Guyou hemisphere-in-a-square projection

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Guyou doubly periodic projection of the world.
The Guyou hemisphere-in-a-square projection with Tissot's indicatrix of deformation. The indicatrix is omitted at the singular points. At those points the deformation is infinite; the indicatrix would be infinite in size.

The Guyou hemisphere-in-a-square projection is a conformal map projection for the hemisphere. It is an oblique aspect of the Peirce quincuncial projection.


The projection was developed by Émile Guyou [fr] of France in 1887.[1][2]

Formal description[edit]

The projection can be computed as an oblique aspect of the Peirce quincuncial projection by rotating the axis 45 degrees. It can also be computed by rotating the coordinates −45 degrees before computing the stereographic projection; this projection is then remapped into a square whose coordinates are then rotated 45 degrees.[3]

The projection is conformal except for the four corners of each hemisphere’s square. Like other conformal polygonal projections, the Guyou is a Schwarz–Christoffel mapping.


Its properties are very similar to those of the Peirce quincuncial:

  • Each hemisphere is represented as a square, the sphere as a rectangle of aspect ratio 2:1.
  • The part where the exaggeration of scale amounts to double that at the centre of each square is only 9% of the area of the sphere, against 13% for the Mercator and 50% for the stereographic[4]
  • The curvature of lines representing great circles is, in every case, very slight, over the greater part of their length.[4]
  • It is conformal everywhere except at the corners of the square that corresponds to each hemisphere, where two meridians change direction abruptly twice each; the Equator is represented by a horizontal line.
  • It can be tessellated in all directions.

Related projections[edit]

See also[edit]


  1. ^ E. Guyou (1887) "Nouveau système de projection de la sphère: Généralisation de la projection de Mercator", Annales Hydrographiques, Ser. 2, Vol. 9, 16–35.
  2. ^ Snyder, John P. (1993). Flattening the Earth. University of Chicago. ISBN 0-226-76746-9.
  3. ^ L.P. Lee (1976). "Conformal Projections based on Elliptic Functions". Cartographica. 13 (Monograph 16, supplement No. 1 to Canadian Cartographer).
  4. ^ a b C.S. Peirce (December 1879). "A Quincuncial Projection of the Sphere". American Journal of Mathematics. The Johns Hopkins University Press. 2 (4): 394–396. doi:10.2307/2369491. JSTOR 2369491.