Using the technique of splitting a trapezoid into two smaller ones of equal area, they then figured out how long it took Jupiter to travel half that distance.
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They can be an attractive alternative to the monotony of choropleth maps, especially with the proliferation of tile-grid maps (an equal-area cartogram useful for comparing states) — but they're just an alternative.
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"By incorporating the Peters projection maps -- an equal area representation -- into classrooms, we are opening the door for students to view the world in a different light," says Natacha Scott, social studies director at Boston Public Schools.
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Early Babylonian mathematicians who lived between 1800 B.C. and 1600 B.C. had figured out, for example, how to calculate the area of a trapezoid, and even how to divide a trapezoid into two smaller trapezoids of equal area.
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Cylindrical equal-area projection of the world; standard parallel at 40°N. The Lambert cylindrical equal-area projection with Tissot's indicatrix of deformation Cylindrical equal-area projection with oblique orientation In cartography, the cylindrical equal-area projection is a family of cylindrical, equal-area map projections.
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Lambert did not give names to any of his projections but they are now known as: # Lambert conformal conic # Transverse Mercator # Lambert azimuthal equal area # Lagrange projection # Lambert cylindrical equal area # Transverse cylindrical equal area # Lambert conical equal area The first three of these are of great importance.Corresponding to the Lambert azimuthal equal-area projection, there is a Lambert zenithal equal-area projection. The Times Atlas of the World (1967), Boston: Houghton Mifflin, Plate 3 et passim. Further details may be found at map projections and in several texts.
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Lambert cylindrical equal-area projection of the world Lambert cylindrical equal-area projection of the world, central meridian at 160°W to focus the map on the oceans. Lambert cylindrical equal-area projection with Tissot's indicatrix of deformation In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is a cylindrical equal-area projection. This projection is undistorted along the equator, which is its standard parallel, but distortion increases rapidly towards the poles. Like any cylindrical projection, it stretches parallels increasingly away from the equator.
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The HEALPix projection is an equal-area hybrid combination of: the Lambert cylindrical equal-area projection, for the equatorial regions of the sphere; and an interrupted Collignon projection, for the polar regions.
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Weisstein, Eric W. "Hammer–Aitoff Equal-Area Projection." From MathWorld—A Wolfram Web Resource Visually, the Aitoff and Hammer projections are very similar. The Hammer has seen more use because of its equal-area property. The Mollweide projection is another equal-area projection of similar aspect, though with straight parallels of latitude, unlike the Hammer's curved parallels.
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Snyder equal-area projection is used in the ISEA (Icosahedral Snyder Equal Area) discrete global grids. The first projection studies was conducted by John P. Snyder in the 1990s. Snyder, J. P. (1992), “An Equal-Area Map Projection for Polyhedral Globes”, Cartographica, 29(1), 10-21. urn:doi:10.3138/27H7-8K88-4882-1752. It is a modified Lambert azimuthal equal-area projection, most adequate to the polyhedral globe, a truncated icosahedron with 32 same-area faces (20 hexagons and 12 pentagons).
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Equal-area representation of Parliament results with each hexagon representing one seat Equal-area representation of the State Assemblies' results with each hexagon representing one seat Pie chart representing proportion of parliament seats won by contesting parties.
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Its equal-area property makes it useful for presenting spatial distribution of phenomena.
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A stereographic projection is conformal and perspective but not equal area or equidistant.
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The equal-area Mollweide projection 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. Several equivalent projections were developed in an attempt to minimize the distortion of countries and continents of planet Earth, keeping the area constant. Equivalent projections are widely used for thematic maps showing scenario distribution such as population, farmland distribution, forested areas, etc.
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PROJ guide's "Icosahedral Snyder Equal Area", proj.org/operations/projections/isea.html D. Carr et al. (1997), "ISEA discrete global grids"; in "Statistical Computing and Statistical Graphics Newsletter" vol. 8. For non-exact approximations (to equal-area) it can be replaced by Gnomonic projection, as in H3 Uber.github.
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Corrections to address these predictive deficiencies have since been made e.g., equal area rule, principle of corresponding states.
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As there are no squares in the hyperbolic plane, their role needs to be taken by regular quadrilaterals, meaning quadrilaterals with all sides congruent and all angles congruent (but these angles are strictly smaller than right angles). There exist, in the hyperbolic plane, (countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area, which, however, are constructed simultaneously. There is no method for starting with a regular quadrilateral and constructing the circle of equal area, and there is no method for starting with a circle and constructing a regular quadrilateral of equal area (even when the circle has small enough radius such that a regular quadrilateral of equal area exists).
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The HEALPix projection combines an equal-area cylindrical projection in equatorial regions with the Collignon projection in polar areas.
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The associated software package HEALPix implements the algorithm. The HEALPix projection (as a general class of spherical projections) is represented by the keyword HPX in the FITS standard for writing astronomical data files. It was approved as part of the official FITS World Coordinate System (WCS) by the IAU FITS Working Group on April 26, 2006. The spherical projection combines a cylindrical equal area projection, the Lambert cylindrical equal-area projection, for the equatorial regions of the sphere and a pseudocylindrical equal area projection, an interrupted Collignon projection, for the polar regions.
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The name Hobo–Dyer is derived from Bronstein and Abramms's first names (Howard and Bob) and Dyer's surname. The original ODT map is printed on two sides, one side with north upwards and the other with south upwards. That, together with its equal-area presentation, is intended to present a different perspective compared with more common non- equal area, north-up maps. The goal is similar to that of other equal-area projections (such as the Gall–Peters projection), but the Hobo-Dyer is billed by the publisher as "more visually satisfying".
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The Strebe 1995 projection with Tissot's indicatrices of distortion. Circles spaced at 30° intervals. 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.
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Since the triangle and lune are both formed by subtracting equal areas from equal area, they are themselves equal in area..
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Geodesic grids may use the dual polyhedron of the geodesic polyhedron, which is the Goldberg polyhedron. Goldberg polyhedra are made up of hexagons and (if based on the icosahedron) 12 pentagons. One implementation that uses an icosahedron as the base polyhedron, hexagonal cells, and the Snyder equal-area projection is known as the Icosahedron Snyder Equal Area (ISEA) grid.
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Monsky's proof combines combinatorial and algebraic techniques, and in outline is as follows: A square can be divided into an even number of triangles of equal area (left), but only into an odd number of approximately equal area triangles (right). #Take the square to be the unit square with vertices at (0,0), (0,1), (1,0) and (1,1). If there is a dissection into n triangles of equal area then the area of each triangle is 1/n. #Colour each point in the square with one of three colours, depending on the 2-adic valuation of its coordinates.
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In 1772 he created the Lambert conformal conic and Lambert azimuthal equal-area projections. The Albers equal-area conic projection features no distortion along standard parallels. It was invented by Heinrich Albers in 1805. In 1715 Herman Moll published the Beaver Map, one of the most famous early maps of North America, which he copied from a 1698 work by Nicolas de Fer.
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The projection shares many characteristics with other members of the family such as the Lambert cylindrical equal-area projection, whose standard parallel is the equator, and the Gall–Peters projection, whose standard parallels are 45°N and 45°S. While equal-area, distortion of shape increases in the Behrmann projection according to distance from the standard parallels. This projection is not equidistant.
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Lambert azimuthal equal-area projection of the world. The center is 0° N 0° E. The antipode is 0° N 180° E, near Kiribati in the Pacific Ocean. That point is represented by the entire circular boundary of the map, and the ocean around that point appears along the entire boundary. The Lambert azimuthal equal-area projection with Tissot's indicatrix of deformation.
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Snyder, John P. (1993). Flattening the Earth: Two Thousand Years of Map Projections p. 165. Chicago and London: The University of Chicago Press. . (Summary of the Peters controversy.) Some of the oldest projections are equal- area (the sinusoidal projection is also known as the "Mercator equal-area projection"), and hundreds have been described, refuting any implication that Peters's map is special in that regard.
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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.
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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.
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The invention of the Lambert cylindrical equal-area projection is attributed to the Swiss mathematician Johann Heinrich Lambert in 1772. Variations of it appeared over the years by inventors who stretched the height of the Lambert and compressed the width commensurately in various ratios. See Named specializations table. The Tobler hyperelliptical projection, first described by Tobler in 1973, is a further generalization of the cylindrical equal-area family.
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The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude is given by φ): The only cylindrical projections that preserve area have a north-south compression precisely the reciprocal of east-west stretching (cos φ): equal-area cylindrical (with many named specializations such as Gall–Peters or Gall orthographic, Behrmann, and Lambert cylindrical equal-area). This divides north-south distances by a factor equal to the secant of the latitude, preserving area but heavily distorting shapes. Any particular cylindrical equal-area map has a pair of identical latitudes of opposite sign (or else the equator) at which the east–west scale matches the north–south scale.
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Up to latitudes 40°44′11.8″N/S, the map is projected according to the sinusoidal projection’s transformation. The higher latitudes are the top sections of a Mollweide projection, grafted to the sinusoidal midsection where the scale of the two projections matches. This grafting results in a kink in the meridians along the parallel of the graft. The projection’s equal-area property follows from the fact that its source projections are themselves both equal-area.
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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. Tobler favored the parameterization shown with the top illustration; that is, α = 0, k = 2.5, and γ = 1.183136.
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Equal area representation implies that a region of interest in a particular portion of the map will share the same proportion of area as in any other part of the map.
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Many properties of regular polygons are invariant under affine transformations, and affine- regular polygons share the same properties. For instance, an affine-regular quadrilateral can be equidissected into m equal-area triangles if and only if m is even, by affine invariance of equidissection and Monsky's theorem on equidissections of squares.. More generally an n-gon with n > 4 may be equidissected into m equal-area triangles if and only if m is a multiple of n..
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As with any pseudocylindrical projection, in the projection’s normal aspect,The Tobler Hyperelliptical Projection on the Center for Spatially Integrated Social Science's site 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"Superellipse" in MathWorld encyclopedia or Lamé curves or sometimes as hyperellipses. The curve is described by .
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Albers projection of the world with standard parallels 20°N and 50°N. The Albers projection with standard parallels 15°N and 45°N, with Tissot's indicatrix of deformation An Albers projection shows areas accurately, but distorts shapes. The Albers equal-area conic projection, or Albers projection (named after Heinrich C. Albers), 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.
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The Gall–Peters projection of the world map 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 projection is named after James Gall and Arno Peters.
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As in crystallography, planes are typically plotted by their poles. Unlike crystallography, the southern hemisphere is used instead of the northern one (because the geological features in question lie below the Earth's surface). In this context the stereographic projection is often referred to as the equal-angle lower-hemisphere projection. The equal-area lower-hemisphere projection defined by the Lambert azimuthal equal- area projection is also used, especially when the plot is to be subjected to subsequent statistical analysis such as density contouring.
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Sinusoidal projection of the world. The sinusoidal projection with Tissot's indicatrix of deformation Jean Cossin, Carte cosmographique ou Universelle description du monde, Dieppe, 1570 The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson–Flamsteed or the Mercator equal-area projection. Jean Cossin of Dieppe was one of the first mapmakers to use the sinusoidal, appearing in a world map of 1570.Jean Cossin, Carte cosmographique ou Universelle description du monde avec le vrai traict des vents, 1570.
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This yields a coarse-resolution equal area grid called the resolution 1 grid. It consists of 20 hexagons on the surface of the sphere and 12 pentagons centered on the 12 vertices of the icosahedron.
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The second book, which contains ten propositions, studies parabolic segments exclusively. It examines these segments by substituting them with rectangles of equal area; an exchange made possible by results obtained in the Quadrature of the Parabola.
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For instance, the regular octagon can be tiled by two squares and four 45° rhombi. In a generalization of Monsky's theorem, proved that no zonogon has an equidissection into an odd number of equal-area triangles.
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For squares on two sides of an arbitrary triangle it yields a parallelogram of equal area over the third side and if the two sides are the legs of a right angle the parallelogram over the third side will be square as well. For a right-angled triangle, two parallelograms attached to the legs of the right angle yield a rectangle of equal area on the third side and again if the two parallelograms are squares then the rectangle on the third side will be a square as well.
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The quad sphere projection does not produce singularities at the poles or elsewhere, as do some other equal-area mapping schemes. Distortion is moderate over the entire sphere, so that at no point are shapes altered beyond recognition.
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The Wiechel projection is an azimuthal, equal-area projection, and a novelty map presented by William H. Wiechel in 1879. It is also a modified azimuthal projection. Distortion of direction, shape, and distance is considerable in the edges.
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Goode homolosine projection of the world. Tissot indicatrix on Goode homolosine projection, 15° graticule. The Goode homolosine projection (or interrupted Goode homolosine projection) is a pseudocylindrical, equal-area, composite map projection used for world maps. Normally it is presented with multiple interruptions.
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The Equal Earth compared to similar equal-area pseudocylindrical projections. The first known thematic map published using the Equal Earth projection is a map of the global mean temperature anomaly for July 2018, produced by the NASA’s Goddard Institute for Space Studies.
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Quartic authalic projection of the world. 15° graticule. In cartography, the quartic authalic projection is an equal-area projection developed by Karl Siemon in 1937 and independently by O.S. Adams in 1944. The meridians in this projection are fourth-order polynomial curves.
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Boggs eumorphic projection of the world. Tissot indicatrix on Boggs eumorphic projection, 15° graticule, gradations every 10° of angular deformation. The Boggs eumorphic projection is a pseudocylindrical, equal-area map projection used for world maps. Normally it is presented with multiple interruptions.
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A dissection into triangles of equal area is called an equidissection. Most polygons cannot be equidissected, and those that can often have restrictions on the possible numbers of triangles. For example, Monsky's theorem states that there is no odd equidissection of a square.
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For example, in a Ricci-flat manifold, a circle in Euclidean space may be deformed into an ellipse with equal area. This is due to Weyl curvature. Ricci-flat manifolds often have restricted holonomy groups. Important cases include Calabi–Yau manifolds and hyperkähler manifolds.
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As stated by Carr at al. article, page 32: : The S in ISEA refers to John P. Snyder. He came out of retirement specifically to address projection problems with the original EMAP grid (see Snyder, 1992). He developed the equal area projection that underlies the gridding system.
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Eckert IV projection of the world. Eckert IV projection with Tissot's indicatrices of distortion. 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.
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The converse is not necessarily true. The counterexamples are equirectangular and equal-area cylindrical projections (of normal aspects). These projections expand meridian-wise and parallel-wise by different ratios respectively. Thus, parallels and meridians cross rectangularly on the map, but these projections do not preserve other angles; i.e.
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There is no discernible segmentation visible. The axis in Emucaris is about one fifth of the pygidium and has a pattern of polygons of approximately equal area. The border of the cephalon is inconspicuous, and the border of the pygidium is longer than the somites. This border lacks ornamentation.
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However, representing area ratios correctly necessarily distorts shapes more than many maps that are not equal-area. The Mercator projection, developed for navigational purposes, has often been used in world maps where other projections would have been more appropriate.Bauer, H.A. (1942). "Globes, Maps, and Skyways (Air Education Series)".
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Animation demonstrating the alternative use of a Schmidt net to produce a Lambert equal-area projection using a polar azimuth. The Schmidt net is not an appropriate grid for representing the Earth's northern or southern hemisphere (because the lines would not correspond to meridians or parallels in such a projection). However, it can be used as a scalar measuring device for projecting latitude-longitude points onto a blank circle of the same size, to produce a Lambert equal-area projection with the azimuth at the north or south pole. The intersection of the parallels with the outer circle can be used as a de facto protractor for plotting a point's longitude as the angle in the polar projection.
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The projection he promoted is a specific parameterization of the cylindrical equal-area projection. In response, a 1989 resolution by seven North American geographical groups disparaged using cylindrical projections for general-purpose world maps, which would include both the Mercator and the Gall–Peters.American Cartographer. 1989. 16(3): 222–223.
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Eichstruth is a municipality in the district of Eichsfeld in Thuringia, Germany. By area, it is the smallest municipality in what was East Germany, although there are 35 rural municipalities ("Gemeinden") and one city ("Stadt"), Arnis, in what was West Germany that have lesser or equal area (as of 31 December 2012).
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Schmidt net, used for making plots of the Lambert azimuthal projection. The Schmidt net is a manual drafting method for the Lambert azimuthal equal-area projection using graph paper. It results in one lateral hemisphere of the Earth with the grid of parallels and meridians. The method is common in the geophysical sciences.
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Lambert was the first mathematician to address the general properties of map projections (of a spherical earth). In particular he was the first to discuss the properties of conformality and equal area preservation and to point out that they were mutually exclusive. (Snyder 1993. p77). In 1772, Lambert publishedLambert, Johann Heinrich. 1772.
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Earth's authalic ("equal area") radius is the radius of a hypothetical perfect sphere that has the same surface area as the reference ellipsoid. The IUGG denotes the authalic radius as . A closed-form solution exists for a spheroid:Snyder, J.P. (1987). Map Projections – A Working Manual (US Geological Survey Professional Paper 1395) p. 16–17.
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A tri-axial equivalent of the Mercator projection was developed by John P. Snyder. Equidistant map projections of a tri-axial ellipsoid were developed by Paweł Pędzich. Conic Projections of a tri-axial ellipsoid were developed by Maxim Nyrtsov. Equal- area cylindrical and azimuthal projections of the tri-axial ellipsoid were developed by Maxim Nyrtsov.
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Researchers in structural geology use the Lambert azimuthal projection to plot lineation and foliation in rocks, slickensides in faults, and other linear and planar features. In this context the projection is called the equal-area hemispherical projection. The Schmidt net is often used to sketch out the Lambert azimuthal projection for these purposes.Borradaile (2003).
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Hammer projection of the world The Hammer projection with Tissot's indicatrix of deformation 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.
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A regular hexagonal grid This honeycomb forms a circle packing, with circles centered on each hexagon. The honeycomb conjecture states that a regular hexagonal grid or honeycomb is the best way to divide a surface into regions of equal area with the least total perimeter. The conjecture was proven in 1999 by mathematician Thomas C. Hales.
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In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square does not have an odd equidissection. The problem was posed by Fred Richman in the American Mathematical Monthly in 1965, and was proved by Paul Monsky in 1970.
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The Lambert azimuthal projection was originally conceived as an equal-area map projection. It is now also used in disciplines such as geology to plot directional data, as follows. A direction in three-dimensional space corresponds to a line through the origin. The set of all such lines is itself a space, called the real projective plane in mathematics.
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In a wide sense, all projections with the point of perspective at infinity (and therefore parallel projecting lines) are considered as orthographic, regardless of the surface onto which they are projected. These kinds of projections distort angles and areas close to the poles. An example of an orthographic projection onto a cylinder is the Lambert cylindrical equal-area projection.
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The Eckert projections are six pseudocylindrical map projections devised by Max Eckert-Greifendorff, who presented them in 1906. The latitudes are parallel lines in all six projections. The projections come in pairs; in the odd-numbered projections, the latitudes are equally spaced, while their even- numbered counterparts are equal-area. The three pairs are distinguished by the shapes of the meridians.
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As a polyconic projection, the parallels are arcs of circles that are not concentric. The points of no distortion are on the central meridian at 44°N/S latitude. Meridians are convex away from the straight central meridian, and parallels are gently concave away from the equator. The projection is neither equal-area nor conformal; rather, it is a compromise projection.
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Because of great land area distortions, it is not well suited for general world maps. Therefore, Mercator himself used the equal-area sinusoidal projection to show relative areas. However, despite such distortions, the Mercator projection was, especially in the late 19th and early 20th centuries, perhaps the most common projection used in world maps, despite being much criticized for this use.Kellaway, G.P. (1946).
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The authalic (equal area) radius of the Clarke 1866 ellipsoid is . The resulting arcminute is . The United States chose five significant digits for its nautical mile, 6080.2 feet, whereas the United Kingdom chose four significant digits for its Admiralty mile, 6080 feet. In 1929, the international nautical mile was defined by the First International Extraordinary Hydrographic Conference in Monaco as exactly 1,852 metres.
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Natural Earth projection of the world. The natural Earth projection with Tissot's indicatrix of deformation The natural Earth projection is a pseudocylindrical map projection designed by Tom Patterson and introduced in 2012. It is neither conformal nor equal-area. It was designed in Flex Projector, a specialized software application that offers a graphical approach for the creation of new projections.
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Mollweide Glacier () is a steep glacier south of Mount Kowalczyk, descending west from Hobbs Ridge into Blue Glacier, in Victoria Land, Antarctica. The name is one of a group in the area associated with surveying applied in 1993 by the New Zealand Geographic Board; this glacier was named from the Mollweide projection, an equal area map projection with the parallels and central meridian being straight lines.
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In turn, if undulating boundary is observed, there should be changes in crustal thickness. Global study of residual Bouguer anomaly data indicates that crustal thickness of Mars varies from 5.8 km to 102 km. Two major peaks at 32 km and 58 km are identified from an equal-area histogram of crustal thickness. These two peaks are linked to the crustal dichotomy of Mars.
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This map shows the antipode of each point on Earth's surface—the points where the blue and yellow overlap are land antipodes; most land has its antipodes in the ocean. This map uses the Lambert azimuthal equal-area projection. The yellow areas are the reflections through Earth's center of land masses of the opposite Western Hemisphere. The same map, from the perspective of the Western Hemisphere.
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Tobler hyperelliptical projection of the world; α = 0, γ = 1.18314, k = 2.5 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.
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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.
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Data from the Smithsonian Astrophysical Observatory was also used which included camera (Baker–Nunn) and some laser ranging. Doppler satellite ground stations providing data for WGS 72 development Worldwide geometric satellite triangulation network, BC-4 cameras The surface gravity field used in the Unified WGS Solution consisted of a set of 410 10° × 10° equal area mean free air gravity anomalies determined solely from terrestrial data.
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The greener, the more areal inflation or deflation. The Ortelius oval projection is a map projection used for world maps largely in the late 16th and early 17th century. It is neither conformal nor equal-area but instead offers a compromise presentation. It is similar in structure to a pseudocylindrical projection but does not qualify as one because the meridians are not equally spaced along the parallels.
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This is the location where the king Thirumalai Naicker excavated the soil to fabricate the bricks required for constructing his palace, Thirumalai Nayakkar Mahal. The pit that was thus formed is seen as tank now. It is approximately 305 m long and 290 m wide, nearly equal area to that of Meenakshi Amman Temple. Built in 1645 A.D.,this is the biggest tank in Tamil Nadu.
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Small-scale DLGs are sold in state units and are cast on either the Albers equal-area conic projection system or the geographic coordinate system of latitude and longitude, depending on the distribution format. All DLGs are referenced to the North American Datum of 1927 (NAD27) or the North American Datum of 1983 (NAD83). USGS DLGs are topologically structured for use in mapping and geographic information system (GIS) applications.
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The potential difference between the ionosphere and the Earth is maintained by thunderstorms, with lightning strikes delivering negative charges from the atmosphere to the ground. World map showing frequency of lightning strikes, in flashes per km² per year (equal-area projection). Lightning strikes most frequently in the Democratic Republic of the Congo. Combined 1995–2003 data from the Optical Transient Detector and 1998–2003 data from the Lightning Imaging Sensor.
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Almost directly on the heels of taungya came another form of teak exploitation. This system, which eventually came to be known as the Myanmar Selection System reduced the British Empire's dependency on the local Karen people. This method entails dividing an area of forest into thirty sections of roughly equal area, one of which is logged every thirty years.Hla Maung Thein, Kanzaki Mamoru, Fukushima Maki, and Yazar Minn.
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The rearrangement proof (click to view animation)The two large squares shown in the figure each contain four identical triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem, Q.E.D.Benson, Donald. The Moment of Proof : Mathematical Epiphanies, pp.
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Map of the cumulative tracks of all tropical cyclones during the 1985–2005 time period. The Pacific Ocean west of the International Date Line sees more tropical cyclones than any other basin, while there is almost no activity in the southern hemisphere between Africa and 160˚W. Equal-area projection. Worldwide, tropical cyclone activity peaks in late summer, when the difference between temperatures aloft and sea surface temperatures is the greatest.
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Each of the Archimedes' quadruplets (green) have equal area to each other and to Archimedes' twin circles In geometry, Archimedes' quadruplets are four congruent circles associated with an arbelos. Introduced by Frank Power in the summer of 1998, each have the same area as Archimedes' twin circles, making them Archimedean circles. Online catalogue of Archimedean circlesClayton W. Dodge, Thomas Schoch, Peter Y. Woo, Paul Yiu (1999). "Those Ubiquitous Archimedean Circles". PDF.
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He calculated the latitude where the two projections had the same scale and therefore should be merged by overlaying the two projections on each other. Above and below 40° 44’ 11.8” N and S respectively, the homolographic projection is used. Between those two latitudes the sinusoidal projection is inserted. The Interrupted Homolosine projection is a pseudo-cylindrical, equal area projection. Initially, Goode’s Homolosine projection did not have universal appeal.
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Dissection of a square and equilateral triangle into each other. No such dissection exists for the cube and regular tetrahedron. In two dimensions, the Wallace–Bolyai–Gerwien theorem states that any two polygons of equal area can be cut up into polygonal pieces and reassembled into each other. David Hilbert became interested in this result as a way to axiomatize area, in connection with Hilbert's axioms for Euclidean geometry.
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The Robinson projection is neither equal-area nor conformal, abandoning both for a compromise. The creator felt that this produced a better overall view than could be achieved by adhering to either. The meridians curve gently, avoiding extremes, but thereby stretch the poles into long lines instead of leaving them as points. Hence, distortion close to the poles is severe, but quickly declines to moderate levels moving away from them.
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Hobo–Dyer projection of the world. Tissot's indicatrices of deformation The Hobo–Dyer map projection is a cylindrical equal-area projection, with standard parallels (there is no north-south or east-west distortion) at 37.5° north and south of the equator. The map was commissioned in 2002 by Bob Abramms and Howard Bronstein of ODT Inc. and drafted by cartographer Mick Dyer, as a modification of the 1910 Behrmann projection.
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Behrmann projection of the world Tissot's indicatrices of deformation The Behrmann projection is a cylindrical map projection described by Walter Behrmann in 1910. It is a member of the cylindrical equal-area projection family. Members of the family differ by their standard parallels, which are parallels along which the projection has no distortion. In the case of the Behrmann projection, the standard parallels are 30°N and 30°S.
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They are in diameter and are capable of being scaled up to long. Each cylinder has six equal-area stripes that run the length of the cylinder; three are transparent windows, three are habitable "land" surfaces. Furthermore, an outer agricultural ring, in diameter, rotates at a different speed to support farming. The habitat's industrial manufacturing block is located in the middle, to allow for minimized gravity for some manufacturing processes.
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In general, triangular and hexagonal grids are constructed so as to better approach the goals of equal-area (or nearly so) plus more seamless coverage across the poles, which tends to be a problem area for square or rectangular grids since in these cases, the cell width diminishes to nothing at the pole and those cells adjacent to the pole then become 3- rather than 4-sided. Criteria for optimal discrete global gridding have been proposed by both Goodchild and KimerlingCriteria and Measures for the Comparison of Global Geocoding Systems, Keith C. Clarke, University of California in which equal area cells are deemed of prime importance. Quadtrees are a specialised form of grid in which the resolution of the grid is varied according to the nature and complexity of the data to be fitted, across the 2-d space. Polar grids utilize the polar coordinate system, using circles of a prescribed radius that are divided into sectors of a certain angle.
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Since he could not find a map that satisfied him, Peters developed one himself. In 1974 he announced the Peters World Map, claiming it was the most accurate representation of the world. The map engendered controversy. The map projection Peters claimed to have developed had been presented more than a century earlier by the Reverend James Gall, and, despite Peters's claims, the projection was not the first or only equal-area projection.
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HEALPix H=4, K=3 projection of the world. HEALPix (sometimes written as Healpix), an acronym for Hierarchical Equal Area isoLatitude Pixelisation of a 2-sphere, refers to either an algorithm for pixelisation of the 2-sphere or to the associated class of map projections. The pixelisation algorithm was devised in 1997 by Krzysztof M. Górski at the Theoretical Astrophysics Center in Copenhagen, Denmark, and first published as a preprint in 1998.
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350px The graph shows the variation of the scale factors for the above three examples. The top plot shows the isotropic Mercator scale function: the scale on the parallel is the same as the scale on the meridian. The other plots show the meridian scale factor for the Equirectangular projection (h=1) and for the Lambert equal area projection. These last two projections have a parallel scale identical to that of the Mercator plot.
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The simplest is a manual tool with adjustable points similar to a caliper. It can determine how short a distance between two impressions on the skin can be distinguished. To differentiate between two points and one point of equal area (the sum of the areas of the two points equals the area of the third point), Dr. Sidney Weinstein created the three- point esthesiometer. A scale on the instrument gives readings in millimeter gradients.
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Stereographic projection or fisheye projection can be used to form a little planet panorama by pointing the virtual camera straight down and setting the field of view large enough to show the whole ground and some of the areas above it; pointing the virtual camera upwards creates a tunnel effect. Conformality of the stereographic projection may produce more visually pleasing result than equal area fisheye projection as discussed in the stereo- graphic projection's article.
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The hemisphere can then be plotted as a disk of radius using the Lambert azimuthal projection. Thus the Lambert azimuthal projection lets us plot directions as points in a disk. Due to the equal-area property of the projection, one can integrate over regions of the real projective plane (the space of directions) by integrating over the corresponding regions on the disk. This is useful for statistical analysis of directional data, including random rigid rotation.
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Researchers in structural geology use the Lambert azimuthal projection to plot crystallographic axes and faces, lineation and foliation in rocks, slickensides in faults, and other linear and planar features. In this context the projection is called the equal-area hemispherical projection. There is also an equal-angle hemispherical projection defined by stereographic projection. The discussion here has emphasized the lower hemisphere z ≤ 0, but some disciplines prefer the upper hemisphere z ≥ 0.
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In mapmaking, a quadrilateralized spherical cube, or quad sphere for short, is an equal-area mapping and binning scheme for data collected on a spherical surface (either that of the Earth or the celestial sphere). It was first proposed in 1975 by Chan and O'Neill for the Naval Environmental Prediction Research Facility. This scheme is also often called the COBE sky cube, because it was designed to hold data from the Cosmic Background Explorer (COBE) project.
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However, stereographic fisheye lenses are typically more expensive to manufacture. Image remapping software, such as Panotools, allows the automatic remapping of photos from an equal-area fisheye to a stereographic projection. The stereographic projection has been used to map spherical panoramas, starting with Horace Bénédict de Saussure's in 1779. This results in effects known as a little planet (when the center of projection is the nadir) and a tube (when the center of projection is the zenith).
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In all of these formulas, λ is the longitude from the central meridian and φ is the latitude. Three years later, Ernst Hermann Heinrich Hammer suggested the use of the Lambert azimuthal equal-area projection in the same manner as Aitoff, producing the Hammer projection. While Hammer was careful to cite Aitoff, some authors have mistakenly referred to the Hammer projection as the Aitoff projection.Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp.
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The individual cells of a grid system can also be useful as units of aggregation, for example as a precursor to data analysis, presentation, mapping, etc. For some applications (e.g., statistical analysis), equal-area cells may be preferred, although for others this may not be a prime consideration. In computer science, one often needs to find out all cells a ray is passing through in a grid (for raytracing or collision detection); this is called "grid traversal".
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Numerous parcels of land were transferred between the two countries during the construction period, 1935–1938. At the end, each nation had ceded an equal area of land to the other. The Boundary Treaty of 1970 transferred an area of Mexican territory to the U.S., near Presidio and Hidalgo, Texas, to build flood control channels. In exchange, the U.S. ceded other land to Mexico, including five parcels near Presidio, the Horcon Tract and Beaver Island near Roma, Texas.
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Van der Grinten projection of the world The Van der Grinten projection with Tissot's indicatrix of deformation The van der Grinten projection is a compromise map projection, which means that it is neither equal-area nor conformal. Unlike perspective projections, the van der Grinten projection is an arbitrary geometric construction on the plane. Van der Grinten projects the entire Earth into a circle. It largely preserves the familiar shapes of the Mercator projection while modestly reducing Mercator's distortion.
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Equal-area representation of the results with each hexagon representing one seat These are the election results of the 2008 Malaysian general election by state constituency. State assembly elections were held in Malaysia on 8 March 2008 as part of the general elections. These members of the legislative assembly (MLAs) representing their constituency from the first sitting of respective state legislative assembly to its dissolution. The state legislature election deposit was set at RM 5,000 per candidate.
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Each median divides the area of the triangle in half; hence the name, and hence a triangular object of uniform density would balance on any median. (Any other lines which divide the area of the triangle into two equal parts do not pass through the centroid.)Dunn, J. A., and Pretty, J. E., "Halving a triangle," Mathematical Gazette 56, May 1972, 105-108. DOI 10.2307/3615256 The three medians divide the triangle into six smaller triangles of equal area.
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Bonne projection of the world, standard parallel at 45°N. Bonne projection with Tissot's indicatrix of deformation. World map by Bernard Sylvanus, 1511 The Bonne projection is a pseudoconical equal-area map projection, sometimes called a dépôt de la guerre, modified Flamsteed, or a Sylvanus projection. Although named after Rigobert Bonne (1727–1795), the projection was in use prior to his birth, in 1511 by Sylvano, Honter in 1561, De l'Isle before 1700 and Coronelli in 1696.
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The fin and rudder are straight edged and of about equal area, the latter extending to the bottom of the fuselage. The fuselage has an oval cross section, increasing in depth forwards to behind the cockpit where the wing is mounted. Both seats, placed in tandem, are forward of the wing root under a canopy. A skid from nose to a monowheel below the wing root leading edge forms the undercarriage, assisted by a small tail bumper.
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Based on the principles of the projection, precise, but lengthy, mathematical formulas were later developed for calculating this projection by computer for a spherical earth. The Chamberlin trimetric projection is neither conformal nor equal-area. Rather, the projection was conceived to minimize distortion of distances everywhere with the side-effect of balancing between areal equivalence and conformality. This projection is not appropriate for mapping the entire sphere because the outer boundary would loop and overlap itself in most configurations.
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It is an area- preserving (equal-area) map, which can be seen by computing the area element of the sphere when parametrized by the inverse of the projection. In Cartesian coordinates it is :dA = dX \; dY. This means that measuring the area of a region on the sphere is tantamount to measuring the area of the corresponding region on the disk. On the other hand, the projection does not preserve angular relationships among curves on the sphere.
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Triangular world maps are also possible using the same method. The name is derived from "authalic" and "graph". The method used to construct the projection ensures that the 96 regions of the sphere that are used to define the projection each have the correct area, but the projection does not qualify as equal-area because the method does not control area at infinitesimal scales or even within those regions. The AuthaGraph world map can be tiled in any direction without visible seams.
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For a geometrical interpretation, consider a rectangle with sides of length and , hence it has perimeter and area . Similarly, a square with all sides of length has the perimeter and the same area as the rectangle. The simplest non-trivial case of the AM–GM inequality implies for the perimeters that and that only the square has the smallest perimeter amongst all rectangles of equal area. Extensions of the AM–GM inequality are available to include weights or generalized means.
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There are a class of hierarchical DGG's named by the Open Geospatial Consortium (OGC) as "Discrete Global Grid Systems" (DGGS), that must to satisfy 18 requirements. Among them, what best distinguishes this class from other hierarchical DGGs, is the Requirement-8, "For each successive level of grid refinement, and for each cell geometry, (...) Cells that are equal area (...) within the specified level of precision".Open Geospatial Consortium (2017), "Topic 21: Discrete Global Grid Systems Abstract Specification". Document 15-104r5 version 1.0.
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The greener, the more areal inflation or deflation. 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.
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Eckert VI projection of the world The Eckert VI projection is an equal-area pseudocylindrical map projection. The length of polar line is half that of the equator, and lines of longitude are sinusoids. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. In 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.
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For smaller-scale maps, such as those spanning continents or the entire world, many projections are in common use according to their fitness for the purpose, such as Winkel tripel, Robinson and Mollweide. Reference maps of the world often appear on compromise projections. Due to distortions inherent in any map of the world, the choice of projection becomes largely one of aesthetics. Thematic maps normally require an equal area projection so that phenomena per unit area are shown in correct proportion.
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Dewan Rakyat results of the 2013 Malaysian General Election (equal-area representation) These are the election results of the 2008 Malaysian general election by parliamentary constituency. These members of parliament (MPs) representing their constituency from the first sitting of 12th Malaysian Parliament to its dissolution. The parliamentary election deposit was set at RM 10,000 per candidate. Similar to previous elections, the election deposit will be forfeited if the particular candidate had failed to secure at least 12.5% or one-eighth of the votes.
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Gall's main work as an astronomer was with the constellations. As part of this work he developed the Gall orthographic projection, a derivative of the Lambert cylindrical equal-area projection, to project the celestial sphere onto flat paper in a manner that avoided distorting the shapes of the constellations. He also applied this technique to terrestrial mapmaking as a way to make a flat map of the round Earth. Gall Orthographic was re-invented by Arno Peters in 1967 and adopted by organisations such as UNESCO.
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The rate of deforestation has been declining since the 1970s when there was considerable hydroelectric and agricultural development. Since 2000, the annual rate has been approximately 6,200 hectares per year. In 2010, British Columbia introduced the Zero Net Deforestation Act so an equal area of trees is planted for carbon storage to offset any forest land that is permanently cleared for another use. Since 1850, ecosystem conversion to agriculture, reservoirs, urban areas, and other land uses has occurred on a total of 2% of the province.
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They also discovered "Blanche's Dissection", a method of dividing a square into rectangles of equal area but different dimensions. They modelled these using abstract electrical networks, an approach that yielded not only solutions to the original problem, but techniques with wider applications to the field of electrical networks. They published their results—under their own names—in 1940. Tutte, who is believed to have contributed the most work under Descartes's name, kept up the pretence for years, refusing to acknowledge even in private that she was fictitious.
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As a result of these criticisms, modern atlases no longer use the Mercator projection for world maps or for areas distant from the equator, preferring other cylindrical projections, or forms of equal-area projection. The Mercator projection is still commonly used for areas near the equator, however, where distortion is minimal. It is also frequently found in maps of time zones. Arno Peters stirred controversy beginning in 1972 when he proposed what is now usually called the Gall–Peters projection to remedy the problems of the Mercator.
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Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Miklós Laczkovich in 1990; the decomposition makes heavy use of the axiom of choice and is therefore non-constructive. Laczkovich estimated the number of pieces in his decomposition at roughly 1050. More recently, gave a completely constructive solution using Borel pieces.
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The Times Atlas of the World (1967), Boston: Houghton Mifflin, Plate 3, et passim. The Lambert azimuthal projection is used as a map projection in cartography. For example, the National Atlas of the US uses a Lambert azimuthal equal-area projection to display information in the online Map Maker application, and the European Environment Agency recommends its usage for European mapping for statistical analysis and display. It is also used in scientific disciplines such as geology for plotting the orientations of lines in three-dimensional space.
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This implies the area of the sphere is equal to the area of the cylinder minus its caps. This result would eventually lead to the Lambert cylindrical equal-area projection, a way of mapping the world that accurately represents areas. Archimedes was particularly proud of this latter result, and so he asked for a sketch of a sphere inscribed in a cylinder to be inscribed on his grave. Later, Roman philosopher Marcus Tullius Cicero discovered the tomb, which had been overgrown by surrounding vegetation.
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We have seen that by partitioning the disk into an infinite number of pieces we can reassemble the pieces into a rectangle. A remarkable fact discovered relatively recently is that we can dissect the disk into a large but finite number of pieces and then reassemble the pieces into a square of equal area. This is called Tarski's circle-squaring problem. The nature of Laczkovich's proof is such that it proves the existence of such a partition (in fact, of many such partitions) but does not exhibit any particular partition.
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Concentration of dots gives a glance at the distribution of stress direction. :2) Fault Motion Fault movement is resolved into three components (as in 3D), which are vertical transverse, horizontal transverse and lateral components, by trigonometric relation with the measured dips and trends. Net slip is shown more clearly which paves the way to understanding the deformation. 3) Individual fault geometry represented on a stereonet :3) Individual Fault Geometry Fault planes are represented by lines in stereonets (equal area lower hemisphere projection), while rakes on them are indicated by dots sitting on the lines.
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The Service’s Wetlands Geodatabase contains five units (map areas) that are populated with digital vector data and raster images. These units include the conterminous U.S., Alaska, Hawaii, Puerto Rico and the U.S. Virgin Islands, and the Pacific Trust Territories. Each unit of the geodatabase contains seamless digital map data in ArcSDE geodatabase format. Data are in a single standard projection (Albers Equal-Area Conic Projection), horizontal planar units in meters, horizontal planar datum is the North American Datum of 1983 (also called NAD83), and minimum coordinate precision of one centimeter.
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A leak of pressure into the volume between the pistons would transform the machine into an effective single piston with equal area on each side, thus defeating the intensifier effect. A mechanically compact and popular form of intensifier is the concentric cylinder form, as illustrated. In this design, one piston and cylinder are reversed: instead of the large diameter piston driving a smaller piston, it instead drives a smaller moving cylinder that fits over a fixed piston. This design is compact, and again may be made in little over twice the stroke.
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Chapter five considers Monsky's theorem on the impossibility of partitioning a square into an odd number of equal-area triangles, and its proof using the 2-adic valuation, and chapter six applies Galois theory to more general problems of tiling polygons by congruent triangles, such as the impossibility of tiling a square with 30-60-90 right triangles. The final chapter returns to the topic of the first, with material on László Rédei's generalization of Hajós's theorem. Appendices cover background material on lattice theory, exact sequences, free abelian groups, and the theory of cyclotomic polynomials.
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HEALPix (sometimes written as Healpix), an acronym for Hierarchical Equal Area isoLatitude Pixelisation of a 2-sphere, can refer to either an algorithm for pixelization of the 2-sphere, an associated software package, or an associated class of map projections. Healpix is widely used for cosmological random map generation. The original motivation for devising HEALPix was one of necessity. NASA's WMAP and the European Space Agency’s mission Planck produce multi-frequency data sets sufficient for the construction of full-sky maps of the microwave sky at an angular resolution of a few arc minutes.
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For this reason, Sperner's lemma can also be used in root-finding algorithms and fair division algorithms; see Simmons–Su protocols. Sperner's lemma is one of the key ingredients of the proof of Monsky's theorem, that a square cannot be cut into an odd number of equal-area triangles. Sperner's lemma can be used to find a competitive equilibrium in an exchange economy, although there are more efficient ways to find it. Fifty years after first publishing it, Sperner presented a survey on the development, influence and applications of his combinatorial lemma.
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In particular, it is impossible to dissect a circle and make a square using pieces that could be cut with an idealized pair of scissors (that is, having Jordan curve boundary). The pieces used in Laczkovich's proof are non-measurable subsets. Laczkovich actually proved the reassembly can be done using translations only; rotations are not required. Along the way, he also proved that any simple polygon in the plane can be decomposed into finitely many pieces and reassembled using translations only to form a square of equal area.
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His doctorate was on map projections, again a topic proposed by Korkin, the degree being awarded in 1896. The work, on equal area plane projections of the sphere, built on ideas of Euler, Joseph Louis Lagrange and Chebyshev. Grave became professor at Kharkov in 1897 and, from 1902, he was appointed professor at the University of Kyiv, where he remained for the rest of his life. Grave is considered as the founder of the Kyiv school of algebra which was to become the centre for algebra in the USSR.
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Stein's doctoral research was in topology, but his research interests later shifted to abstract algebra and combinatorics. In combinatorics, he is known for formulating the tripod packing problem. The tripods of this problem are infinite polycubes, the unions of the lattice cubes along three axis-parallel rays, and they have also been called "Stein corners" in honor of his contributions to this problem. Stein is also known as one of the independent discoverers of Fáry's theorem, and for his contributions to equidissection, the partition of polygons into triangles of equal area.
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Development of Cahill-butterfly's quasi-octant projection circa 1915 The projection is neither conformal nor equal-area,[A conformal projection has the following properties: the scale at any point is the same in all directions.. and the angle between any two lines on the map must be the same as on the earth. An equal area projection preserves relationships between corresponding areas..] bound by circular arcs, with no meridians and no parallels, in which the spherical surface of the earth is divided into eight octants, each flattened into the shape of a Reuleaux triangle. If transferred to an elastic support, it would be possible to cover with them the surface of a model of the earth's globe.. The eight triangles are oriented in a similar way as per two four-leaf cloversProiezioni a ottanti side by side, being the earth poles in the center of each clove. One of the sides of the eight triangles, (the one opposite to the center of the pseudo clover), is one fourth of the equator, the remaining two (those that converge to the center of the pseudo clover), are part of the two meridians that with the equator dissect the globe in the eight octants.
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Collignon projection of the world. 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. For the smallest choices of the parameters chosen for this projection, the sphere may be mapped either to a single diamond, a pair of squares, or a triangle. The projection is used in the polar areas as part of the HEALPix spherical projection, which is widely used in physical cosmology in making maps of the cosmic microwave background, in particular by the WMAP and Planck space missions.
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The Biological Dynamics of Forest Fragments Project (BDFFP), was born out of the SLOSS (single large or several small reserves of equal area) debate in the mid - 1970s about the application of the theory of island biogeography to conservation planning. The debate determined that the species richness and the rate of growth increase as the area of a reserve increases. It also determined that the shape of a reserve is very important to the species diversity. Reserves with a large surface area to volume ratio tend to be affected more by edge effects than reserves with a small surface area to volume ratio.
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The rectangles on the hoist side in Swedish flags had the proportions 4:5 and required a mark of the same shape, while Norwegian flags had squares on the hoist side, and hence a square union mark. As a result of the diagonal division, both nations’ colours were of equal area. The union mark was introduced by royal order in council on 20 June 1844. It had been proposed by a joint committee from both countries, appointed in 1839 with the mandate of discussing the symbols of the Union to ensure that they would reflect the equal status of the two united kingdoms.
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Thus the sinusoidal part of the isotherm is replaced by a horizontal line (red line in Fig. 1). According to the Maxwell construction (or "equal area rule"), the height of the horizontal line is such that the two green areas in Fig. 1 are equal. The direct quote from James Clerk Maxwell which became the Maxwell construction: “Now let us suppose the medium to pass from B to F along the hypothetical curve BCDEF in a state always homogeneous, and to return along the straight line path FB in the form of a mixture of liquid and vapour.
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Stereographic projection of the spherical panorama of the Last Supper sculpture by Michele Vedani in Esino Lario, Lombardy, Italy during Wikimania 2016 "Vue circulaire des montagnes qu'on découvre du sommet du Glacier de Buet", Horace-Benedict de Saussure, Voyage dans les Alpes, précédés d'un essai sur l'histoire naturelle des environs de Geneve. Neuchatel, 1779–96, pl. 8. Some fisheye lenses use a stereographic projection to capture a wide-angle view.Samyang 8 mm 3.5 Fisheye CS Compared to more traditional fisheye lenses which use an equal-area projection, areas close to the edge retain their shape, and straight lines are less curved.
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Kangacaris differs from Emucaris fava in having three thorax segments, while Emucaris has four. When disregarding the border, Kangacaris has an inverted egg-shape, while the pygidium of Emucaris is a triangle with a rounded termination. The axis of the pygidium of Kangacaris is ⅓ of the width of the body and clearly segmented, while in Emucaris it is one fifth and has a pattern of polygons of approximately equal area. The axis of the pygidium of Emucaris terminates in a spine that ends at the outer rim of the border, while Kangacaris lacks a terminal spine.
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The European grid is a proposed, multipurpose Pan-European mapping standard. It is based on the ETRS89 Lambert Azimuthal Equal-Area projection coordinate reference system, with the centre of the projection at the point 52° N, 10° E and false easting: x0 = 4321000 m, false northing: y0 = 3210000 m (CRS identifier in Inspire: ETRS89-LAEA). The grid is designated as Grid_ETRS89-LAEA5210. For identification of an individual resolution level, the name is extended by identification of cell size in metres (example: _100K). The origin of Grid_ETRS89-LAEA5210 coincides with the false origin of the ETRS89-LAEA coordinate reference system (x=0, y=0).
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The standard defines the requirements of an hierarchical DGG, including how to operate the grid. Any DGG that satisfies these requirements can be named DGGS. "A DGGS specification SHALL include a DGGS Reference Frame and the associated Functional Algorithms as defined by the DGGS Core Conceptual Data Model".Section 6.1, "DGGS Core Data Model Overview", of the DGGS standard : For an Earth grid system to be compliant with this Abstract Specification it must define a hierarchical tessellation of equal area cells that both partition the entire Earth at multiple levels of granularity and provide a global spatial reference frame.
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Mollweide projection of the world The Mollweide projection with Tissot's indicatrix of deformation 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 projection was first published by mathematician and astronomer Karl (or Carl) Brandan Mollweide (1774–1825) of Leipzig in 1805.
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Equal-area projection of constituencies The Conservative Party won a landslide victory securing 365 seats out of 650, giving them an overall majority of 80 seats in the House of Commons. They gained seats in several Labour Party strongholds in Northern England that were held by the party for decades, which had formed the 'red wall'. The constituency of Bishop Auckland elected a Conservative MP for the first time in its 134-year history. In the worst result for the party in 84 years, Labour won 202 seats, a loss of 60 compared to the previous election.
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Eckert II projection of the world The Eckert II projection is an equal-area pseudocylindrical map projection. In the equatorial aspect (where the equator is shown as the horizontal axis) 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.
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The following applications of shear mapping were noted by William Kingdon Clifford: :"A succession of shears will enable us to reduce any figure bounded by straight lines to a triangle of equal area." :"... we may shear any triangle into a right-angled triangle, and this will not alter its area. Thus the area of any triangle is half the area of the rectangle on the same base and with height equal to the perpendicular on the base from the opposite angle."William Kingdon Clifford (1885) Common Sense and the Exact Sciences, page 113 The area-preserving property of a shear mapping can be used for results involving area.
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The maps shown here are based on this relationship; they show a Lambert azimuthal equal-area projection of the Earth, in yellow, overlaid on which is another map, in blue, shifted horizontally by 180° of longitude and inverted about the Equator with respect to latitude. Noon at one place is midnight at the other (ignoring daylight saving time and irregularly shaped time zones) and, with the exception of the tropics, the longest day at one point corresponds to the shortest day at the other, and midwinter at one point coincides with midsummer at the other. Sunrise and sunset do not quite oppose each other at antipodes due to refraction of sunlight.
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If the two sums of areas of opposite triangles are equal ( Area(BCL) + Area(DAL) = Area(LAB) + Area(DLC) ), then the point L is located on the Newton line, that is the line which connects E and F. For a parallelogram the Newton line does not exist since both midpoints of the diagonals coincide with point of intersection of the diagonals. Moreover the area identity of the theorem holds in this case for any inner point of the quadrilateral. The converse of Anne's theorem is true as well, that is for any point on the Newton line which is an inner point of the quadrilateral, the area identity holds.
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For that reason, the voltage waveform can usually be approximated by the fundamental frequency of voltage. If this approximation is used, current harmonics produce no effect on the real power transferred to the load. An intuitive way to see this comes from sketching the voltage wave at fundamental frequency and overlaying a current harmonic with no phase shift (in order to more easily observe the following phenomenon). What can be observed is that for every period of voltage, there is equal area above the horizontal axis and below the current harmonic wave as there is below the axis and above the current harmonic wave.
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The projection maps meridians to vertical straight lines of constant spacing (for meridional intervals of constant spacing), and circles of latitude to horizontal straight lines of constant spacing (for constant intervals of parallels). The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.
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Yale University Map Department It is intended as an illustrated guide for sailors and attempts to include all the new transatlantic discoveries. Isolario contains an oval depiction of the world, a type of map invented by Bordone Columbia University photo of Bordone's map and later formalized into the equal-area elliptical Mollweide projection three centuries later. Bordone's map shows a very distorted Mondo Novo (New World), displaying only the northern regions of South America. North America, depicted as a large island, is labeled Terra del Laboratore (Land of the worker), almost certainly a reference to the slave trading in the area in those days (and hence the name Labrador).
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In 2007, R. E. Schwartz showed that outer billiards has some unbounded orbits when defined relative to the Penrose Kite, thus answering the original Moser-Neumann question in the affirmative. The Penrose kite is the convex quadrilateral from the kites-and-darts Penrose tilings. Subsequently, Schwartz showed that outer billiards has unbounded orbits when defined relative to any irrational kite. An irrational kite is a quadrilateral with the following property: One of the diagonals of the quadrilateral divides the region into two triangles of equal area and the other diagonal divides the region into two triangles whose areas are not rational multiples of each other.
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In the mutually agreed minor adjustments, part of the EEZ of the small Greek islands of Strofades, Othoni, and Mathraki was traded off for an equal area elsewhere (i.e part of the EEZ of Italy's Calabria). According to Dendias, this creates an extremely favorable legal precedent for Greece in its dispute with Turkey. The United States, the Libyan House of Representatives, and the LNA led by Halifa Haftar, welcomed the EEZ agreement between Greece and Italy, with the US calling it "exemplary" and "an example of how these things should be done [in the region]", while reiterating their opposition to the Turkey-GNA EEZ agreement.
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The Maxwell equal area rule can also be derived from an assumption of equal chemical potential μ of coexisting liquid and vapour phases. On the isotherm shown in the above plot, points a and c are the only pair of points which fulfill the equilibrium condition of having equal pressure, temperature and chemical potential. It follows that systems with volumes intermediate between these two points will consist of a mixture of the pure liquid and gas with specific volumes equal to the pure liquid and gas phases at points a and c. The van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV.
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World map showing the frequency of lightning strikes, in flashes per km² per year (equal-area projection), from combined 1995–2003 data from the Optical Transient Detector and 1998–2003 data from the Lightning Imaging Sensor. Lightning is not distributed evenly around Earth, as shown in the map. On Earth, the lightning frequency is approximately 44 (± 5) times per second, or nearly 1.4 billion flashes per year and the average duration is 0.2 seconds made up from a number of much shorter flashes (strokes) of around 60 to 70 microseconds. Many factors affect the frequency, distribution, strength and physical properties of a typical lightning flash in a particular region of the world.
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If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1. In every polygon with perimeter p and area A , the isoperimetric inequality p^2 > 4\pi A holds.Dergiades, Nikolaos, "An elementary proof of the isoperimetric inequality", Forum Mathematicorum 2, 2002, 129–130. For any two simple polygons of equal area, the Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon.
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In 1842 Clausen was hired by the staff of the Tartu Observatory, becoming its director in 1866-1872. Works by Clausen include studies on the stability of Solar system, comet movement, ABC telegraph code and calculation of 250 decimals of Pi (later, only 248 were confirmed to be correct). In 1840 he discovered the Von Staudt–Clausen theorem. Also in 1840 he also found two compass and straightedge constructions of lunes with equal area to a square, adding to three (including the lune of Hippocrates) known to the ancient Greek mathematician Hippocrates of Chios; it was later shown that these five lunes are the only possible solutions to this problem.. Translated from Postnikov's 1963 Russian book on Galois theory.
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Werner projection of the world Woodcut from 1536 by Oronce Finé showing the Werner projection The Werner projection is a pseudoconic equal-area map projection sometimes called the Stab-Werner or Stabius-Werner projection. Like other heart-shaped projections, it is also categorized as cordiform. Stab- Werner refers to two originators: Johannes Werner (1466-1528), a parish priest in Nuremberg, refined and promoted this projection that had been developed earlier by Johannes Stabius (Stab) of Vienna around 1500. The projection is a limiting form of the Bonne projection, having its standard parallel at one of the poles (90°N/S)... Distances along each parallel and along the central meridian are correct, as are all distances from the north pole.
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This ROSAT image is an Aitoff-Hammer equal-area map in galactic coordinates with the Galactic center in the middle of the 0.25 keV diffuse X-ray background. In addition to discrete sources which stand out against the sky, there is good evidence for a diffuse X-ray background. During more than a decade of observations of X-ray emission from the Sun, evidence of the existence of an isotropic X-ray background flux was obtained in 1956. This background flux is rather consistently observed over a wide range of energies. The early high-energy end of the spectrum for this diffuse X-ray background was obtained by instruments on board Ranger 3 and Ranger 5.
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MVA 220/155 kV phase-shifting transformer. A phase angle regulating transformer, phase angle regulator (PAR, American usage), phase-shifting transformer, phase shifter (West coast American usage), or quadrature booster (quad booster, British usage), is a specialised form of transformer used to control the flow of real power on three-phase electric transmission networks. For an alternating current transmission line, power flow through the line is proportional to the sine of the difference in the phase angle of the voltage between the transmitting end and the receiving end of the line.The "equal area criteria" for power system stability requires that this angle be less than 90 degrees, so for practical purposes this angle will be measurably less than 90 degrees.
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Precisely, an inflection point of a doubly continuously differentiable (C^2) curve on the surface of a sphere is a point p with the following property: let I be the connected component containing p of the intersection of the curve with its tangent great circle at p. (For most curves I will just be p itself, but it could also be an arc of the great circle.) Then, for p to be an inflection point, every neighborhood of I must contain points of the curve that belong to both of the hemispheres separated by this great circle. The theorem states that every C^2 curve that partitions the sphere into two equal-area components has at least four inflection points in this sense.
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The Geneva Accord bases the International Border between the States of Palestine and Israel on the June 4th 1967 lines, in accordance with UNSC Resolution 242UNSC Resolution 242, 22 November 1967 and UNSC Resolution 338,UNSC Resolution 338, 22 October 1973 with reciprocal modifications in the form of landswaps on a 1:1 basis. Israel will annex several areas currently densely populated by Jewish settlements near the Green Line (such as Gush Etzion). In return for areas annexed by Israel from the West Bank, the Palestinians will receive territory of equal area and quality adjacent mostly to the Gaza Strip. The State of Israel will assume responsibility for resettling the Israelis living in what would be determined as Palestinian sovereign territory such as Ariel and other settlements.
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The aircraft was designed by Dr Ulrich Hütter as a 40% sized, scaled-down version of the Dornier Do 17's fuselage and wing panels without the twin-engine nacelles, and built by Schempp-Hirth. The airframe was entirely of wood and used a retractable tricycle landing gear - one of the earliest non-Heinkel- built German airframe designs to use such an arrangement. Power was supplied by a Hirth HM 60 inverted, air-cooled inline four-cylinder engine mounted within the fuselage near the wings. Other than the engine installation, the only other unusual feature of the aircraft was its all-new, full four-surface cruciform tail, which included a large ventral fin/rudder unit of equal area to the dorsal surface.
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The state is bordered by Canada's Yukon and British Columbia to the east (making it the only state to border a Canadian territory), the Gulf of Alaska and the Pacific Ocean to the south and southwest, the Bering Sea, Bering Strait, and Chukchi Sea to the west and the Arctic Ocean to the north. Alaska's territorial waters touch Russia's territorial waters in the Bering Strait, as the Russian Big Diomede Island and Alaskan Little Diomede Island are only apart. Alaska has a longer coastline than all the other U.S. states combined. 48 contiguous states (Albers equal-area conic projection) At in area, Alaska is by far the largest state in the United States, and is more than twice the size of the second-largest U.S. state, Texas.
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The isotherms are straight and vertical, isobars are straight and horizontal and dry adiabats are also straight and have a 45 degree inclination to the left while moist adiabats are curved. Wind barbs are often plotted at the side of the diagram to indicate the winds at different heights they are used to save space with symbols to help in charts. However, using this configuration sacrifices the equal-area property of the original Clausius–Clapeyron relation requirements between the temperature of the environment and the temperature of a parcel of air lifted/lowered. Although it permits to analyze the cloud cover and the stability of the airmass, it thus does not permit to calculate the Convective Available Potential Energy (CAPE).
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James Randi (2001) That Dratted Triangle, proof by Martin Gardner Congruence of edge lengths allows rotation of the selected triangles to form three equal-area parallelograms, which bisect into six triangles of equal size to the original interior triangle. An early exhibit of this geometrical construction and area computation was given by Robert Potts in 1859 in his Euclidean geometry textbook.Robert Potts (1859) Euclid's Elements of Geometry, Fifth school edition, problems 59 and 100, pages 78 & 80 via Internet Archive According to Cook and Wood (2004), this triangle puzzled Richard Feynman in a dinner conversation; they go on to give four different proofsR.J. Cook & G.V. Wood (2004) "Feynman's Triangle", Mathematical Gazette 88:299–302 A more general result is known as Routh's theorem.
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The Storch V had a straight edged wing with about 17° of sweep on the leading edge and with only slight taper and dihedral. It was built around a plywood skinned D-box leading edge spar and a second spar near mid-chord and was fabric covered. Broad, lobate ailerons were hinged at right angles to the line of flight, protruding beyond the trailing edges and carrying small trim tabs not fitted to the Storch IV. Broad, low endplate fins and rudders of about equal area, cambered on their inner surfaces provided directional stability and control. Their profiles were lower and simpler than those on the Storch IV. The rudders could work together for steering and in opposition for braking.
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As results, a and b represent the maximum and minimum scale factors at the point, which is the same thing as the semimajor and semiminor axes of the Tissot ellipse; s represents the amount of inflation or deflation in area (also given by a ∙ b); and ω represents the maximum angular distortion at the point. For the Mercator projection, and any other conformal projection, h = k and θ′ = 90° so that each ellipse degenerates into a circle with the radius h = k being equal to the scale factor in any direction at that point. For the sinusoidal projection, and any other equal-area projection, the semi-major axis of the ellipse is the reciprocal of the semi-minor axis so that every ellipse has the same area even though their eccentricities vary.
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Equal angle grids (the class that includes c-squares) have the advantage that transformation of spatial data (for example as simple coordinates of latitude and longitude) in and out of the grid notation can be simple, since the latitude–longitude grid is itself equal angle. On the actual surface of the globe, the cells are approximately "square" only adjacent to the equator, and become progressively narrower and tapered (also with curved northern and southern boundaries) as they approach the poles, and cells adjoining the poles are unique in possessing three faces rather than four. By contrast, equal area grids attempt to preserve a constant area for all cells at the same hierarchical level (resolution), at the expense of losing concordance with familiar lines of latitude and/or longitude.
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Normally it takes about 40,000 sq > ft of grazing land for 1 cow/steer (for milk/meat) or 2 goats (for > milk/meat/wool), or 2 sheep (for milk/meat/wool). [In contrast] With > [biointensive farming] and maximizing the edible calorie output in your > vegan diet design, one person’s complete balanced diet can be grown on about > 4,000 sq ft—a much smaller area. The challenge [to growing animals for food] > is that by 2014, 90% of the world’s people will only have access to about > 4,500 sq ft of farmable land per person, if they leave an equal area in a > wild state to protect plant and animal genetic diversity and the world’s > ecosystems! As you will see from the information that follows on the land > requirements for incorporating livestock, this becomes a challenge.
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Oberwolfach in 2009 Paul Monsky (born June 17, 1936) is an American mathematician and professor at Brandeis University. After earning a bachelor's degree from Swarthmore College, he received his Ph.D. in 1962 from the University of Chicago under the supervision of Walter Lewis Baily, Jr. He has introduced the Monsky–Washnitzer cohomology and he has worked intensively on Hilbert–Kunz functions and Hilbert–Kunz multiplicity. In 2007, Monsky and Holger Brenner gave an example showing that tight closure does not commute with localization. Monsky's theorem, the statement that a square cannot be divided into an odd number of equal-area triangles, is named after Monsky, who published the first proof of it in 1970.. In the mid-1970s, Monsky stopped paying U.S. federal income tax in protest against military spending.
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Newport Arch, a 3rd- century Roman gate The Romans conquered this part of Britain in AD 48 and shortly afterwards built a legionary fortress high on a hill overlooking the natural lake formed by the widening of the River Witham (the modern day Brayford Pool) and at the northern end of the Fosse Way Roman road (A46). The Celtic name Lindon was subsequently Latinised to Lindum and given the title Colonia when it was converted into a settlement for army veterans. The conversion to a colonia was made when the legion moved on to York (Eboracum) in AD 71\. Lindum colonia or more fully, Colonia Domitiana Lindensium, after the Emperor Domitian who ruled at the time, was established within the walls of the hilltop fortress with the addition of an extension of about equal area, down the hillside to the waterside below.
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How the Earth is projected onto a cylinder The projection was invented by the Swiss mathematician Johann Heinrich Lambert and described in his 1772 treatise, Beiträge zum Gebrauche der Mathematik und deren Anwendung, part III, section 6: Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten, translated as, Notes and Comments on the Composition of Terrestrial and Celestial Maps. Lambert's projection is the basis for the cylindrical equal-area projection family. Lambert chose the equator as the parallel of no distortion. By multiplying the projection's height by some factor and dividing the width by the same factor, the regions of no distortion can be moved to any desired pair of parallels north and south of the equator. These variations, particularly the Gall–Peters projection, are more commonly encountered in maps than Lambert’s original projection due to their lower distortion overall.
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Flow coefficients are determined by comparing the actual flow of a test piece to the theoretical flow of a perfect orifice of equal area. Thus the flow coefficient should be a close measure of efficiency. It cannot be exact because the L/D does not indicate the actual minimum size of the duct. A real orifice plate showing how the fluid would actually flow A theoretical orifice plate showing perfect flow which is used as a standard for comparing the efficiencies of real flows An orifice with a flow coefficient of 0.59 would flow the same amount of fluid as a perfect orifice with 59% of its area or 59% of the flow of a perfect orifice with the same area (orifice plates of the type shown would have a coefficient of between 0.58 and 0.62 depending on the precise details of construction and the surrounding installation).
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Roman north wall of Lindum Colonia The Ninth Legion, Hispana was probably moved from Lincoln to found the fortress at York around 71 AD Then, after a probable short occupation by the Second Legion, who had moved to Chester by 77-78 AD"Jones" (2002) pg 31 the Legionary fort would have been left on a care and maintenance basis. The exact date that it was converted into a colonia is unknown, but a generally favoured date is 86 AD."Jones" (2002) pg. 51 This was an important settlement for retired legionaries, established by the emperor Domitian within the walls and using the street grid of the hilltop fortress, with the addition of an extension of about equal area, down the hillside to the waterside below. The town became a major flourishing settlement, accessible from the sea both through the River Trent and through the River Witham.
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500px Square or rectangular grids are frequently used for purposes such as translating spatial information expressed in Cartesian coordinates (latitude and longitude) into and out of the grid system. Such grids may or may not be aligned with the grid lines of latitude and longitude; for example, Marsden Squares, World Meteorological Organization squares, c-squares and others are aligned, while Universal Transverse Mercator coordinate system and various national grid based systems such as the British national grid reference system are not. In general, these grids fall into two classes, those that are "equal angle", that have cell sizes that are constant in degrees of latitude and longitude but are unequal in area (particularly with varying latitude), or those that are "equal area" (statistical grids), that have cell sizes that are constant in distance on the ground (e.g. 100 km, 10 km) but not in degrees of longitude, in particular.
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The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime, but F_5 is composite and so are all other Fermat numbers that have been verified as of 2017. A regular n-gon is constructible using straightedge and compass if and only if the odd prime factors of n (if any) are distinct Fermat primes. Likewise, a regular n-gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of n are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form 2^a3^b+1. It is possible to partition any convex polygon into n smaller convex polygons of equal area and equal perimeter, when n is a power of a prime number, but this is not known for other values of n.
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The Equal Earth projection (2018), an increasingly popular equal-area pseudocylindrical projection for world maps Because the Earth is (nearly) spherical, any planar representation (a map) requires it to be flattened in some way, known as a projection. Most map projections are implemented using mathematical formulas and computer algorithms based on geographic coordinates (latitude, longitude). All projections generate distortions such that shapes and areas cannot both be conserved simultaneously, and distances can never all be preserved. The mapmaker must choose a suitable map projection according to the space to be mapped and the purpose of the map; this decision process becomes increasingly important as the scope of the map increases; while a variety of projections would be indistinguishable on a city street map, there are dozens of drastically different ways of projecting the entire world, with extreme variations in the type, degree, and location of distortion.
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A tennis ball In geometry, the tennis ball theorem states that any smooth curve on the surface of a sphere that divides the sphere into two equal-area subsets without touching or crossing itself must have at least four inflection points, points at which the curve does not consistently bend to only one side of its tangent line. The tennis ball theorem was first published under this name by Vladimir Arnold in 1994, and is often attributed to Arnold, but a closely related result appears earlier in a 1968 paper by Beniamino Segre, and the tennis ball theorem itself is a special case of a theorem in a 1977 paper by Joel L. Weiner. The name of the theorem comes from the standard shape of a tennis ball, whose seam forms a curve that meets the conditions of the theorem; the same kind of curve is also used for the seams on baseballs.
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1040) showed that two lunes, formed on the two sides of a right triangle, whose outer boundaries are semicircles and whose inner boundaries are formed by the circumcircle of the triangle, then the areas of these two lunes added together are equal to the area of the triangle. The lunes formed in this way from a right triangle are known as the lunes of Alhazen.Hippocrates' Squaring of the Lune at cut-the- knot, accessed 2012-01-12.. The quadrature of the lune of Hippocrates is the special case of this result for an isosceles right triangle.. In the mid-20th century, two Russian mathematicians, Nikolai Chebotaryov and his student Anatoly Dorodnov, completely classified the lunes that are constructible by compass and straightedge and that have equal area to a given square. All such lunes can be specified by the two angles formed by the inner and outer arcs on their respective circles; in this notation, for instance, the lune of Hippocrates would have the inner and outer angles (90°, 180°).
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Folding initiates by shortening; limb lengthening and rotation and hinge migration, cause synclinal deflection below its original position accompanied by the flow of ductile material beneath the synclinal trough to the anticlinal core; resulting in increased amplitude of the anticlinal fold.Mitra, S. (2002) Fold-accommodation faults, American Association of Petroleum Geologists Bulletin, 86, 4, 671-693 Further compression dominated by hinge migration, yields tightening of folds and space accommodation issues within the anticlinal core; leading to the formation of disharmonic folds .Hardy, S. and Finch, E. (2005). Discrete-element modeling of detachment folding. Basin Research, 17, 507-520Mitra, S. and Namson, J. (1989) Equal-area balancing, American Journal of Science, 289, 563-599 Epard and Groshong, (1994) recognize a similar pattern to disharmonic folding they label it second-order shortening.Epard, J. L. and Groshong, R. H., Jr. (1994) Kinematic model of detachment folding including limb rotation, fixed hinges and layer- parallel strain, Tectonophysics 247, 85-103 Basic models and experiments Storti, F., Salvini, F., and McClay, K. (1997).
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During the Japanese Occupation, Europeans and Eurasian Singaporeans were generally spared the harsher treatments by the Japanese than other racial groups; however many of them became an increasing nuisance for their activist efforts, in particular the Catholic societies, who fostered strong community bonds with the local Chinese. From December 1943 to April 1944, a combination of a collapsing currency, rising food and continued social activism culminated in the reactionary and punitive land acquisition strategy which relocated about 400 Roman Catholic Chinese and 300 European/Eurasian families (of which most were land-owners and many Chinese households also ran small businesses or shop-keeps from their homes) that forcibly acquired land and fixed property from homeowners in exchange for an equal area of dry land two miles from the town of Bahau in Negeri Sembilan state in Malaya. Propagandised in the ironic manner of sending them off to live happier, better lives as a purely Catholic community where they could run their own affairs despite the reality that it was meant as a punishment. The responsibility for administering the affairs of the settlement was mockingly bestowed to prominent activist for Chinese welfare under occupation Roman Catholic Bishop, Monseigneur Adrian Devals.
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