Guyou hemisphere-in-a-square projection
The Guyou hemisphere-in-a-square projection is a conformal map projection for the hemisphere (except for the four angles of the square where the conformality fails). It is an oblique aspect of the Peirce quincuncial projection. When it is used to represent the entire sphere it is known as the Guyou doubly periodic projection (see also doubly periodic function).
It was developed by Émile Guyou of France in 1887 (Snyder, 1989).
Formal description 
It 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 (Lee, 1976). When both hemispheres are mapped with it, and placed next to each other, it is called Guyou doubly periodic.
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 (Peirce, 1879)
- The curvature of lines representing great circles is, in every case, very slight, over the greater part of their length. (Peirce, 1879)
- 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 
- It is based upon the stereographic projection
- The Adams hemisphere-in-a-square projection and the Peirce quincuncial projection are two of its oblique aspects.
- L.P. Lee (1976). "Conformal Projections based on Elliptic Functions". Cartographica 13 (Monograph 16, supplement No. 1 to Canadian Cartographer).
- 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.
- Snyder, John P. (1993). Flattening the Earth. University of Chicago. ISBN 0-226-76746-9.
- Snyder, John P. (1989). An Album of Map Projections, Professional Paper 1453. US Geological Survey.