Guyou hemisphere-in-a-square projection

Last updated
Guyou doubly periodic projection of the world. Guyou doubly periodic projection SW.JPG
Guyou doubly periodic projection of the world.
The Guyou hemisphere-in-a-square projection with Tissot's indicatrix of deformation. The indicatrix is omitted at the singular points. At those points the deformation is infinite; the indicatrix would be infinite in size. Guyou with Tissot's Indicatrices of Distortion.svg
The Guyou hemisphere-in-a-square projection with Tissot's indicatrix of deformation. The indicatrix is omitted at the singular points. At those points the deformation is infinite; the indicatrix would be infinite in size.

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.

Contents

History

The projection was developed by Émile Guyou  [ fr ] of France in 1887. [1] [2]

Formal description

The projection can be computed as an oblique aspect of the Peirce quincuncial projection by rotating the axis 45 degrees. It can also be computed by rotating the coordinates 45 degrees before computing the stereographic projection; this projection is then remapped into a square whose coordinates are then rotated 45 degrees. [3]

The projection is conformal except for the four corners of each hemisphere's square. Like other conformal polygonal projections, the Guyou is a Schwarz–Christoffel mapping.

Properties

Its properties are very similar to those of the Peirce quincuncial:

See also

Related Research Articles

<span class="mw-page-title-main">Latitude</span> Geographic coordinate specifying north–south position

In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole, with 0° at the Equator. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude and longitude are used together as a coordinate pair to specify a location on the surface of the Earth.

<span class="mw-page-title-main">Mercator projection</span> Cylindrical conformal map projection

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 it is unique in representing north as up and south as down everywhere while preserving local directions and shapes. The map is thereby conformal. 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 increasing latitude to become infinite at the poles. As a result, landmasses such as Greenland, Antarctica, Canada and Russia appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

<span class="mw-page-title-main">Conformal map</span> Mathematical function which preserves angles

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

<span class="mw-page-title-main">Map projection</span> Systematic representation of the surface of a sphere or ellipsoid onto a plane

In cartography, a map projection is any of a broad set of transformations employed to represent the curved two-dimensional surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography.

<span class="mw-page-title-main">Stereographic projection</span> Particular mapping that projects a sphere onto a plane

In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere, onto a plane perpendicular to the diameter through the point. It is a smooth, bijective function from the entire sphere except the center of projection to the entire plane. It maps circles on the sphere to circles or lines on the plane, and is conformal, meaning that it preserves angles at which curves meet and thus locally approximately preserves shapes. It is neither isometric nor equiareal.

<span class="mw-page-title-main">Projected coordinate system</span> Cartesian geographic coordinate system

A projected coordinate system, also known as a projected coordinate reference system, a planar coordinate system, or grid reference system, is a type of spatial reference system that represents locations on the Earth using cartesian coordinates (x,y) on a planar surface created by a particular map projection. Each projected coordinate system, such as "Universal Transverse Mercator WGS 84 Zone 26N," is defined by a choice of map projection (with specific parameters), a choice of geodetic datum to bind the coordinate system to real locations on the earth, an origin point, and a choice of unit of measure. Hundreds of projected coordinate systems have been specified for various purposes in various regions.

<span class="mw-page-title-main">Planisphere</span>

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.

<span class="mw-page-title-main">Transverse Mercator projection</span> Adaptation of the standard Mercator projection

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 Universal Transverse Mercator. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

<span class="mw-page-title-main">Oblique Mercator projection</span> Map projection

The oblique Mercator map projection is an adaptation of the standard Mercator projection. The oblique version is sometimes used in national mapping systems. When paired with a suitable geodetic datum, the oblique Mercator delivers high accuracy in zones less than a few degrees in arbitrary directional extent.

<span class="mw-page-title-main">Universal Transverse Mercator coordinate system</span> System for assigning planar coordinates to locations on the surface of the Earth.

The Universal Transverse Mercator (UTM) is a map projection system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means it ignores altitude and treats the earth as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the x, y coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system.

<span class="mw-page-title-main">Universal polar stereographic coordinate system</span>

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.

<span class="mw-page-title-main">Tissot's indicatrix</span> Characterization of distortion in map protections

In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

<span class="mw-page-title-main">Peirce quincuncial projection</span> Conformal map 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 square. The name quincuncial refers to this arrangement: the north pole at the center and quarters of the south pole in the corners form a quincunx pattern like the pips on the five face of a traditional die. The projection has the distinctive property that it forms a seamless square tiling of the plane, conformal except at four singular points along the equator.

<span class="mw-page-title-main">Adams hemisphere-in-a-square projection</span> Conformal map 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.

<span class="mw-page-title-main">Web Mercator projection</span> Mercator variant map projection

Web Mercator, Google Web Mercator, Spherical Mercator, WGS 84 Web Mercator or WGS 84/Pseudo-Mercator is a variant of the Mercator map projection and is the de facto standard for Web mapping applications. It rose to prominence when Google Maps adopted it in 2005. It is used by virtually all major online map providers, including Google Maps, CARTO, Mapbox, Bing Maps, OpenStreetMap, Mapquest, Esri, and many others. Its official EPSG identifier is EPSG:3857, although others have been used historically.

<span class="mw-page-title-main">AuthaGraph projection</span> Polyhedral compromise map projection

AuthaGraph is an approximately equal-area world map projection invented by Japanese architect Hajime Narukawa in 1999. The map is made by equally dividing a spherical surface into 96 triangles, transferring it to a tetrahedron while maintaining area proportions, and unfolding it onto a rectangle: it is a polyhedral map projection. The map substantially preserves sizes and shapes of all continents and oceans while it reduces distortions of their shapes, as inspired by the Dymaxion map. The projection does not have some of the major distortions of the Mercator projection, like the expansion of countries in far northern latitudes, and allows for Antarctica to be displayed accurately and in whole. Triangular world maps are also possible using the same method. The name is derived from "authalic" and "graph".

<span class="mw-page-title-main">Stereographic map projection</span> Type of conformal map projection

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.

<span class="mw-page-title-main">Spherical conic</span>

In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section in the plane, and as in the planar case, a spherical conic can be defined as the locus of points the sum or difference of whose great-circle distances to two foci is constant. By taking the antipodal point to one focus, every spherical ellipse is also a spherical hyperbola, and vice versa. As a space curve, a spherical conic is a quartic, though its orthogonal projections in three principal axes are planar conics. Like planar conics, spherical conics also satisfy a "reflection property": the great-circle arcs from the two foci to any point on the conic have the tangent and normal to the conic at that point as their angle bisectors.

<span class="mw-page-title-main">Polyhedral map projection</span> Type of 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 map projection is Buckminster Fuller's Dymaxion map. When the spherical polyhedron faces are transformed to the faces of an ordinary polyhedron instead of laid flat in a plane, the result is a polyhedral globe.

References

  1. E. Guyou (1887) "Nouveau système de projection de la sphère: Généralisation de la projection de Mercator", Annales Hydrographiques, Ser. 2, Vol. 9, 16–35. https://www.retronews.fr/journal/annales-hydrographiques/1-janvier-1887/1877/4868382/23
  2. Snyder, John P. (1993). Flattening the Earth. University of Chicago. ISBN   0-226-76746-9.
  3. L.P. Lee (1976). "Conformal Projections based on Elliptic Functions". Cartographica. 13 (Monograph 16, supplement No. 1 to Canadian Cartographer).
  4. 1 2 C.S. Peirce (December 1879). "A Quincuncial Projection of the Sphere". American Journal of Mathematics. The Johns Hopkins University Press. 2 (4): 394–396. doi:10.2307/2369491. JSTOR   2369491.