The **Tobler hyperelliptical projection** is a family of equal-area pseudocylindrical projections that may be used for world maps. Waldo R. Tobler introduced the construction in 1973 as the *hyperelliptical* projection, now usually known as the Tobler hyperelliptical projection.^{ [1] }

As with any pseudocylindrical projection, in the projection’s normal aspect,^{ [2] } the parallels of latitude are parallel, straight lines. Their spacing is calculated to provide the equal-area property. The projection blends the cylindrical equal-area projection with meridians of longitude that follow a particular kind of curve known as * superellipses *^{ [3] } or Lamé curves or sometimes as hyperellipses. The curve is described by *x*^{k} + *y*^{k} = *γ*^{k}. The relative weight of the cylindrical equal-area projection is given as *α*, ranging from all cylindrical equal-area with *α* = 1 to all hyperellipses with *α* = 0.

When *α* = 0 and *k* = 1 the projection degenerates to the Collignon projection; when *α* = 0, *k* = 2, and *γ* ≈ 1.2731 the projection becomes the Mollweide projection.^{ [4] } Tobler favored the parameterization shown with the top illustration; that is, *α* = 0, *k* = 2.5, and *γ* = 1.183136.

The **Gall–Peters projection** is a rectangular map projection that maps all areas such that they have the correct sizes relative to each other. Like any equal-area projection, it achieves this goal by distorting most shapes. The projection is a particular example of the cylindrical equal-area projection with latitudes 45° north and south as the regions on the map that have no distortion.

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. 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 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. The study of map projections is the characterization of the 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.

A **superellipse**, also known as a **Lamé curve** after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape.

**Johann Heinrich Lambert** was a Swiss polymath who made important contributions to the subjects of mathematics, physics, philosophy, astronomy and map projections.

In the mathematical field of differential geometry, the **Riemann curvature tensor** or **Riemann–Christoffel tensor** is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measure the failure of second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is *flat*, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In plane Euclidean geometry, a **rhombus** is a quadrilateral whose four sides all have the same length. Another name is **equilateral quadrilateral**, since equilateral means that all of its sides are equal in length. The rhombus is often called a **diamond**, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a **lozenge**, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

The **Robinson projection** is a map projection of a world map which shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of readily showing the whole globe as a flat image.

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.

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 **Goode homolosine projection** is a pseudocylindrical, equal-area, composite map projection used for world maps. Normally it is presented with multiple interruptions. Its equal-area property makes it useful for presenting spatial distribution of phenomena.

A **Lambert conformal conic projection** (**LCC**) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication *Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten*.

In mathematics, the **differential geometry of surfaces** deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: *extrinsically*, relating to their embedding in Euclidean space and *intrinsically*, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

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

In mathematics, a **superelliptic curve** is an algebraic curve defined by an equation of the form

The **Boggs eumorphic projection** is a pseudocylindrical, equal-area map projection used for world maps. Normally it is presented with multiple interruptions. Its equal-area property makes it useful for presenting spatial distribution of phenomena. The projection was developed in 1929 by Samuel Whittemore Boggs (1889–1954) to provide an alternative to the Mercator projection for portraying global areal relationships. Boggs was geographer for the United States Department of State from 1924 until his death. The Boggs eumorphic projection has been used occasionally in textbooks and atlases.

The **Equal Earth map projection** is an equal-area pseudocylindrical projection for world maps, invented by Bojan Šavrič, Bernhard Jenny, and Tom Patterson in 2018. It is inspired by the widely used Robinson projection, but unlike the Robinson projection, retains the relative size of areas. The projection equations are simple to implement and fast to evaluate.

In differential geometry a **translation surface** is a surface that is generated by translations:

In map projection, **equal-area maps** preserve area measure, generally distorting shapes in order to do that. Equal-area maps are also called **equivalent** or **authalic**.

- ↑ Snyder, John P. (1993).
*Flattening the Earth: 2000 Years of Map Projections*. Chicago: University of Chicago Press. p. 220. - ↑ The Tobler Hyperelliptical Projection on the Center for Spatially Integrated Social Science's site
- ↑ "Superellipse" in MathWorld encyclopedia
- ↑ Tobler, Waldo (1973). "The hyperelliptical and other new pseudocylindrical equal area map projections".
*Journal of Geophysical Research*.**78**(11): 1753–1759. Bibcode:1973JGR....78.1753T. CiteSeerX 10.1.1.495.6424 . doi:10.1029/JB078i011p01753.

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