embroiled in controversy from the start, the map is nonetheless widely used in the British school system and is promoted by the United Nations Educational and Scientific Cultural Organization (UNESCO) because of its ability to communicate visually the actual relative sizes of the various regions of the planet."
  4. Joanna Walters (March 19, 2017). "Boston public schools map switch aims to amend 500 years of distortion". The Guardian . Retrieved March 19, 2017.
  5. 1 2 3 Crampton, Jeremy (1994). "Cartography's defining moment: The Peters projection controversy, 1974–1990". Cartographica. 31 (4): 16–32. doi:10.3138/1821-6811-l372-345p.
  6. Snyder, John P. (1989). An Album of Map Projections p. 19. Washington, D.C.: U.S. Geological Survey Professional Paper 1453. (Mathematical properties of the Gall–Peters and related projections.)
  7. 1 2 3 Monmonier, Mark (2004). Rhumb Lines and Map Wars: A Social History of the Mercator Projection p. 152. Chicago: The University of Chicago Press. (Thorough treatment of the social history of the Mercator projection and Gall–Peters projections.)
  8. Smyth, C. Piazzi. (1870). On an Equal-Surface Projection and its Anthropological Applications. Edinburgh: Edmonton & Douglas. (Monograph describing an equal-area cylindric projection and its virtues, specifically disparaging Mercator's projection.)
  9. "Archived copy". Archived from the original on 2010-06-22. Retrieved 2011-11-14.CS1 maint: archived copy as title (link)
  10. "Ncc Friendship Press' Peters Projection Map".
  11. "The New Flat Earth". May 1983.
  12. Ncc Friendship Press' Peters Projection Map
  13. Maling, D.H. (1993). Coordinate Systems and Map Projections, second edition, second printing, p. 431. Oxford: Pergamon Press. ISBN   0-08-037234-1.
  14. 1 2 Edwards, Trystan (1953). A New Map of the World. London: B.T. Batsford LTD.
  15. Hinks, Arthur R. (1912). Map Projections p. 29. London: Cambridge University Press.
  16. Steers, J.A. (1927). An Introduction to the Study of Map Projections 9th ed. p. 154. London: The University of London Press.
  17. 1 2 Kellaway, G.P. (1946). Map Projections p. 37–38. London: Methuen & Co. LTD. (Mentions allegations of an "imperialistic motive".)
  18. Abelson, C.E. (1954). Common Map Projections p. 4. Sevenoaks: W.H. Smith & Sons.
  19. Chamberlin, Wellman (1947). The Round Earth on Flat Paper p. 99. Washington, D.C.: The National Geographic Society.
  20. Fisher, Irving (1943). "A World Map on a Regular Icosahedron by Gnomonic Projection." Geographical Review 33 (4): 605. (A jab at Mercator and proposal for a replacement.)
  21. Stewart, John Q. (1943). "The Use and Abuse of Map Projections". Geographical Review 33 (4): 590. (Noting quixotic promotions of new projections.)
  22. Bauer, H.A. (1942). "Globes, Maps, and Skyways (Air Education Series)". New York. p. 28
  23. Miller, Osborn Maitland (1942). "Notes on Cylindrical World Map Projections". Geographical Review. 43 (3): 405–409. doi:10.2307/211756. JSTOR   211756.
  24. Raisz, Erwin Josephus. (1938). General Cartography. New York: McGraw-Hill. 2d ed., 1948. p. 87.
  25. Robinson, Arthur Howard. (1960). Elements of Cartography, second edition. New York: John Wiley and Sons. p. 82.
  26. 1 2 3 Snyder, John P. (1993). Flattening the Earth: Two Thousand Years of Map Projections p. 157. Chicago and London: The University of Chicago Press. ISBN   0-226-76746-9. (Summary of the Peters controversy.)
  27. (1977) The Bulletin 25 (17) pp. 126–127. Bonn: Federal Republic of Germany Press and Information Office.
  28. 1 2 3 Snyder, John P. (1993). Flattening the Earth: Two Thousand Years of Map Projections p. 165. Chicago and London: The University of Chicago Press. ISBN   0-226-76746-9. (Summary of the Peters controversy.)
  29. Bowyer, T.D.; German, G.A. (1959). A Guide to Map Projections p. 16. London: John Murray.
  30. Snyder, J.P. (1988). "Social Consciousness and World Maps". Christian Century. 105 (2): 190–192. doi:10.1559/152304085783915063.
  31. Robinson, Arthur H. (1985). "Arno Peters and His New Cartography". American Cartographer. 12 (2): 103–111. doi:10.1559/152304085783915063.
  32. Canters, Frank; Decleir, Hugo (1989). The World in Perspective: A Directory of World Map Projections, p. 36. Chichester: John Wiley & Sons LTD. ISBN   0-471-92147-5. (Distortion of distance in world maps. Gall–Peters is structurally very similar to Behrmann.)
  33. Peters, Arno (1983). Die Neue Kartographie/The New Cartography (in German and English). Klagenfurt, Austria: Carinthia University; New York: Friendship Press.
  34. 1 2 3 Arno Peters: Radical Map, Remarkable Man. (A DVD documentary.) 2008.
  35. Robinson, Arthur (1990). "Rectangular World Maps—No!". Professional Geographer. 42 (1): 101–104. doi:10.1111/j.0033-0124.1990.00101.x.
  36. American Cartographer. 1989. 16(3): 222–223.
  37. Harley, J.B. (1991). "Can There Be a Cartographic Ethics?". Cartographic Perspectives. 10 (10): 9–16. doi:10.14714/CP10.1053.

Further reading