Tobler hyperelliptical projection

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Tobler hyperelliptical projection of the world; a = 0, g = 1.18314, k = 2.5 Tobler hyperelliptical projection SW.jpg
Tobler hyperelliptical projection of the world; α = 0, γ = 1.18314, k = 2.5
The Tobler hyperelliptical projection with Tissot's indicatrix of deformation; a = 0, k = 3 Tobler Hyperelliptical with Tissot's Indicatrices of Distortion.svg
The Tobler hyperelliptical projection with Tissot's indicatrix of deformation; α = 0, k = 3

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]

World map map of the surface of the Earth

A world map is a map of most or all of the surface of the Earth. World maps form a distinctive category of maps due to the problem of projection. Maps by necessity distort the presentation of the earth's surface. These distortions reach extremes in a world map. The many ways of projecting the earth reflect diverse technical and aesthetic goals for world maps.

Waldo R. Tobler American geographer

Waldo Rudolph Tobler was an American-Swiss geographer and cartographer. Tobler's idea that "Everything is related to everything else, but near things are more related than distant things" is referred to as the "first law of geography." He has proposed a second law as well: "The phenomenon external to an area of interest affects what goes on inside". Tobler was an active Professor Emeritus at the University of California, Santa Barbara Department of Geography until his death.

Contents

Overview

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 xk + yk = γk. 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.

Circle of latitude Geographic notion

A circle of latitude on Earth is an abstract east–west circle connecting all locations around Earth at a given latitude.

Latitude The angle between zenith at a point and the plane of the equator

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.

Cylindrical equal-area projection

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

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.

In mathematics, a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class. Degeneracy is the condition of being a degenerate case.

The Collignon projection is an equal-area pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881.

Mollweide projection map projection

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.

See also

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Mercator projection map projection

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Superellipse closed curve resembling an ellipse

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Goode homolosine projection map projection

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Eckert VI projection

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Eckert II projection

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In mathematics, a superelliptic curve is a plane curve with an equation of the form

Loximuthal projection

In cartography, the loximuthal projection is a map projection introduced by Karl Siemon in 1935, and independently in 1966 by Waldo R. Tobler, who named it. It is characterized by the fact that loxodromes from one chosen central point are shown straight lines, correct in azimuth from the center, and are "true to scale" in the sense that distances measured along such lines are proportional to lengths of the corresponding rhumb lines on the surface of the earth. It is neither an equal-area projection nor conformal.

Boggs eumorphic projection

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.

Equal Earth projection pseudocylindrical, equal-area map projection

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.

References

  1. Snyder, John P. (1993). Flattening the Earth: 2000 Years of Map Projections. Chicago: University of Chicago Press. p. 220.
  2. The Tobler Hyperelliptical Projection on the Center for Spatially Integrated Social Science's site
  3. "Superellipse" in MathWorld encyclopedia
  4. 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.