Lee conformal world in a tetrahedron

{{Short description|Polyhedral conformal map projection}}

File:Lee Conformal World in a Tetrahedron projection.png

File:Lee Tetrahedral (triangular) with Tissot's Indicatrices of Distortion.svg.]]

File:Lee tetrahedral projection tessellated.jpg

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.{{cite journal

| title = Some Conformal Projections Based on Elliptic Functions

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

| year = 1965

| journal = Geographical Review

| volume = 55 | issue = 4 | pages = 563–580

| doi = 10.2307/212415 | jstor = 212415

}} {{pb}}

{{cite journal

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

| year = 1973

| title = The Conformal Tetrahedric Projection with some Practical Applications

| journal = The Cartographic Journal

| volume = 10 | number = 1 | pages = 22–28

| doi = 10.1179/caj.1973.10.1.22

}} {{pb}}

{{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].

Coordinates from a spherical datum can be transformed into Lee conformal projection coordinates with the following formulas, where $\backslash lambda$ is the longitude and $\backslash varphi$ the latitude:

: $2\; \backslash operatorname\{sm\}w\backslash ,\backslash operatorname\{cm\}w\; =\; 2^\{5/6\}\backslash exp(i\backslash lambda)\; \backslash tan\backslash bigl(\backslash tfrac14\backslash pi\; -\; \backslash tfrac12\backslash varphi\backslash bigr)$

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.