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

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 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.

• The Adams hemisphere-in-a-square projection and the Peirce quincuncial projection are different aspects of the same underlying Schwarz–Christoffel mapping. Such mappings are transformations of half a stereographic projection.

• List of map projections