# Peirce quincuncial projection

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The Peirce quincuncial projection [1] is a conformal map projection developed by Charles Sanders Peirce in 1879. The projection has the distinctive property that it can be tiled ad infinitum on the plane, with edge-crossings being completely smooth except for four singular points per tile. The projection has seen use in digital photography for portraying 360° views. The description quincuncial refers to the arrangement of four quadrants of the globe around the center hemisphere in an overall square pattern. Typically the projection is oriented such that the north pole lies at the center.

## History

The maturation of complex analysis 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), [2] 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. [3] 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 projection with a pole at infinity, by means of an elliptic function". [4] The Peirce quincuncial is really a projection of the hemisphere, but its tessellation properties (see below) permit its use for the entire sphere. The projection maps the interior of a circle onto the interior of a square by means of the Schwarz–Christoffel mapping, as follows: [5]

${\displaystyle \operatorname {sd} \left({\sqrt {2}}w,1/{\sqrt {2}}\right)={\sqrt {2}}r}$

where sd is the ratio of two Jacobi elliptic functions: sn/dn; w is the mapped point on the plane as a complex number (w = x + iy); and r is the stereographic projection with a scale of 1/2 at the center. An elliptic integral of the first kind can be used to solve for w. The comma notation used for sd(u,k) means that 1/2 is the modulus for the elliptic function ratio, as opposed to the parameter [which would be written sd(u|m)] or the amplitude [which would be written sd(u\α)]. The mapping has a scale factor of 1/2 at the center, like the generating stereographic projection.

## 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 projection and 50% for the stereographic projection.
• 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 singularities 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 the example to the right). 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. [6]

## Known uses

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. [6] 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. [7]

In transverse aspect, one hemisphere becomes the Adams hemisphere-in-a-square 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, and on the equator at 155°E, in the Arctic, Atlantic, and Pacific oceans, and in Antarctica. [8] That great circle divides the traditional Western and Eastern hemispheres.

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. [8] That great circle divides the traditional western and eastern hemispheres.

## Related Research Articles

In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications.

The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator, but accelerates with latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

A geographic coordinate system is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector. A common choice of coordinates is latitude, longitude and elevation. To specify a location on a plane requires a map projection.

In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.

In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. Every distinct map projection distorts in a distinct way, by definition. The study of map projections is the characterization of these distortions. There is no limit to the number of possible map projections. Projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds. However, "map projection" refers specifically to a cartographic projection.

In geometry, the stereographic projection is a particular mapping (function) 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 preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

In mathematics, the open unit disk around P, is the set of points whose distance from P is less than 1:

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form

In astronomy, a planisphere is a star chart analog computing instrument in the form of two adjustable disks that rotate on a common pivot. It can be adjusted to display the visible stars for any time and date. It is an instrument to assist in learning how to recognize stars and constellations. The astrolabe, an instrument that has its origins in Hellenistic astronomy, is a predecessor of the modern planisphere. The term planisphere contrasts with armillary sphere, where the celestial sphere is represented by a three-dimensional framework of rings.

The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the UTM. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

A gnomonic map projection displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achieved by casting surface points of the sphere onto a tangent plane, each landing where a ray from the center of the sphere passes through the point on the surface and then on to the plane. No distortion occurs at the tangent point, but distortion increases rapidly away from it. Less than half of the sphere can be projected onto a finite map. Consequently, a rectilinear photographic lens, which is based on the gnomonic principle, cannot image more than 180 degrees.

The universal polar stereographic (UPS) coordinate system is used in conjunction with the universal transverse Mercator (UTM) coordinate system to locate positions on the surface of the earth. Like the UTM coordinate system, the UPS coordinate system uses a metric-based cartesian grid laid out on a conformally projected surface. UPS covers the Earth's polar regions, specifically the areas north of 84°N and south of 80°S, which are not covered by the UTM grids, plus an additional 30 minutes of latitude extending into UTM grid to provide some overlap between the two systems.

A pole figure is a graphical representation of the orientation of objects in space. For example, pole figures in the form of stereographic projections are used to represent the orientation distribution of crystallographic lattice planes in crystallography and texture analysis in materials science.

The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the Lambert zenithal equal-area 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 Adams hemisphere-in-a-square is a conformal map projection for a hemisphere. It is a transverse version of the Peirce quincuncial projection, and is named after American cartographer Oscar Sherman Adams, who published it in 1925. When it is used to represent the entire sphere it is known as the Adams doubly periodic projection. Like many conformal projections, conformality fails at certain points, in this case at the four corners.

In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth is preserved in the image of the projection, i.e. the projection is a conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, then their images on a map with a conformal projection cross at a 39° angle.

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers.

The stereographic projection, also known as the planisphere projection or the azimuthal conformal projection, is a conformal map projection whose use dates back to antiquity. Like the orthographic projection and gnomonic projection, the stereographic projection is an azimuthal projection, and when on a sphere, also a perspective projection.

## References

1. A Quincuncial Projection of the Sphere by Charles Sanders Peirce. 1890.
I. Frischauf. Bemerkungen zu C. S. Peirce Quincuncial Projection. (Tr., Comments on C. S. Peirce Quincuncial Projection.)
A Treatise on Projections by Thomas Craig. U.S. Government Printing Office, 1882. p 132
Science, Volume 11. Moses King, 1900. p 186
2. (Lee, 1976) gives 1877 as the year in which the projection was conceived, citing "US Coast Survey Report for the Year Ending with June 1877", 191192.
3. Lee, L. P. (1976). "Conformal Projections Based on Jacobian Elliptic Functions". Cartographica. 13: 67–101. doi:10.3138/X687-1574-4325-WM62.CS1 maint: ref=harv (link)
4. Peirce, C.S. (1879). "A quincuncial projection of the sphere". American Journal of Mathematics. 2 (4): 394–396. doi:10.2307/2369491. JSTOR   2369491.
5. Lee, L.P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica. pp. 67–69.
6. Snyder, John P. (1989). An Album of Map Projections, Professional Paper 1453 (PDF). US Geological Survey. pp. 190, 236.
7. German, Daniel; d'Angelo, Pablo; Gross, Michael; Postle, Bruno (June 2007). "New Methods to Project Panoramas for Practical and Aesthetic Purposes". Proceedings of Computational Aesthetics 2007. Banff: Eurographics. pp. 15–22.
8. Carlos A. Furuti. Map Projections:Conformal Projections.
• Peirce, C. S. (1877/1879), "Appendix No. 15. A Quincuncial Projection of the Sphere", Report of the Superintendent of the United States Coast Survey Showing the Progress of the Survey for Fiscal Year Ending with June 1877, pp. 191–194 followed by 25 progress sketches including (25th) the illustration (the map itself). Full Report submitted to the Senate December 26, 1877 and published 1880 (see further below).
• Article first published December 1879, American Journal of Mathematics2 (4): 394–397 (without the sketches except final map), Google Books Eprint (Google version of map is partly botched), JSTOR Eprint, doi:10.2307/2369491. AJM version reprinted in Writings of Charles S. Peirce4:68–71.
• Article reprinted 1880 including publication of all sketches, in the full Report, by the U.S. Government Printing Office, Washington, D.C. NOAA PDF Eprint, link goes to Peirce's article on Report's p. 191, PDF's p. 215. NOAA's PDF lacks the sketches and map and includes broken link to their planned online location, NOAA's Historical Map and Chart Collection, where they do not seem to be as of 7/19/2010. Google Books Eprint (Google botched the sketches and partly botched the illustration (the map itself).) Note: Other Google edition of 1877 Coast Survey Report completely omits the pages of sketches including the illustration (the map).