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The Lee conformal world in a tetrahedron is a polyhedral, conformal
The Lee conformal world in a tetrahedron is a polyhedral, conformal map projection
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...
that projects the globe onto a tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
using Dixon elliptic functions
In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions ( doubly periodic meromorphic functions on the complex plane) that map from each regular hexagon in a hexagonal tiling to the whole complex plane. Because these f ...
. 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
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
infinitely in the plane. It was developed by L. P. Lee in 1965.
Coordinates from a spherical datum
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted. ...
can be transformed into Lee conformal projection coordinates with the following formulas, where is the longitude and the latitude:
:
where
:
and sm and cm are Dixon elliptic functions
In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions ( doubly periodic meromorphic functions on the complex plane) that map from each regular hexagon in a hexagonal tiling to the whole complex plane. Because these f ...
.
Since there is no elementary expression for these functions, Lee suggests using the 28th degree MacLaurin series
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ( ...
.
See also
*List of map projections
This is a summary of map projections that have articles of their own on Wikipedia or that are otherwise notable
Notability is the property
of being worthy of notice, having fame, or being considered to be of a high degree of interest, signif ...
* AuthaGraph projection
AuthaGraph is an approximately Map projection#Equal-area, equal-area world map map projection, projection invented by Japanese architect Hajime Narukawa in 1999. The map is made by equally dividing a spherical surface into 96 triangles, transferr ...
, another tetrahedral projection, 1999
* Dymaxion map
The Dymaxion map or Fuller map is a projection of a world map onto the surface of an icosahedron, which can be unfolded and flattened to two dimensions. The flat map is heavily interrupted in order to preserve shapes and sizes.
The projection wa ...
, 1943
* Peirce quincuncial projection
The Peirce quincuncial projection is the conformal map projection from the sphere to an unfolded square dihedron, developed by Charles Sanders Peirce in 1879. Each octant projects onto an isosceles right triangle, and these are arranged into a s ...
, 1879
* Polyhedral map projection
A polyhedral map projection is a map projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the polyhedron is transformed to a polygon or other shape in the plane. The best-known polyhedral ...
, earliest known is by Leonardo da Vinci, 1514
References
Conformal projections {{cartography-stub