The **Hammer retroazimuthal projection** is a modified azimuthal proposed by Ernst Hermann Heinrich Hammer in 1910. As a retroazimuthal projection, azimuths (directions) are correct from any point to the designated center point.^{ [1] } Additionally, all distances from the center of the map are proportional to what they are on the globe. In whole-world presentation, the back and front hemispheres overlap, making the projection a non-injective function. The back hemisphere can be rotated 180° to avoid overlap, but in this case, any azimuths measured from the back hemisphere must be corrected.

Given a radius *R* for the projecting globe, the projection is defined as:

where

and

The latitude and longitude of the point to be plotted are *φ* and *λ* respectively, and the center point to which all azimuths are to be correct is given as *φ*_{1} and *λ*_{0}.

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.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In navigation, a **rhumb line**, **rhumb**, or **loxodrome** is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north.

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.

The use of **orthographic projection in cartography** dates back to antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection, in which the sphere is projected onto a tangent plane or secant plane. The *point of perspective* for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.

The **Craig retroazimuthal** map projection was created by James Ireland Craig in 1909. It is a modified cylindrical projection. As a retroazimuthal projection, it preserves directions from everywhere to one location of interest that is configured during construction of the projection. The projection is sometimes known as the **Mecca projection** because Craig, who had worked in Egypt as a cartographer, created it to help Muslims find their qibla. In such maps, Mecca is the configurable location of interest.

The **Mollweide projection** is an equal-area, pseudocylindrical map projection generally used for global maps of the world or night sky. It is also known as the **Babinet projection**, **homalographic projection**, **homolographic projection**, and **elliptical projection**. The projection trades accuracy of angle and shape for accuracy of proportions in area, and as such is used where that property is needed, such as maps depicting global distributions.

The **azimuthal equidistant projection** is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point. A useful application for this type of projection is a polar projection which shows all meridians as straight, with distances from the pole represented correctly. The flag of the United Nations contains an example of a polar azimuthal equidistant projection.

The **scale** of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

The **Littrow projection** is a map projection developed by Joseph Johann von Littrow in 1833. It is the only conformal, retroazimuthal map projection. As a retroazimuthal projection, the Littrow shows directions, or azimuths, correctly from any point to the center of the map.

The **Albers equal-area conic projection**, or **Albers projection**, is a conic, equal area map projection that uses two standard parallels. Although scale and shape are not preserved, distortion is minimal between the standard parallels.

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 **Aitoff projection** is a modified azimuthal map projection proposed by David A. Aitoff in 1889. Based on the equatorial form of the azimuthal equidistant projection, Aitoff first halves longitudes, then projects according to the azimuthal equidistant, and then stretches the result horizontally into a 2:1 ellipse to compensate for having halved the longitudes. Expressed simply:

The **Hammer projection** is an equal-area map projection described by Ernst Hammer in 1892. Using the same 2:1 elliptical outer shape as the Mollweide projection, Hammer intended to reduce distortion in the regions of the outer meridians, where it is extreme in the Mollweide.

In cartography, the **cylindrical equal-area projection** is a family of cylindrical, equal-area map projections.

The **armadillo projection** is a map projection used for world maps. It is neither conformal nor equal-area but instead affords a view evoking a perspective projection while showing most of the globe instead of the half or less that a perspective would. The projection was presented in 1943 by Erwin Raisz (1893–1968) as part of a series of "orthoapsidal" projections, which are perspectives of the globe projected onto various surfaces. This one in the series has the globe projected onto half a torus. Raisz singled it out and named it the "armadillo" projection.

The **American polyconic** map projection is a map projection used for maps of the United States and regions of the United States beginning early in the 19th century. It belongs to the polyconic projection class, which consists of map projections whose parallels are non-concentric circular arcs except for the equator, which is straight. Often the American polyconic is simply called the **polyconic projection**.

The **Nicolosi globular projection** is a map projection invented about the year 1,000 by the Iranian polymath al-Biruni. As a circular representation of a hemisphere, it is called *globular* because it evokes a globe. It can only display one hemisphere at a time and so normally appears as a "double hemispheric" presentation in world maps. The projection came into use in the Western world starting in 1660, reaching its most common use in the 19th century. As a "compromise" projection, it preserves no particular properties, instead giving a balance of distortions.

The **Strebe 1995 projection**, **Strebe projection**, **Strebe lenticular equal-area projection**, or **Strebe equal-area polyconic projection** is an equal-area map projection presented by Daniel "daan" Strebe in 1994. Strebe designed the projection to keep all areas proportionally correct in size; to push as much of the inevitable distortion as feasible away from the continental masses and into the Pacific Ocean; to keep a familiar equatorial orientation; and to do all this without slicing up the map.

The **Eckert-Greifendorff projection** is an equal-area map projection described by Max Eckert-Greifendorff in 1935. Unlike his previous six projections, It is not pseudocylindrical.

- ↑ Snyder, John P. (1993).
*Flattening the Earth: Two Thousand Years of Map Projections*. Chicago: University of Chicago Press. pp. 228–229. ISBN 0-226-76747-7 . Retrieved 2011-11-14.

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