The Guyou hemisphere-in-a-square projection is a conformal

Conformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

 map projection

Map projection

A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...

 for the hemisphere (except for four points where the conformality fails). It is an oblique aspect of the Peirce quincuncial projection

Peirce quincuncial projection

The Peirce quincuncial projection is a conformal map projection that presents the sphere as a square...

. When it is used to represent the entire sphere it is known as the Guyou doubly periodic projection (see also doubly periodic function).

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.

Properties

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
  Stereographic projection
  The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...
  
  
• The Adams hemisphere-in-a-square projection
  Adams hemisphere-in-a-square projection
  The Adams-hemisphere-in-a-square is a conformal map projection for a hemisphere . It is a transverse version of the Peirce quincuncial projection. When it is used to represent the entire sphere it is known as the Adams doubly periodic projection....
  
   and the Peirce quincuncial projection
  Peirce quincuncial projection
  The Peirce quincuncial projection is a conformal map projection that presents the sphere as a square...
  
  are two of its oblique aspects.