Cox conformal projection

#projections #maps #code #d3.js

18 August 2017

 
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Cox Projection, South aspect
Philippe Rivière, 2017

Jacques-François Cox (1898-1972) is a Belgian astronomer, who will become rector of the Free University of Brussels (ULB). In 1935, he describes, in the Bulletin de la Classe des Sciences published by the Royal Academy, a “representation of the entire surface of the earth in an equilateral triangle.”

Three years later, Joseph Antoine Magis, who works as a computer at the Royal Observatory of Belgium, offers a formula and numerical tables that allow its precise computation with the means of the times.

He suggests an application, to draw the Gulf stream:

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Sketch of a map of the Gulf stream
J. Magis, 1938

This projection is conformal, meaning that it preserves the angles: two paths that cross on the planet with a certain angle are represented on the plane with the same angle.

This is made at the expense of surfaces, which as a consequence cannot be equivalent. But, as Magis notes, the area distorsion is less, in the central region of the map, than it is on the traditional conformal projections (Mercator’s, Lambert’s). “Though it was imagined as a way to solve a theoretical problem, concludes Magis, this canvas therefore has a practical interest in mathematical geography for the proper representation of equatorial regions, notably for Insulindia and for the Belgian Congo.” However, the complexity of the necessary computations will in practice prohibit its use.

In 1976, Laurence Patrick Lee, computer at the Department of Survey and Land Information of New Zealand, drawing on the work of the American geographer Oscar S. Adams, publishes a mathematical tour de force in which he proposes a general approach for conformal projections on the Platonic solids (the tetrahedron, the cube, etc). His article, more than a hundred pages long, warrants a special issue of the scientific journal Cartographica. He shows how some of these projections are linked to one of Dixon’s elliptic functions named sm, and produces numeric tables.

In 2011, the mathematician and computer science researcher Malcolm Douglas McIlroy introduces a more precise computation method, based on two Taylor series. This method is the one that we are currently implementing for d3.js.

In a classical view, ignoring Antarctica:

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Cox Projection, South aspect, ignoring Antarctica
Philippe Rivière, 2017

Here is the North aspect, which looks a bit grotesque:

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Projection Cox, North aspect
Philippe Rivière, 2017

By assembling several of these triangles, one can obtain interesting wallpaper or tiling effects.

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Six equilateral triangles in a chain dance
Philippe Rivière, 2017

Bibliography

— J. F. Cox, “Représentation de la surface entière de la terre dans un triangle équilatéral,” Bulletin de la Classe des Sciences, Académie Royale de Belgique, 5, 21: 66-71. Bruxelles, 1935. [I’m still looking for a copy of this article!]

— J. Magis, “Calcul du canevas de la représentation conforme de la sphère entière dans un triangle équilatéral,” Bulletin Geodésique, 247-256. Paris, 1938.

— L. P. Lee, “Conformal Projections Based On Dixon Elliptic Functions,” Cartographica: The International Journal for Geographic Information and Geovisualization, Vol. 13, # 1, June 1976.

— M. D. McIlroy, Wallpaper Maps (PDF), 2011.