Guyou hemisphere-in-a-square projection

The Guyou hemisphere-in-a-square projection is a conformal map projection for the hemisphere (except for four points 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).

History

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.

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
* The Adams hemisphere-in-a-square projection and the Peirce quincuncial projection are two of its oblique aspects.

References

*cite journal | author=L.P. Lee | title = Conformal Projections based on Elliptic Functions | year=1976 | journal = Cartographica | volume=13 | issue = Monograph 16, supplement No. 1 to Canadian Cartographer
*cite journal | author=C.S. Peirce | title = A Quincuncial Projection of the Sphere | year=1879 | pages =394--396 | journal=American Journal of Mathematics | volume = 2 | issue = 4| month = Dec | doi = 10.2307/2369491
* cite book
last = Snyder
first = John P.
title = Flattening the Earth
publisher = University of Chicago
date = 1993
isbn = 0-226-76746-9
* cite book
last = Snyder
first = John P.
title = An Album of Map Projections, Professional Paper 1453
publisher = US Geological Survey
date = 1989