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. [1]
Strebe first presented the projection at a joint meeting of the Canadian Cartographic Association and the North American Cartographic Information Society (NACIS) in August 1994. [2] Its final formulation was completed in 1995. The projection has been available in the map projection software Geocart since Geocart 1.2, released in October 1994. [3]
The projection is arrived at by a series of steps, each of which preserves areas. Because each step preserves areas, the entire process preserves areas. The steps use a technique invented by Strebe called "substitute deprojection" [3] or "Strebe's transformation". [4] First, the Eckert IV projection is computed. Then, pretending that the Eckert projection is actually a shrunken portion of the Mollweide projection, the Eckert is "deprojected" back onto the sphere using the inverse transformation of the Mollweide projection. This yields a full-sphere-to-partial-sphere map. Then this mapped sphere is projected back to the plane using the Hammer projection. While the projections named here are the ones that define the Strebe 1995 projection, the substitute deprojection principle is not constrained to particular projections.
The projection as described can be formulated as follows: [3]
where is solved iteratively:
In these formulae, represents longitude and represents latitude.
Strebe's preferred arrangement is to set , as shown, and 11°E as the central meridian to avoid dividing eastern Siberia's Chukchi Peninsula. However, s can be modified to change the appearance without destroying the equal-area property.
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a given point in space is specified by three real numbers: the radial distancer along the radial line connecting the point to the fixed point of origin; the polar angleθ between the radial line and a polar axis; and the azimuthal angleφ as the angle of rotation of the radial line around the polar axis. (See graphic re the "physics convention".) Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates.
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics contains a list of common spherical harmonics.
In probability theory, the Borel–Kolmogorov paradox is a paradox relating to conditional probability with respect to an event of probability zero. It is named after Émile Borel and Andrey Kolmogorov.
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The Mollweide projection is an equal-area, pseudocylindrical map projection generally used for maps of the world or celestial sphere. 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.
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.
Great-circle navigation or orthodromic navigation is the practice of navigating a vessel along a great circle. Such routes yield the shortest distance between two points on the globe.
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Space-oblique Mercator projection is a map projection devised in the 1970s for preparing maps from Earth-survey satellite data. It is a generalization of the oblique Mercator projection that incorporates the time evolution of a given satellite ground track to optimize its representation on the map. The oblique Mercator projection, on the other hand, optimizes for a given geodesic.
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the regular solid harmonics, which are well-defined at the origin and the irregular solid harmonics, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:
Wagner VI is a pseudocylindrical whole Earth map projection. Like the Robinson projection, it is a compromise projection, not having any special attributes other than a pleasing, low distortion appearance. Wagner VI is equivalent to the Kavrayskiy VII horizontally elongated by a factor of ⁄. This elongation results in proper preservation of shapes near the equator but slightly more distortion overall. The aspect ratio of this projection is 2:1, as formed by the ratio of the equator to the central meridian. This matches the ratio of Earth’s equator to any meridian.
The Eckert IV projection is an equal-area pseudocylindrical map projection. The length of the polar lines is half that of the equator, and lines of longitude are semiellipses, or portions of ellipses. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, meridians are the same whereas parallels differ. Odd-numbered projections have parallels spaced equally, whereas even-numbered projections have parallels spaced to preserve area. Eckert IV is paired with Eckert III.
The Eckert II projection is an equal-area pseudocylindrical map projection. In the equatorial aspect the network of longitude and latitude lines consists solely of straight lines, and the outer boundary has the distinctive shape of an elongated hexagon. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, the meridians have the same shape, and the odd-numbered projection has equally spaced parallels, whereas the even-numbered projection has parallels spaced to preserve area. The pair to Eckert II is the Eckert I 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.
The Equal Earth map projection is an equal-area pseudocylindrical global map projection, 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 cartography, an equivalent, authalic, or equal-area projection is a map projection that preserves relative area measure between any and all map regions. Equivalent projections are widely used for thematic maps showing scenario distribution such as population, farmland distribution, forested areas, and so forth, because an equal-area map does not change apparent density of the phenomenon being mapped.
