Peirce quincuncial projection
Encyclopedia
The Peirce quincuncial projection is a conformal map projection
(except for four points where its conformality fails) that presents the sphere as a square. It was developed by Charles Sanders Peirce in 1879.
led to general techniques for conformal mapping, where points of a flat surface are handled as numbers on the complex plane
.
While working at the U.S. Coast and Geodetic Survey, the American philosopher Charles Sanders Peirce published his projection in 1879 (Peirce 1879), having been inspired by H.A. Schwarz's 1869 conformal transformation of a circle onto a polygon of n sides (known as the Schwarz–Christoffel mapping). In the normal aspect, Peirce's projection presents the northern hemisphere
in a square; the southern hemisphere
is split into four isosceles triangles symmetrically surrounding the first one, akin to star-like projections. In effect, the whole map is a square, inspiring Peirce to call his projection quincuncial, after the arrangement of five items in a quincunx
.
After Peirce presented his projection, two other cartographers developed similar projections of the hemisphere (or the whole sphere, after a suitable rearrangement) on a square: Guyou in 1887 and Adams in 1925 (Lee, 1976). The three projections are transversal versions of each other (see related projections below).
with a pole at infinity, by means of an elliptic function" (Peirce, 1879). The Peirce quincuncial is really a projection of the hemisphere, but its tessellation properties (see below) permit its use for the entire sphere. Peirce's projection maps the interior of a circle (corresponding to each hemisphere, which were created by projecting them using the stereographic projection) onto the interior of a square (using the Schwarz–Christoffel mapping) (Lee, 1976).
A point P on the Earth's surface, a distance p from the north pole
with longitude
θ and latitude
λ is first mapped to a point (p, θ) of the plane through the equator, viewed as the complex plane with coordinate w; this w coordinate is then mapped to another point (x, y) of the complex plane (given the coordinate z) by an elliptic function of the first kind. Using Gudermann's notation for Jacobi's elliptic functions
, the relationships are
Furthermore, the four triangles of the second hemisphere of Peirce quincuncial projection can be rearranged as another square that is placed next to the square that corresponds to the first hemisphere, resulting in a rectangle with aspect ratio of 2:1; this arrangement is equivalent to the transverse aspect of the Guyou hemisphere-in-a-square projection
(Snyder, 1993).
Like many other projections based upon complex numbers, the Peirce quincuncial has been rarely used for geographic purposes. One of the few recorded cases is in 1946, when it was used by the U.S. Coast and Geodetic Survey world map of air routes (Snyder, 1993).
It has been used recently to present spherical panoramas for practical as well as aesthetic purposes, where it can present the entire sphere with most areas being recognizable (German et al. 2007).
(the pole is placed at the corner of the square). Its four singularities are at the north pole, the south pole, on the equator at 25°W
, and on the equator at 155°E, in the Arctic, Atlantic, and Pacific oceans, and in Antarctica. That great circle divides the traditional western and eastern hemisphere
.
In oblique aspect (45 degrees) of one hemisphere becomes the Guyou hemisphere-in-a-square projection
(the pole is placed in the middle of the edge of the square). Its four singularities are at 45 degrees north and south latitude on the great circle composed of the 20°W
meridian and the 160°E meridians, in the Atlantic and Pacific oceans. That great circle divides the traditional western and eastern hemispheres.
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...
(except for four points where its conformality fails) that presents the sphere as a square. It was developed by Charles Sanders Peirce in 1879.
History
The maturation of complex analysisComplex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
led to general techniques for conformal mapping, where points of a flat surface are handled as numbers on the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
.
While working at the U.S. Coast and Geodetic Survey, the American philosopher Charles Sanders Peirce published his projection in 1879 (Peirce 1879), having been inspired by H.A. Schwarz's 1869 conformal transformation of a circle onto a polygon of n sides (known as the Schwarz–Christoffel mapping). In the normal aspect, Peirce's projection presents the northern hemisphere
Northern Hemisphere
The Northern Hemisphere is the half of a planet that is north of its equator—the word hemisphere literally means “half sphere”. It is also that half of the celestial sphere north of the celestial equator...
in a square; the southern hemisphere
Southern Hemisphere
The Southern Hemisphere is the part of Earth that lies south of the equator. The word hemisphere literally means 'half ball' or "half sphere"...
is split into four isosceles triangles symmetrically surrounding the first one, akin to star-like projections. In effect, the whole map is a square, inspiring Peirce to call his projection quincuncial, after the arrangement of five items in a quincunx
Quincunx
A quincunx is a geometric pattern consisting of five points arranged in a cross, that is five coplanar points, four of them forming a square or rectangle and a fifth at its center...
.
After Peirce presented his projection, two other cartographers developed similar projections of the hemisphere (or the whole sphere, after a suitable rearrangement) on a square: Guyou in 1887 and Adams in 1925 (Lee, 1976). The three projections are transversal versions of each other (see related projections below).
Formal description
The Peirce quincuncial projection is "formed by transforming the stereographic projectionStereographic projection
The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...
with a pole at infinity, by means of an elliptic function" (Peirce, 1879). The Peirce quincuncial is really a projection of the hemisphere, but its tessellation properties (see below) permit its use for the entire sphere. Peirce's projection maps the interior of a circle (corresponding to each hemisphere, which were created by projecting them using the stereographic projection) onto the interior of a square (using the Schwarz–Christoffel mapping) (Lee, 1976).
A point P on the Earth's surface, a distance p from the north pole
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...
with longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
θ and latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
λ is first mapped to a point (p, θ) of the plane through the equator, viewed as the complex plane with coordinate w; this w coordinate is then mapped to another point (x, y) of the complex plane (given the coordinate z) by an elliptic function of the first kind. Using Gudermann's notation for Jacobi's elliptic functions
Jacobi's elliptic functions
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications...
, the relationships are
Properties
According to Peirce, his projection has the following properties (Peirce, 1879):- The sphere is presented in a square.
- The part where the exaggeration of scale amounts to double that at the centre is only 9% of the area of the sphere, against 13% for the Mercator and 50% for the stereographic
- 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 four corners of the inner hemisphere (thus the midpoints of edges of the projection), where the equator and four meridians change direction abruptly (the equator is represented by a square). These are singularitiesMathematical singularityIn mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
where differentiability fails. - It can be tessellated in all directions.
Tiled Peirce quincuncial maps
The projection tessellates the plane; i.e., repeated copies can completely cover (tile) an arbitrary area, each copy's features exactly matching those of its neighbors. See this image for an example.
Furthermore, the four triangles of the second hemisphere of Peirce quincuncial projection can be rearranged as another square that is placed next to the square that corresponds to the first hemisphere, resulting in a rectangle with aspect ratio of 2:1; this arrangement is equivalent to the transverse aspect of the Guyou hemisphere-in-a-square projection
Guyou hemisphere-in-a-square projection
The Guyou hemisphere-in-a-square projection is a conformal map projection for the hemisphere . It is an oblique aspect of the Peirce quincuncial projection...
(Snyder, 1993).
Known uses
It has been used recently to present spherical panoramas for practical as well as aesthetic purposes, where it can present the entire sphere with most areas being recognizable (German et al. 2007).
Related projections
In transverse aspect, one hemisphere becomes the Adams hemisphere-in-a-square projectionAdams hemisphere-in-a-square projection
The Adams-hemisphere-in-a-square is a conformal map projection for a hemisphere . It is a transverse version of the Peirce quincuncial projection. When it is used to represent the entire sphere it is known as the Adams doubly periodic projection....
(the pole is placed at the corner of the square). Its four singularities are at the north pole, the south pole, on the equator at 25°W
25th meridian west
The meridian 25° west of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, Greenland, the Atlantic Ocean, Cape Verde Islands, the Southern Ocean, and Antarctica to the South Pole....
, and on the equator at 155°E, in the Arctic, Atlantic, and Pacific oceans, and in Antarctica. That great circle divides the traditional western and eastern hemisphere
Eastern Hemisphere
The Eastern Hemisphere, also Eastern hemisphere or eastern hemisphere, is a geographical term for the half of the Earth that is east of the Prime Meridian and west of 180° longitude. It is also used to refer to Europe, Asia, Africa, and Australasia, vis-à-vis the Western Hemisphere, which includes...
.
In oblique aspect (45 degrees) of one hemisphere becomes the Guyou hemisphere-in-a-square projection
Guyou hemisphere-in-a-square projection
The Guyou hemisphere-in-a-square projection is a conformal map projection for the hemisphere . It is an oblique aspect of the Peirce quincuncial projection...
(the pole is placed in the middle of the edge of the square). Its four singularities are at 45 degrees north and south latitude on the great circle composed of the 20°W
20th meridian west
The meridian 20° west of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, Greenland, Iceland, the Atlantic Ocean, the Southern Ocean, and Antarctica to the South Pole....
meridian and the 160°E meridians, in the Atlantic and Pacific oceans. That great circle divides the traditional western and eastern hemispheres.