Guyou hemisphere-in-a-square projection

The projection was developed by {{Ill|Émile Guyou|fr}} of France in 1887.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

{{cite book

| last=Snyder

| author-link=John P. Snyder

| first=John P.

| title=Flattening the Earth

| publisher=University of Chicago

| year=1993

| isbn=0-226-76746-9}}

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.{{cite book

| last = Lee | first = L. P. | author-link = Laurence Patrick Lee

| year = 1976

| title = Conformal Projections Based on Elliptic Functions

| location = Toronto | publisher = B. V. Gutsell, York University

| series = Cartographica Monographs | volume = 16

| url = https://archive.org/details/conformalproject0000leel | url-access = limited

| isbn = 0-919870-16-3

}} Supplement No. 1 to [https://www.utpjournals.press/toc/cart/13/1 The Canadian Cartographer 13].

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

{{cite journal

| author=C.S. Peirce

| title=A Quincuncial Projection of the Sphere

| date=December 1879

| pages=394–396

| journal=American Journal of Mathematics

| volume=2 | issue=4

| doi=10.2307/2369491

| publisher=The Johns Hopkins University Press

| jstor=2369491}}

• The curvature of lines representing great circles is, in every case, very slight, over the greater part of their length.
• 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

{{Reflist|30em}}

{{Map projections}}

Category:Conformal projections