Lee conformal world in a tetrahedron explained

The Lee conformal world in a tetrahedron is a polyhedral, conformal map projection that projects the globe onto a tetrahedron using Dixon elliptic functions. It is conformal everywhere except for the four singularities at the vertices of the polyhedron. Because of the nature of polyhedra, this map projection can be tessellated infinitely in the plane. It was developed by Laurence Patrick Lee in 1965.^[1]

Coordinates from a spherical datum can be transformed into Lee conformal projection coordinates with the following formulas,^[1] where is the longitude and the latitude:

2 \operatorname{sm}w\operatorname{cm}w = 2 ^{5/6} \exp(i λ ) \tan l (\tfrac 14 \pi - \tfrac 12 \phi r ) where w = x + y i

and sm and cm are Dixon elliptic functions.

Since there is no elementary expression for these functions, Lee suggests using the 28th degree MacLaurin series.^[1]

See also

• List of map projections
• AuthaGraph projection, another tetrahedral projection, 1999
• Dymaxion map, 1943
• Peirce quincuncial projection, 1879
• Polyhedral map projection, earliest known is by Leonardo da Vinci, 1514

Notes and References

1. Some Conformal Projections Based on Elliptic Functions. Lee . L.P. . Laurence Patrick Lee. 1965. Geographical Review. 55 . 4 . 563–580. 10.2307/212415 . 212415. Lee . L. P. . Laurence Patrick Lee. 1973. The Conformal Tetrahedric Projection with some Practical Applications. The Cartographic Journal. 10 . 1 . 22–28. 10.1179/caj.1973.10.1.22. Book: Lee, L. P. . Laurence Patrick Lee. 1976. Conformal Projections Based on Elliptic Functions. Toronto . B. V. Gutsell, York University. Cartographica Monographs . 16. limited. 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.