Equal Earth projection
The Equal Earth map projection is an equal-area pseudocylindrical projection for world maps, invented by Bojan Šavrič, Bernhard Jenny, Tom Patterson in 2018. It is inspired by the used Robinson projection, but unlike the Robinson projection, retains the relative size of areas; the projection equations are simple to implement and fast to evaluate. The projection is formulated as the polynomial equations x = 2 3 λ cos θ 3 y = A 4 θ 9 + A 3 θ 7 + A 2 θ 3 + A 1 θ where sin θ = 3 2 sin ϕ A 1 = 1.340264, A 2 = − 0.081106, A 3 = 0.000893, A 4 = 0.003796 and φ refers to latitude and λ to longitude. The features of the Equal Earth projection include: The curved sides of the projection suggest the spherical form of Earth. Straight parallels make it easy to compare how far south places are from the equator. Meridians are evenly spaced along any line of latitude. Software for implementing the projection is easy to write and executes efficiently; the Equal Earth map projection was created by Bojan Šavrič, Tom Patterson, Bernhard Jenny, as detailed in a 2018 publication for International Journal of Geographical Information Science.
According to the creators:We created it to provide a visually pleasing alternative to the Gall–Peters projection, which some schools and organizations have adopted out of concern for fairness—they need a world map showing continents and countries at their true sizes relative to each other. 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. Official website
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; 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, 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. List of map projections Media related to Maps with Behrmann projection at Wikimedia Commons Table of examples and properties of all common projections, from radicalcartography.net An interactive Java Applet to study the metric deformations of the Berhrmann Projection
In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map. A single indicatrix describes the distortion at a single point; because distortion varies across a map Tissot's indicatrices are placed across a map to illustrate the spatial change in distortion. A common scheme places them at each intersection of displayed parallels; these schematics are important in the study of map projections, both to illustrate distortion and to provide the basis for the calculations that represent the magnitude of distortion at each point. There is a one-to-one correspondence between the Tissot indicatrix and the metric tensor of the map projection coordinate conversion.
Tissot's theory was developed in the context of cartographic analysis. The geometric model represents the Earth, comes in the form of a sphere or ellipsoid. Tissot's indicatrices illustrate linear and areal distortions of maps: A map distorts distances wherever the quotient between the lengths of an infinitesimally short line as projected onto the projection surface, as it is on the Earth model, deviates from unity; the quotient is called the scale factor. Unless the projection is conformal at the point being considered, the scale factor varies by direction around the point. A map distorts angles wherever the angles measured on the model of the Earth are not conserved in the projection; this is expressed by an ellipse of distortion, not a circle. A map distorts areas wherever areas measured in the model of the Earth are not conserved in the projection; this is expressed by ellipses of distortion. In conformal maps, where each point preserves angles projected from the geometric model, the Tissot's indicatrices are all circles of size varying by location also with varying orientation.
In equal-area projections, where area proportions between objects are conserved, the Tissot's indicatrices all have the same area, though their shapes and orientations vary with location. In arbitrary projections, both area and shape vary across the map. In the adjacent image, ABCD is a circle with unit area defined in a spherical or ellipsoidal model of the Earth, A′B′C′D′ is the Tissot's indicatrix that results from its projection on the plane. Segment OA is transformed in OA′, segment OB is transformed in OB′. Linear scale is not conserved along these two directions, since OA′ is not equal to OA and OB′ is not equal to OB. Angle MOA, in the unit area circle, is transformed in angle M′OA′ in the distortion ellipse; because M ′ OA ′ ≠ MOA, we know. The area of circle ABCD is, by definition, equal to 1; because the area of ellipse A′B′ is less than 1, a distortion of area has occurred. In dealing with a Tissot indicatrix, different notions of radius come into play; the first is the infinitesimal radius of the original circle.
The resulting ellipse of distortion will have infinitesimal radius, but by the mathematics of differentials, the ratios of these infinitesimal values are finite. So, for example, if the resulting ellipse of distortion is the same size of infinitesimal as on the sphere its radius is considered to be 1. Lastly, the size that the indicatrix gets drawn for human inspection on the map is arbitrary; when a network of indicatrices is drawn on a map, they are all scaled by the same arbitrary amount so that their sizes are proportionally correct. Like M in the diagram, the axes from O along the parallel and along the meridian may undergo a change of length and a rotation when projecting, it is common in the literature to represent scale along the meridian as h and scale along the parallel as k, for a given point. The angle between meridian and parallel might have changed from 90° to some other value. Indeed, unless the map is conformal, all angles except the one subtended by the semi-major axis and semi-minor axis of the ellipse might have changed.
A particular angle will have changed the most, value of that maximum change is known as the angular deformation, denoted as θ′. Which angle, how it is oriented do not figure prominently in distortion analysis, it is the value of the change, significant. The values of h, k, θ′ can be computed as follows. H = 1 R 2 + 2 k = 1 R cos φ 2
Eckert IV projection
The Eckert IV projection is an equal-area pseudocylindrical map projection. The length of the polar lines is half that of the equator, lines of longitude are semiellipses, or portions of ellipses, it was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. In each pair, the meridians have the same shape, the odd-numbered projection has spaced parallels, whereas the even-numbered projection has parallels spaced to preserve area; the pair to Eckert IV is the Eckert III projection. Given a radius of sphere R, central meridian λ0 and a point with geographical latitude φ and longitude λ, plane coordinates x and y can be computed using the following formulas: x = 2 4 π + π 2 R ≈ 0.422 2382 R, y = 2 π 4 + π R sin θ ≈ 1.326 5004 R sin θ, where θ + sin θ cos θ + 2 sin θ = sin φ. Θ can be solved for numerically using Newton's method. Θ = arcsin ≈ arcsin φ = arcsin λ = λ 0 + x 4 π + π 2 2 R ≈ λ 0 + x 0.422 2382 R List of map projections Eckert II projection Eckert VI projection Max Eckert-Greifendorff Eckert IV projection at Mathworld
A map projection is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane. Maps cannot be created without map projections. All map projections distort the surface in some fashion. Depending on the purpose of the map, some distortions are acceptable and others are not. There is no limit to the number of possible map projections. More the surfaces of planetary bodies can be mapped if they are too irregular to be modeled well with a sphere or ellipsoid. More projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, manifolds. However, "map projection" refers to a cartographic projection. Maps can be more useful than globes in many situations: they are more compact and easier to store; these useful traits of maps motivate the development of map projections. However, Carl Friedrich Gauss's Theorema Egregium proved that a sphere's surface cannot be represented on a plane without distortion.
The same applies to other reference surfaces used as models for the Earth, such as oblate spheroids and geoids. Since any map projection is a representation of one of those surfaces on a plane, all map projections distort; every distinct map projection distorts in a distinct way. The study of map projections is the characterization of these distortions. Projection is not limited to perspective projections, such as those resulting from casting a shadow on a screen, or the rectilinear image produced by a pinhole camera on a flat film plate. Rather, any mathematical function transforming coordinates from the curved surface to the plane is a projection. Few projections in actual use are perspective. For simplicity, most of this article assumes. In reality, the Earth and other large celestial bodies are better modeled as oblate spheroids, whereas small objects such as asteroids have irregular shapes. Io is better modeled by triaxial prolated spheroid with small eccentricities. Haumea's shape is a Jacobi ellipsoid, with its major axis twice as long as its minor and with its middle axis one and half times as long as its minor.
These other surfaces can be mapped as well. Therefore, more a map projection is any method of "flattening" a continuous curved surface onto a plane. Many properties can be measured on the Earth's surface independent of its geography; some of these properties are: Area Shape Direction Bearing Distance ScaleMap projections can be constructed to preserve at least one of these properties, though only in a limited way for most. Each projection compromises, or approximates basic metric properties in different ways; the purpose of the map determines. Because many purposes exist for maps, a diversity of projections have been created to suit those purposes. Another consideration in the configuration of a projection is its compatibility with data sets to be used on the map. Data sets are geographic information. Different datums assign different coordinates to the same location, so in large scale maps, such as those from national mapping systems, it is important to match the datum to the projection; the slight differences in coordinate assignation between different datums is not a concern for world maps or other vast territories, where such differences get shrunk to imperceptibility.
The classical way of showing the distortion inherent in a projection is to use Tissot's indicatrix. For a given point, using the scale factor h along the meridian, the scale factor k along the parallel, the angle θ′ between them, Nicolas Tissot described how to construct an ellipse that characterizes the amount and orientation of the components of distortion. By spacing the ellipses along the meridians and parallels, the network of indicatrices shows how distortion varies across the map; the creation of a map projection involves two steps: Selection of a model for the shape of the Earth or planetary body. Because the Earth's actual shape is irregular, information is lost in this step. Transformation of geographic coordinates to Cartesian or polar plane coordinates. In large-scale maps, Cartesian coordinates have a simple relation to eastings and northings defined as a grid superimposed on the projection. In small-scale maps and northings are not meaningful, grids are not superimposed; some of the simplest map projections are literal projections, as obtained by placing a light source at some definite point relative to the globe and projecting its features onto a specified surface.
This is not the case for most projections, which are defined only in terms of mathematical formulae that have no direct geometric interpretation. However, picturing the light source-globe model can be helpful in understanding the basic concept of a map projection A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a developable surface; the cylinder and the plane are all developable surfaces. The sphere and ellipsoid do not have developable surfaces, so any projection of them onto a plane will have to dis
A cadastre is a comprehensive land recording of the real estate or real property's metes-and-bounds of a country. In most countries, legal systems have developed around the original administrative systems and use the cadastre to define the dimensions and location of land parcels described in legal documentation; the cadastre is a fundamental source of data in lawsuits between landowners. In the United States, Cadastral Survey within the Bureau of Land Management maintains records of all public lands; such surveys require detailed investigation of the history of land use, legal accounts, other documents. Land registration and cadastre complement each other. A cadastre includes details of the ownership, the tenure, the precise location, the dimensions, the cultivations if rural, the value of individual parcels of land. Cadastres are used by many nations around the world, some in conjunction with other records, such as a title register; the International Federation of Surveyors defines cadastre as follows: A Cadastre is a parcel based, up-to-date land information system containing a record of interests in land.
It includes a geometric description of land parcels linked to other records describing the nature of the interests, the ownership or control of those interests, the value of the parcel and its improvements. The word cadastre came into English through French from Late Latin capitastrum, a register of the poll tax, the Greek katástikhon, a list or register, from katà stíkhon —literally, "down the line", in the sense of "line by line" along the directions and distances between the corners mentioned and marked by monuments in the metes and bounds; the word forms the adjective cadastral, used in public administration for ownership and taxation purposes. The terminology for cadastral divisions may include counties, ridings, sections, lots and city blocks. Other languages have kept the original t sound in the second syllable. In modern Greek, though, it has been replaced by ktimatologio; some of the earliest cadastres were ordered by Roman Emperors to recover state owned lands, appropriated by private individuals, thereby recover income from such holdings.
One such cadastre was done in AD 77 in Campania, a surviving stone marker of the survey reads "The Emperor Vespasian, in the eighth year of his tribunician power, so as to restore the state lands which the Emperor Augustus had given to the soldiers of Legion II Gallica, but which for some years had been occupied by private individuals, ordered a survey map to be set up with a record on each'century' of the annual rental". In this way Vespasian was able to reimpose taxation uncollected on these lands. With the fall of Rome the use of cadastral maps discontinued. Medieval practice used written descriptions of the extent of land rather than using more precise surveys. Only in the sixteenth and early seventeenth centuries did the use of cadastral maps resume, beginning in the Netherlands. With the emergence of capitalism in Renaissance Europe the need for cadastral maps reemerged as a tool to determine and express control of land as a means of production; this took place first in land disputes and spread to governmental practice as a means of more precise tax assessment.
Cadastral surveys document the boundaries of land ownership, by the production of documents, sketches, plans and maps. They were used to ensure reliable facts for land valuation and taxation. An example from early England is the Domesday Book in 1086. Napoleon established a comprehensive cadastral system for France, regarded as the forerunner of most modern versions; the Public Lands Survey System is a cadastral survey of the United States originating in legislation from 1785, after international recognition of the United States. The Dominion Land Survey is a similar cadastral survey conducted in Western Canada begun in 1871 after the creation of the Dominion of Canada in 1867. Both cadastral surveys are made relative to principal meridian and baselines; these cadastral surveys divided the surveyed areas into townships, square land areas of 36 square miles. These townships are divided into sections, each one-mile square. Unlike in Europe this cadastral survey preceded settlement and as a result influenced settlement patterns.
Properties are rectangular, boundary lines run on cardinal bearings, parcel dimensions are in fractions or multiples of chains. Land descriptions in Western North America are principally based on these land surveys. Cadastral survey information is a base element in Geographic Information Systems or Land Information Systems used to assess and manage land and built infrastructure; such systems are employed on a variety of other tasks, for example, to track long-term changes over time for geological or ecological studies, where land tenure is a significant part of the scenario. A cadastral map is a map; some cadastral maps show additional details, such as survey district names, unique identifying numbers for parcels, certificate of title numbers, positions of existing structures, section or lot numbers and their respective areas
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; the projection is defined by: x = cos φ y = φ where φ is the latitude, λ is the longitude, λ0 is the central meridian. Scale is constant along the central meridian, east–west scale is constant throughout the map. Therefore, the length of each parallel on the map is proportional to the cosine of the latitude, as it is on the globe; this makes the left and right bounding meridians of the map into half of a sine wave, each mirroring the other. Each meridian is half of a sine wave with only the amplitude differing, giving the projection its name; each is shown on the map as longer than the central meridian, whereas on the globe all are the same length. The true distance between two points on a meridian can be measured on the map as the vertical distance between the parallels that intersect the meridian at those points.
With no distortion along the central meridian and the equator, distances along those lines are correct, as are the angles of intersection of other lines with those two lines. Distortion is lowest throughout the region of the map close to those lines. Similar projections which wrap the east and west parts of the sinusoidal projection around the north pole are the Werner and the intermediate Bonne and Bottomley projections; the MODLAND Integerized Sinusoidal Grid, based on the sinusoidal projection, is a geodesic grid developed by the NASA's Moderate-Resolution Imaging Spectroradiometer science team. List of map projections Gerardus Mercator, Nicolas Sanson, John Flamsteed – mathematicians who developed the technique. Media related to Sinusoidal projection at Wikimedia Commons Pseudocylindrical Projections Table of examples and properties of all common projections, from radicalcartography.net