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 projection was developed by Émile Guyou of France in 1887.^{ [1] }

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.^{ [2] }

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.

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

- Each hemisphere is represented as a square, the sphere as a rectangle of aspect ratio 2:1.
- The part where the exaggeration of scale amounts to double that at the centre of each square is only 9% of the area of the sphere, against 13% for the Mercator and 50% for the stereographic
^{ [3] } - The curvature of lines representing great circles is, in every case, very slight, over the greater part of their length.
^{ [3] } - It is conformal everywhere except at the corners of the square that corresponds to each hemisphere, where two meridians change direction abruptly twice each; the Equator is represented by a horizontal line.
- It can be tessellated in all directions.

- The Adams hemisphere-in-a-square projection and the Peirce quincuncial projection are different aspects of the same underlying Schwarz–Christoffel mapping. Such mappings are transformations of half a stereographic projection.

In complex analysis, the **Riemann mapping theorem** states that if *U* is a non-empty simply connected open subset of the complex number plane **C** which is not all of **C**, then there exists a biholomorphic mapping *f* from *U* onto the open unit disk

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

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 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.

In complex analysis, a **Schwarz–Christoffel mapping** is a conformal transformation of the upper half-plane onto the interior of a simple polygon. Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces and fluid dynamics. They are named after Elwin Bruno Christoffel and Hermann Amandus Schwarz.

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 arbitary directional extent.

The **Universal Transverse Mercator** (**UTM**) is a 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.

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.

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.

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 **Peirce quincuncial projection** 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.

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 **Gall stereographic projection**, presented by James Gall in 1855, is a cylindrical projection. It is neither equal-area nor conformal but instead tries to balance the distortion inherent in any projection.

**Web Mercator**, **Google Web Mercator**, **Spherical Mercator**, **WGS 84 Web Mercator** or **WGS 84/Pseudo-Mercator** is a variant of the Mercator 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, Mapbox, Bing Maps, OpenStreetMap, Mapquest, Esri, and many others. Its official EPSG identifier is EPSG:3857, although others have been used historically.

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.

- ↑ Snyder, John P. (1993).
*Flattening the Earth*. University of Chicago. ISBN 0-226-76746-9. - ↑ L.P. Lee (1976). "Conformal Projections based on Elliptic Functions".
*Cartographica*.**13**(Monograph 16, supplement No. 1 to Canadian Cartographer). - 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.

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