The Spilhaus World Ocean Map in a Square

A story about the "Spilhaus projection" – a map projection that went viral in fall 2018 and is now supported in ArcGIS.

An Old World Ocean Map Goes Viral

In September and October of 2018, three maps went viral on social media and the web. All of them had the same perspective, featured oceans as the main focus, and presented the oceans as one body of water. The maps were based on the so-called "Spilhaus projection" and centered on Antarctica. Though it has recently gained some popularity online, this projection is not new. Many articles recognize Athelstan F. Spilhaus, a South African-American geophysicist and oceanographer, as the author of this projection back in 1942.

A world ocean map based on the “Spilhaus projection” and published by mare, German magazine about oceans.

At  Esri , we received multiple requests to support the projection in ArcGIS during the fall of 2018. To implement a projection into GIS software, we would prefer to have one-to-one forward and inverse equations not only for spherical Earth models, but also for ellipsoids, such as WGS 1984. None of the requesters were able to provide us with any of these equations, so we started looking into the projection, hoping to find some clues to the mathematics behind the map.


Searching for the Equations

A review of Spilhaus's publications

As any good researcher would, we started reviewing Spilhaus's publications on world ocean maps. We found four relevant publications – two discuss Spilhaus's projection, and three show a map using his projection. 

1942: Maps of the Whole World Ocean

Several news articles published in fall 2018 mentioned that Athelstan Spilhaus designed his world ocean map in 1942. Reviewing his publications from that year, no such projection is presented. He did present two world ocean maps that year – one used the August conformal projection and one used the Hammer-Aitoff equal-area projection.

However, Spilhaus did discuss a cut or a line on the surface of a globe, which could serve as the edge of a world ocean map that would intersect the oceans as little as possible.

A satisfactory line, however, is the half great circle the ends of which are on the geographical equator at longitudes 75°W and 150°E and which passes through latitude 70 °N, longitude 165 °W (the neighborhood of Bering Strait).

1979: To See the Oceans, Slice Up the Land

The earliest publication we found that had information about the “Spilhaus projection” was the Smithsonian article "To see the oceans, slice up the land" by Athelstan Spilhaus, published in November 1979.

In this article, Spilhaus first continues the discussion about using a cut or line across the land as the edge of a world ocean map.

So the best poles for whole ocean maps are the substantial antipodal land area in China (near Hankou) and South America (near Cordoba in Argentina) with a cut joining them across the Bering Strait. The Bering Strait, although navigable, is so shallow and narrow that it constitutes no real oceanic connection. These poles and the Bering Strait cut can be used with any existing map projection...

Then, he talks about his new map of the world ocean and reveals that the world ocean map projection was developed in collaboration with two geodesists.

This map is based on a new projection of the world in a square devised by geodesists Robert Hanson and Erwin Schmid, who helped me with the equations and programmed the computers which drew the maps.

In John P. Snyder's table of map projections developed during the twentieth century (Snyder 1993), one cannot find any information about this "new projection of the world in a square." Snyder does mention Erwin Schmid of the U.S. Coast and Geodetic Survey. There is no mention of Robert Hanson in Snyder's book. An online search for Robert Hanson results in a few hits on publications by the U.S. Coast and Geodetic Survey and NOAA’s National Geodetic Survey, but there is no publication that explains this "new projection."

Spilhaus also provides a short description of the projection’s properties in this article.

The distortions at the two corners around the poles in South America and China are very great indeed, but it is in the land that we wish to concentrate maximum distortions. This map has extraordinary additional property of being doubly-periodic. This means that if you had multiple copies, you could match the edges perfectly and repeat the pattern just like decorative tiles. The endlessly repeating mural which would result would tell us that a true map of the world has no edges.

1983: World Ocean Maps: The Proper Places to Interrupt

Athelstan Spilhaus reveals more information about his world ocean map in his next publication in 1983. This time, we learn about the start and end points of the cut or line across the land used for the map edge.

Since the largest distortions occur at the singular points, these are better placed in the two major antipodal land areas on earth: South China (115°E and 30°N) and Argentina (65°W and 30°S).

Spilhaus also tells us more about the projection's distortion, and he reveals that it is a conformal projection.

For a conformal version, I have used a projection of the world in a square, developed for me by Hanson and Schmid. This map is periodic which means that multiple copies can be fitted together to form a continuous pattern of many world maps in a mosaic.

In this article, Spilhaus names the projection the Spilhaus, Hanson, Schmid conformal projection. He also references his Smithsonian article from 1979, and the same year appears on the note about Hanson and Schmid. Since no other reference is added to the descriptions of the projection, it is reasonable to believe that the Smithsonian article from 1979 is the original publication of the projection.

1991: Atlas of the World

The last publication in which the world ocean projection appears is the Atlas of the World by Athelstan Spilhaus from 1991. Here, the projection is presented between composite maps with continental shorelines as natural boundaries. Spilhaus does not provide any description of the map beyond the map caption, where he names the projection World Ocean Map in a Square. Here too, he references his Smithsonian article from 1979.

From Spilhaus's papers, we learned that the projection is conformal and depicts the world in a square. It can be mosaicked into an endless image of the world, which explains why Spilhaus's world ocean map repeatedly displays the Gulf of Mexico, areas in Central America, the west coast of South America, and the area north of the Bering Strait. The edge of the map is a great circle starting in South China at 115°E and 30°N, ending in Argentina at 65°W and 30°S, and passing through the neighborhood of the Bering Strait, possibly at 165°W and 70°N. The start and end points represent the "poles," which have the largest distortion. They are projected into the diagonally opposite corners of the square. The projection was developed in collaboration with Robert Hanson and Erwin Schmid of the former U.S. Coast and Geodetic Survey in 1979, not in 1942 as referenced in the news articles published in fall 2018. Unfortunately, none of the papers contain the equations for the projection.

The Adams projection of the world in a square II

Browsing through  An Album of Map Projections  by J. P. Snyder and P. M. Voxland from 1989, we found the Adams projection of the world in a square II by Oscar S. Adams. This projection has remarkable similarities to the "Spilhaus projection." Both projections are conformal and portray the world in a square. They greatly distort areas near the poles, which appear in two diagonally opposite corners. Distortions in the other two corners are smaller. Just like Spilhaus’s projection, Adams’s projection can also be mosaicked into an endless map of the world. We also found a connection between Oscar S. Adams and the two co-authors of the “Spilhaus projection”, Robert Hanson and Erwin Schmid. Adams worked for the U.S. Coast and Geodetic Survey, the same organization where Hanson and Schmid worked. The only noticeable difference between the projections is that Spilhaus’s projection appears to be in an oblique aspect.

The "Spilhaus projection" (left) published in Spilhaus (1983) and the Adams projection of the world in a square II (right) in Snyder and Voxland (1989).

Could the "Spilhaus projection" really be the Adams projection in an oblique aspect?

The only way to answer this question is to implement the Adams projection of the world in a square II, create an oblique case, and try to georeference one of Spilhaus's world ocean maps to that particular oblique case.

The map we tried to georeference was the "World Ocean Map in a Square" from Spilhaus's 1991 Atlas of the World. It is the largest, highest quality map from the three original publications. Unfortunately, the original map is slightly skewed – it is not a perfect square. To address this, we used an affine transformation for georeferencing. It includes skewing in addition to translation, scaling, and rotation of the map, and it requires only three common points for a unique adjustment. When georeferencing, we tried to ensure that the skewing parameter remained minimal and only in the range required to correct the skewed map back to a square.

The first oblique case we tested was where the "poles" were set to South China at 115°E and 30°N and to Argentina at 65°W and 30°S. The great circle, representing the edge of the map, passed through the neighborhood of the Bering Strait at 165°W and 70°N, as specified by Spilhaus in his 1942 paper. Points on the graticule in the Labrador Sea, Bering Sea, and west of the Southern California coast were used to make the unique adjustment. The resulting map was significantly different. The coast outlines and the graticule did not line up. Georeferencing also introduced additional undesirable skewness, which did not correct the original skewness of the map.

Comparing the Spilhaus map (gray) and the Adams projection in the first oblique aspect (brown). Points on the graticule in the Labrador Sea, Bering Sea, and west of the Southern California coast were used for georeferencing. 

From the results of the first map, it is obvious that the great circle passing through 165°W and 70°N does not represent the edge of the map. Spilhaus never revealed in his publications how Hanson and Schmid created the projection. He used a different term to describe the cut joining the two poles in his 1979 Smithsonian article. Instead of "the neighborhood of the Bering Strait,” he used the term “across the Bering Strait,” which could imply that the cut goes right across the strait.

For the second oblique case, we used the same "poles" in South China and Argentina, but we moved the edge of the map to cut exactly through the Diomede islands in the Bering Strait. The result was shocking. The graticule and coastlines of both maps started lining up, with some differences here and there, especially away from the control points. In addition, the top square edges of the maps were more aligned and skewing of the transformation already corrected some of the original skewness of the map.

Comparing the Spilhaus map (gray) and the Adams projection in the second oblique aspect (brown). The graticule and coastlines of both maps start lining up.

Although the second map we created was closer to the original, the maps still did not quite match. The graticule was still misaligned, and the coastlines, especially those towards the edges of the map and away from the control points, did not line up well with the original Spilhaus map. Furthermore, both square edges remained slightly skewed compared to each other.

Comparing the Spilhaus map (gray) and the Adams projection in the second oblique aspect (brown). Coastlines in Central America (left), Patagonia and Antarctic (middle), and Yellow Sea and Sea of Japan (right) are still substantially offset between the two maps when zoomed in.

Since Spilhaus never revealed the exact location he used in the Bering Strait, we used a trial-and-error method to figure it out. Accounting for the resolution, scale, and generalization of the Spilhaus map and our small-scale data, we found the best fit was a cut going through the Bering Strait at 169°W and 65.3°N. That map best mimicked the original Spilhaus map. The graticule and coastlines lined up nicely after georeferencing. Both square edges appeared to be parallel and the top edges of the two maps lined up.

Comparing the Spilhaus map (gray) and the oblique Adams projection with the cut edge across the Bering Strait at 169°W and 65.3°N (brown). The graticule and coastlines in all three images are aligned between the two maps.

Looking at the results, we can most certainly claim that Spilhaus’s world ocean map was in fact created using an oblique aspect of the Adams projection of the world in a square II. The closest match we found has the edge of the map starting in South China, passing across the Bering Strait, and ending in Argentina.


The Adams Square II Projection in ArcGIS

The next step towards supporting Spilhaus’s world ocean map in ArcGIS was the implementation of the Adams projection of the world in a square II. We used the original publication by Oscar S. Adams from 1929. There, he derives the equations for his projection by first conceptually shrinking the whole world into a hemisphere while maintaining conformality. Then he applies the Guyou projection with elliptic functions to finally project the curved surface onto the plane. 

Adams only presented the forward equations for spherical Earth models. Most of today’s geospatial data is defined based on ellipsoidal models, such as WGS 1984 or GRS 1980. That data needs to be converted back from the projected surface to geographic coordinates and we needed to derive the forward and inverse equations for ellipsoidal Earth models. We achieved that by converting geodetic coordinates to a conformal sphere, conformally shrinking the model to a hemisphere, and resolving a complex elliptic integral of the first kind. 

The Adams projection of the world in a square II is available in ArcGIS Pro 2.5 (ArcGIS 10.8) as the  Adams Square II  projection. This projection has seven projection parameters. The False Easting, False Northing, Longitude of Center, Latitude of Center, and Scale Factor parameters are well-known and used in many other projections in the software. The Azimuth parameter is a direction from the North towards the top "pole" at the center on the conformal sphere. The XY Plane Rotation parameter is the rotation angle of the square map in the projection plane.

The Adams Square II projection in normal aspect available in ArcGIS Pro 2.5 (ArcGIS 10.8).

Additional information about the Adams Square II map projection in ArcGIS is available in the  ArcGIS Pro  or  ArcGIS Desktop  documentation.


The Spilhaus World Ocean Map in ArcGIS

Once the Adams Square II projection was implemented into the software, the final step was deriving the parameters for the projected coordinate system of Spilhaus’s world ocean map. Because most of today’s world data is defined with WGS 1984 geographic coordinates, the implemented Spilhaus map is based on the WGS 1984 geographic coordinate system. To derive the projection parameters, we used the edge of the map passing through the same three points used by Spilhaus, starting in South China at 115°E and 30°N, passing across the Bering Strait at 169°W and 65.3°N, and ending in Argentina at 65°W and 30°S. The only difference is that the edge does not represent a great circle on a sphere, but rather an arbitrary curve on the surface of the WGS 1984 ellipsoid passing through all three points. 

The Spilhaus world ocean map in a square or "Spilhaus projection" is now  available in ArcGIS Pro 2.5 (ArcGIS 10.8)  as the WGS 1984 Spilhaus Ocean Map in Square projected coordinate system. Its WKID is 54099 and it has the following projection parameters:

  • Geographic Coordinate System: GCS WGS 1984
  • Projection: Adams Square II
  • False Easting: 0 m
  • False Northing: 0 m
  • Scale Factor: 1
  • Azimuth: 40.17823482°
  • Longitude of Center: 66.94970198°E
  • Latitude of Center: 49.56371678°S
  • XY Plane Rotation: 45°

The WGS 1984 Spilhaus Ocean Map in a Square in ArcGIS Pro 2.5 (ArcGIS 10.8).

The original Spilhaus world ocean map in a square repeatedly shows the coastlines of the Gulf of Mexico, areas in Central America, the west coast of South America, and the area north of the Bering Strait. You can re-create the same world ocean map in ArcGIS by using the WGS 1984 Spilhaus Ocean Map in Square projected coordinate system as demonstrated in the image below.  John Nelson , cartographer at  Esri , shares a few more examples of the map in his blog post " Spilhaus? More like Thrillhaus ."

Map of ocean currents created by John Nelson using the WGS 1984 Spilhaus Ocean Map in Square projected coordinate system in ArcGIS.

If you would like to see the steps for creating a seamless ocean map, check out  this blog post . You can also download  John's ArcGIS Pro project package  with the map all set up and ready for you to populate with your favorite oceanic data.


References

Adams, O. S. (1925). Elliptic Functions Applied to Conformal World Maps. Washington: U.S. Coast and Geodetic Survey Special Publication 112.

Adams, O. S. (1929). Conformal Projection of the Sphere Within a Square. Washington: U.S. Coast and Geodetic Survey Special Publication 153.

Snyder, J. P. (1993). Flattening the Earth. Two Thousand Years of Map Projections. Chicago and London: University of Chicago Press.

Snyder, J. P. and Voxland, P. M. (1989). An Album of Map Projections. U.S. Geological Survey Professional Paper 1453. Washington, DC: United States Government Printing Office. doi:  10.3133/pp1453 

Spilhaus, A. (1942). "Maps of the whole world ocean." Geographical Review, 32 (3), p. 431–5. doi:  10.2307/210385 

Spilhaus, A. (November 1979). "To see the oceans, slice up the land." Smithsonian, p. 54–63.

Spilhaus, A. (1983). “World ocean maps: The proper places to interrupt.” Proceedings of the American Philosophical Society, 127 (1), p. 50–60.  jstor.org/stable/210385 

Spilhaus, A. (1991). Atlas of the World with Geophysical Boundaries Showing Oceans, Continents and Tectonic Plates in Their Entirety. Independence Square, Philadelphia: The American Philosophical Society.

The "Spilhaus projection" (left) published in Spilhaus (1983) and the Adams projection of the world in a square II (right) in Snyder and Voxland (1989).

Comparing the Spilhaus map (gray) and the Adams projection in the first oblique aspect (brown). Points on the graticule in the Labrador Sea, Bering Sea, and west of the Southern California coast were used for georeferencing. 

Comparing the Spilhaus map (gray) and the Adams projection in the second oblique aspect (brown). The graticule and coastlines of both maps start lining up.

Comparing the Spilhaus map (gray) and the Adams projection in the second oblique aspect (brown). Coastlines in Central America (left), Patagonia and Antarctic (middle), and Yellow Sea and Sea of Japan (right) are still substantially offset between the two maps when zoomed in.

Comparing the Spilhaus map (gray) and the oblique Adams projection with the cut edge across the Bering Strait at 169°W and 65.3°N (brown). The graticule and coastlines in all three images are aligned between the two maps.

The Adams Square II projection in normal aspect available in ArcGIS Pro 2.5 (ArcGIS 10.8).

The WGS 1984 Spilhaus Ocean Map in a Square in ArcGIS Pro 2.5 (ArcGIS 10.8).

Map of ocean currents created by John Nelson using the WGS 1984 Spilhaus Ocean Map in Square projected coordinate system in ArcGIS.