Guyou hemisphere-in-a-square projection

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Template:Short description

File:Guyou doubly periodic projection SW.JPG
Guyou doubly periodic projection of the world.
File:Guyou with Tissot's Indicatrices of Distortion.svg
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.

History

The projection was developed by Template:Ill of France in 1887.[1][2]

Formal description

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.

Properties

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

  • 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

See also

References

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  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. https://www.retronews.fr/journal/annales-hydrographiques/1-janvier-1887/1877/4868382/23
  2. ↑ Script error: No such module "citation/CS1".
  3. ↑ Script error: No such module "citation/CS1". Supplement No. 1 to The Canadian Cartographer 13.
  4. ↑ a b Script error: No such module "Citation/CS1".

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Template:Map projections

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