what does it mean when a map projection distorts the earth
771916
A world map is a map of most or all of the surface of Earth. World maps, because of their scale, must deal with the problem of projection. Maps rendered in two dimensions by necessity distort the display of the three-dimensional surface of the Earth. While this is true of any map, these distortions reach extremes in a world map. Many techniques have been developed to present world maps that address diverse technical and aesthetic goals. Charting a world map requires global knowledge of the Earth, its oceans, and its continents. From prehistory through the Middle Ages, creating an accurate world map would have been impossible because less than half of Earth's coastlines and only a small fraction of its continental interiors were known to any culture. With exploration that began during the European Renaissance, knowledge of the Earth's surface accumulated rapidly, such that most of the world's coastlines had been mapped, at least roughly, by the mid-1700s and the continental interiors by the twentieth century. Maps of the world generally focus either on political features or on physical features. Political maps emphasize territorial boundaries and human settlement. Physical maps show geographical features such as mountains, soil type, or land use. Geological maps show not only the surface, but characteristics of the underlying rock, fault lines, and subsurface structures. Choropleth maps use color hue and intensity to contrast differences between regions, such as demographic or economic statistics. Map projections All world maps are based on one of several map projections, or methods of representing a globe on a plane. All projections distort geographic features, distances, and directions in some way. The various map projections that have been developed provide different ways of balancing accuracy and the unavoidable distortion inherent in making world maps. Perhaps the best-known projection is the Mercator Projection, originally designed as a nautical chart. Thematic maps A thematic map shows geographical information about one or a few focused subjects. These maps "can portray physical, social, political, cultural, economic, sociological, agricultural, or any other aspects of a city, state, region, nation, or continent". Historical maps Early world maps cover depictions of the world from the Iron Age to the Age of Discovery and the emergence of modern geography during the early modern period. Old maps provide information about places that were known in past times, as well as the philosophical and cultural basis of the map, which were often much different from modern cartography. Maps are one means by which scientists distribute their ideas and pass them on to future generations. See also Wikipedia's clickable world map Global Map Globe International Map of the World List of map projections List of world map changes Mappa mundi Maps of the world Rhumbline network Theorema Egregium Time zone References Further reading Edson, Evelyn (2011). The World Map, 1300–1492: The Persistence of Tradition and Transformation. JHU Press. Harvey, P. D. A. (2006). The Hereford world map: medieval world maps and their context. British Library.
World map
62475939
In cartography, an equivalent, authalic, or equal-area projection is a map projection that preserves relative area measure between any and all map regions. Equivalent projections are widely used for thematic maps showing scenario distribution such as population, farmland distribution, forested areas, and so forth, because an equal-area map does not change apparent density of the phenomenon being mapped. By Gauss's Theorema Egregium, an equal-area projection cannot be conformal. This implies that an equal-area projection inevitably distorts shapes. Even though a point or points or a path or paths on a map might have no distortion, the greater the area of the region being mapped, the greater and more obvious the distortion of shapes inevitably becomes. Description In order for a map projection of the sphere to be equal-area, its generating formulae must meet this Cauchy-Riemann-like condition: where is constant throughout the map. Here, represents latitude; represents longitude; and and are the projected (planar) coordinates for a given coordinate pair. For example, the sinusoidal projection is a very simple equal-area projections. Its generating formulæ are: where is the radius of the globe. Computing the partial derivatives, and so with taking the value of the constant . For an equal-area map of the ellipsoid, the corresponding differential condition that must be met is: where is the eccentricity of the ellipsoid of revolution. Statistical grid The term "statistical grid" refers to a discrete grid (global or local) of an equal-area surface representation, used for data visualization, geocode and statistical spatial analysis. List of equal-area projections These are some projections that preserve area: Azimuthal Lambert azimuthal equal-area Wiechel (pseudoazimuthal) Conic Albers Lambert equal-area conic projection Pseudoconical Bonne Bottomley Werner Cylindrical (with latitude of no distortion) Lambert cylindrical equal-area (0°) Behrmann (30°) Hobo–Dyer (37°30′) Gall–Peters (45°) Pseudocylindrical Boggs eumorphic Collignon Eckert II, IV and VI Equal Earth Goode's homolosine Mollweide Sinusoidal Tobler hyperelliptical Other Eckert-Greifendorff McBryde-Thomas Flat-Polar Quartic Projection Hammer Strebe 1995 Snyder equal-area projection, used for geodesic grids. See also Authalic latitude Authalic radius Equiareal map (mathematics) Measure-preserving dynamical system Geodesic polygon area References
Equal-area projection
15346132
In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth (a sphere or an ellipsoid) is preserved in the image of the projection; that is, the projection is a conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, their images on a map with a conformal projection cross at a 39° angle. Properties A conformal projection can be defined as one that is locally conformal at every point on the map, albeit possibly with singular points where conformality fails. Thus, every small figure is nearly similar to its image on the map. The projection preserves the ratio of two lengths in the small domain. All of the projection's Tissot's indicatrices are circles. Conformal projections preserve only small figures. Large figures are distorted by even conformal projections. In a conformal projection, any small figure is similar to the image, but the ratio of similarity (scale) varies by location, which explains the distortion of the conformal projection. In a conformal projection, parallels and meridians cross rectangularly on the map; but not all maps with this property are conformal. 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. these projections are not conformal. As proven by Leonhard Euler in 1775, a conformal map projection cannot be equal-area, nor can an equal-area map projection be conformal. This is also a consequence of Carl Gauss's 1827 Theorema Egregium [Remarkable Theorem]. A conformal parameterization of a disc-like domain on the sphere is deemed scale-optimal when it minimizes the ratio of maximum to minimum scale across the entire map. This occurs by assigning a unit scale to the boundary of the disc. Chebyshev applied this theorem to create a conformal map for the European part of the Russian Empire, which reduced scale errors to 1/50. List of conformal projections Mercator projection (conformal cylindrical projection) Mercator projection of normal aspect (Every rhumb line is drawn as a straight line on the map.) Transverse Mercator projection Gauss–Krüger coordinate system (This projection preserves lengths on the central meridian on an ellipsoid) Oblique Mercator projection Space-oblique Mercator projection (a modified projection from Oblique Mercator projection for satellite orbits with the Earth rotation within near conformality) Lambert conformal conic projection Oblique conformal conic projection (This projection is sometimes used for long-shaped regions, like as continents of Americas or Japanese archipelago.) Stereographic projection (Conformal azimuthal projection. Every circle on the earth is drawn as a circle or a straight line on the map.) Miller Oblated Stereographic Projection (Modified stereographic projection for continents of Africa and Europe.) GS50 projection (This projection are made from a stereographic projection with an adjustment by a polynomial on complex numbers.) Littrow projection (conformal retro-azimuthal projection) Lagrange projection (a polyconic projection, and a composition of a Lambert conformal conic
Conformal map projection
259906
Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this manner (translated from Latin): Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged. The theorem is "remarkable" because the starting definition of Gaussian curvature makes direct use of position of the surface in space. So it is quite surprising that the result does not depend on its embedding in spite of all bending and twisting deformations undergone. In modern mathematical terminology, the theorem may be stated as follows: Elementary applications A sphere of radius R has constant Gaussian curvature which is equal to 1/R2. At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling. Conversely, the surface of a sphere cannot be unfolded onto a flat plane without distorting the distances. If one were to step on an empty egg shell, its edges have to split in expansion before being flattened. Mathematically, a sphere and a plane are not isometric, even locally. This fact is significant for cartography: it implies that no planar (flat) map of Earth can be perfect, even for a portion of the Earth's surface. Thus every cartographic projection necessarily distorts at least some distances. The catenoid and the helicoid are two very different-looking surfaces. Nevertheless, each of them can be continuously bent into the other: they are locally isometric. It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same. Thus isometry is simply bending and twisting of a surface without internal crumpling or tearing, in other words without extra tension, compression, or shear. An application of the theorem is seen when a flat object is somewhat folded or bent along a line, creating rigidity in the perpendicular direction. This is of practical use in construction, as well as in a common pizza-eating strategy: A flat slice of pizza can be seen as a surface with constant Gaussian curvature 0. Gently bending a slice must then roughly maintain this curvature (assuming the bend is roughly a local isometry). If one bends a slice horizontally along a radius, non-zero principal curvatures are created along the bend, dictating that the other principal curvature at these points must be zero.
Theorema Egregium
5645470
The history of cartography refers to the development and consequences of cartography, or mapmaking technology, throughout human history. Maps have been one of the most important human inventions for millennia, allowing humans to explain and navigate their way through the world. When and how the earliest maps were made is unclear, but maps of local terrain are believed to have been independently invented by many cultures. The earliest surviving maps include cave paintings and etchings on tusk and stone. Maps were produced extensively by ancient Babylon, Greece, Rome, China, and India. The earliest maps ignored the curvature of Earth's surface, both because the shape of the Earth was uncertain and because the curvature is not important across the small areas being mapped. However, since the age of Classical Greece, maps of large regions, and especially of the world, have used projection from a model globe to control how the inevitable distortion gets apportioned on the map. Modern methods of transportation, the use of surveillance aircraft, and more recently the availability of satellite imagery have made documentation of many areas possible that were previously inaccessible. Free online services such as Google Earth have made accurate maps of the world more accessible than ever before. Etymology The English term cartography is modern, borrowed from the French cartographie in the 1840s, itself based on Middle Latin carta "map". Pre-modern era Earliest known maps It is not always clear whether an ancient artifact had been wrought as a map or as something else. The definition of "map" is also not precise. Thus, no single artifact is generally accepted to be the earliest surviving map. Candidates include: A map-like representation of a mountain, river, valleys and routes around Pavlov in the Czech Republic, carved on a mammoth tusk, that has been dated to 25,000 BC. An Aboriginal Australian cylcon that may be as much as 20,000 years old that is thought to depict the Darling River. A map etched on a mammoth bone at Mezhyrich that is about 15,000 years old. Dots dating to 14,500 BC found on the walls of the Lascaux caves map of part of the night sky, including the three bright stars Vega, Deneb, and Altair (the Summer Triangle asterism), as well as the Pleiades star cluster. The Cuevas de El Castillo in Spain that contains a dot map of the Corona Borealis constellation dating from 12,000 BC. A polished chunk of sandstone from a cave in Spanish Navarre, dated to 14,000 BC, that may be symbols for landscape features, such as hills or dwellings, superimposed on animal etchings. Alternatively, it may also represent a spiritual landscape, or simple incisings. Another ancient picture that resembles a map that was created in the late 7th millennium BC in Çatalhöyük, Anatolia, modern Turkey. This wall painting may represent a plan of this Neolithic village; however, recent scholarship has questioned the identification of this painting as a map. The "Saint-Bélec slab" (2200–1600 BC), whose lines and symbols have been argued to represent a cadastral
History of cartography
7654598
John Parr Snyder (12 April 1926 – 28 April 1997) was an American cartographer most known for his work on map projections for the United States Geological Survey (USGS). Educated at Purdue and MIT as a chemical engineer, he had a lifetime interest in map projections as a hobby, but found the calculations tedious without the benefit of expensive calculators or computers. At a cartography conference in 1976, he learned of the need for a map projection that would suit the special needs of Landsat satellite imagery. He had recently been able to purchase a pocket calculator (TI-59) of his own and set to work creating what became known as the space-oblique mercator projection, which he provided to the USGS at no charge. He was subsequently offered a job within the USGS within two years, where his work apparently led him to the eventual publication of the definitive technical guide to map projections entitled Map Projections: A Working Manual among other works. He also authored Flattening the Earth: Two Thousand Years of Map Projections which details the historical development of hundreds of map projections. Snyder developed at least one other projection, called GS50, which uses a complex polynomial to project the 50 U.S. states with minimal distortion. He taught courses on map projection at George Mason University. He was president of the American Cartographic Association (now CaGIS) from 1990–1991 and also served as a secretary to the Washington Map Society. John Snyder died April 28, 1997. Publications An album of map projections. USGS Professional Paper No. 1453. 1989. Bibliography of map projections. USGS Bulletin No. 1856. 1988. Map Projections: A Working Manual USGS Professional Paper 1395. 1987. Map projections used for large-scale quadrangles by the U.S. Geological Survey. USGS Circular No. 982. 1986. Space Oblique Mercator projection mathematical development. USGS Bulletin No. 1518. 1981. See also Snyder equal-area projection References External links Further biographical information Donna Urschel Geography by the Numbers :Staff Member Solves Mystery of Mapping Equations, a biography of J.P. Snyder at the Library of Congress Information and electronic version of Map Projections: A Working Manual About the GS50 projection Obituary posted to CANSPACE forum 1926 births 1997 deaths American cartographers American civil servants Purdue University alumni Massachusetts Institute of Technology alumni 20th-century American geologists United States Geological Survey personnel 20th-century cartographers
John P. Snyder
56494038
The Snake Projection is a continuous map projection typically used as the planar coordinate system for realizing low distortion throughout long linear engineering projects. Details The Snake Projection was originally developed by University College London and Network Rail to provide a continuous low distortion projection for the West Coast Mainline infrastructure works. The parameters defining each Snake Projection are tailored for the specific project; the most typical use is with large-scale linear engineering projects such as rail infrastructure, however the projection is equally applicable to any application requiring a low distortion grid along a linear route (for example pipelines and roads). The name of the projection is derived from the sinuous snake-like nature of the projects it may be designed for. Typical map projection distance distortion characteristics of a Snake Projection are minimal over the whole route within approximately 20 kilometres of the centre line. The principal advantage of the projection is that, for the corridor defining the design space, distances measured on the ground have a nearly one to one relationship with distances in coordinate space (i.e. no scale factor need be applied to convert between distances in grid and distances on the ground). The length of the applicable corridor is variable on a project basis, however when required the projection can extend over several hundreds of kilometres to achieve grid distortion of less than 20 parts per million along the route. The main disadvantage is that away from the design corridor the distortion of the projection is not controlled. The Snake Projection is suited for engineering purposes due to its low distortion characteristics. An example of its differentiation from mapping grids is the 60m increase in length of the London to Birmingham section of the HS2 rail line, purely due to the more accurate grid representation compared to the length when using the national mapping coordinate system British National Grid. Usage The Snake Projection is the engineering coordinate system used for a significant proportion of primary rail routes in the UK, including that of the HS2 London to Birmingham high speed line. For the London to Glasgow West Coast Main Line the distortion in the Snake Projection used is no greater than 20 parts per million within 5 kilometres of either side of the track. Implementation The Snake Projection algorithm converts between geographical and grid coordinates, however the method of technical implementation can vary. One method of implementing a Snake Projection is to define using an NTv2 geodetic transformation coupled with a standard parameterised map projection (such as Transverse Mercator); this is increasing in popularity due to better compatibility with CAD and GIS software. The global EPSG geodetic coordinate system database features several snake projection definitions through the NTv2 approach. Other implementations include those published through the SnakeGrid organisation. See also List of map projections Surveying References Map projections Geodesy Rail infrastructure Surveying Civil engineering
Snake Projection
1785304
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, using it in a world map in 1570. The projection represents the poles as points, as they are on the sphere, but the meridians and continents are distorted. The equator and the central meridian are the most accurate parts of the map, having no distortion at all, and the further away from those that one examines, the greater the distortion. The projection is defined by: where is the latitude, λ is the longitude, and λ is the longitude of the central meridian. Scale is constant along the central meridian, and 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 (MODIS) science team. See also List of map projections Gerardus Mercator, Nicolas Sanson, and John Flamsteed – mathematicians who developed the technique. References External links Pseudocylindrical Projections Table of examples and properties of all common projections, from radicalcartography.net Map projections Equal-area projections
Sinusoidal projection
7354687
The Winkel tripel projection (Winkel III), a modified azimuthal map projection of the world, is one of three projections proposed by German cartographer Oswald Winkel (7 January 1874 – 18 July 1953) in 1921. The projection is the arithmetic mean of the equirectangular projection and the Aitoff projection: The name (German for 'triple') refers to Winkel's goal of minimizing three kinds of distortion: area, direction, and distance. Algorithm where λ is the longitude relative to the central meridian of the projection, φ is the latitude, φ is the standard parallel for the equirectangular projection, sinc is the unnormalized cardinal sine function, and In his proposal, Winkel set A closed-form inverse mapping does not exist, and computing the inverse numerically requires the use of iterative methods. Comparison with other projections David M. Goldberg and J. Richard Gott III showed that the Winkel tripel fares better against several other projections analyzed against their measures of distortion, producing minimal distance, Tissot indicatrix ellipticity and area errors, and the least skew of any of the projections they studied. By a different metric, Capek's "Q", the Winkel tripel ranked ninth among a hundred map projections of the world, behind the common Eckert IV projection and Robinson projections. In 1998, the Winkel tripel projection replaced the Robinson projection as the standard projection for world maps made by the National Geographic Society. Many educational institutes and textbooks soon followed National Geographic's example in adopting the projection, most of which still utilize it. See also List of map projections References External links Table of common projections Map projections
Winkel tripel projection
13980071
The history of longitude describes the centuries-long effort by astronomers, cartographers and navigators to discover a means of determining the longitude of any given place on Earth. The measurement of longitude is important to both cartography and navigation. In particular, for safe ocean navigation, knowledge of both latitude and longitude is required, however latitude can be determined with good accuracy with local astronomical observations. Finding an accurate and practical method of determining longitude took centuries of study and invention by some of the greatest scientists and engineers. Determining longitude relative to the meridian through some fixed location requires that observations be tied to a time scale that is the same at both locations, so the longitude problem reduces to finding a way to coordinate clocks at distant places. Early approaches used astronomical events that could be predicted with great accuracy, such as eclipses, and building clocks, known as chronometers, that could keep time with sufficient accuracy while being transported great distances by ship. Later methods used the telegraph and then radio to synchronize clocks. Today the problem of longitude has been solved to centimeter accuracy through satellite navigation. Longitude before the telescope Eratosthenes in the 3rd century BCE first proposed a system of latitude and longitude for a map of the world. His prime meridian (line of longitude) passed through Alexandria and Rhodes, while his parallels (lines of latitude) were not regularly spaced, but passed through known locations, often at the expense of being straight lines. By the 2nd century BCE Hipparchus was using a systematic coordinate system, based on dividing the circle into 360°, to uniquely specify places on Earth. So longitudes could be expressed as degrees east or west of the primary meridian, as is done today (though the primary meridian is different). He also proposed a method of determining longitude by comparing the local time of a lunar eclipse at two different places, to obtain the difference in longitude between them. This method was not very accurate, given the limitations of the available clocks, and it was seldom done – possibly only once, using the Arbela eclipse of 330 BCE. But the method is sound, and this is the first recognition that longitude can be determined by accurate knowledge of time. Ptolemy, in the 2nd century CE, based his mapping system on estimated distances and directions reported by travellers. Until then, all maps had used a rectangular grid with latitude and longitude as straight lines intersecting at right angles. For large areas this leads to unacceptable distortion, and for his map of the inhabited world, Ptolemy used projections (to use the modern term) with curved parallels that reduced the distortion. No maps (or manuscripts of his work) exist that are older than the 13th century, but in his Geography he gave detailed instructions and latitude and longitude coordinates for hundreds of locations that are sufficient to re-create the maps. While Ptolemy's system is well-founded, the actual data used are of very variable quality, leading to many inaccuracies
History of longitude
276184
Elastography is any of a class of medical imaging modalities that map the elastic properties and stiffness of soft tissue. The main idea is that whether the tissue is hard or soft will give diagnostic information about the presence or status of disease. For example, cancerous tumours will often be harder than the surrounding tissue, and diseased livers are stiffer than healthy ones. The most prominent techniques use ultrasound or magnetic resonance imaging (MRI) to make both the stiffness map and an anatomical image for comparison. Historical background Palpation is the practice of feeling the stiffness of a person's or animal's tissues with the health practitioner's hands. Manual palpation dates back at least to 1500 BC, with the Egyptian Ebers Papyrus and Edwin Smith Papyrus both giving instructions on diagnosis with palpation. In ancient Greece, Hippocrates gave instructions on many forms of diagnosis using palpation, including palpation of the breasts, wounds, bowels, ulcers, uterus, skin, and tumours. In the modern Western world, palpation became considered a respectable method of diagnosis in the 1930s. Since then, the practice of palpation has become widespread, and it is considered an effective method of detecting tumours and other pathologies. Manual palpation has several important limitations: it is limited to tissues accessible to the physician's hand, it is distorted by any intervening tissue, and it is qualitative but not quantitative. Elastography, the measurement of tissue stiffness, seeks to address these challenges. How it works There are numerous elastographic techniques, in development stages from early research to extensive clinical application. Each of these techniques works in a different way. What all methods have in common is that they create a distortion in the tissue, observe and process the tissue response to infer the mechanical properties of the tissue, and then display the results to the operator, usually as an image. Each elastographic method is characterized by the way it does each of these things. Inducing a distortion To image the mechanical properties of tissue, we need to see how it behaves when deformed. There are three main ways of inducing a distortion to observe. These are: Pushing/deforming or vibrating the surface of the body (skin) or organ (prostate) with a probe or a tool, Using acoustic radiation force impulse imaging using ultrasound to remotely create a 'push' inside the tissue, and Using distortions created by normal physiological processes, e.g. pulse or heartbeat. Observing the response The primary way elastographic techniques are categorized is by what imaging modality (type) they use to observe the response. Elastographic techniques use ultrasound, magnetic resonance imaging (MRI) and pressure/stress sensors in tactile imaging (TI) using tactile sensor(s). There are a handful of other methods that exist as well. The observation of the tissue response can take many forms. In terms of the image obtained, it can be 1-D (i.e. a line), 2-D (a plane), 3-D (a volume), or 0-D (a single value), and it can be a video or a single image. In most cases, the result is displayed to the
Elastography
8492267
The General Perspective projection is a map projection. When the Earth is photographed from space, the camera records the view as a perspective projection. When the camera is aimed toward the center of the Earth, the resulting projection is called Vertical Perspective. When aimed in other directions, the resulting projection is called a Tilted Perspective. Perspective and usage The Vertical Perspective is related to the stereographic projection, gnomonic projection, and orthographic projection. These are all true perspective projections, meaning that they result from viewing the globe from some vantage point. They are also azimuthal projections, meaning that the projection surface is a plane tangent to the sphere. This results in correct directions from the center to all other points. The point of perspective, or vantage point, for the General Perspective Projection is at a finite distance. It depicts the earth as it appears from some relatively short distance above the surface, typically a few hundred to a few tens of thousands of kilometers. When tilted, the General Perspective projection, also called the tilted perspective projection, is not azimuthal (see second figure below); directions are not true from the central point, and the projection plane is not tangent to the sphere. Tilted perspectives are common from aerial and low orbit photography, generally taken from at a height measured in kilometers to hundreds of kilometers, rather than the hundreds or thousands of kilometers typical of a vertical perspective. However, Richard Edes Harrison pioneered the use of this projection on strategic maps showing military theaters during WWII. Some prominent Internet mapping tools also use the tilted perspective projection. For example, Google Earth and NASA World Wind show the globe as it appears from space. These applications permit a wide variety of interactive pan and zoom operations, including fly-through simulations, mimicking pictures or movies taken with a hand-held camera from an airplane or spacecraft. History Some forms of the projection were known to the Greeks and Egyptians 2,000 years ago. It was studied by several French and British scientists in the 18th and 19th centuries. However, the projection had little practical value at that time; computationally simpler nonperspective azimuthal projections could be used instead. Space exploration led to a renewed interest in the perspective projection. Now the concern was for a pictorial view from space, not for minimal distortion. A picture taken with a hand-held camera from the window of a spacecraft has a tilted vertical perspective, so the crewed Gemini and Apollo space missions sparked interest in this projection. Mathematics The formulas for the general perspective projection are derived using trigonometry. They are written in terms of longitude (λ) and latitude (φ) on the sphere. Define the radius of the sphere R and the center point (and origin) of the projection (λ0, φ0). The equations for the orthographic projection onto the (x, y) tangent plane reduce to the following: where is the angular distance and denotes the distance from the perspective point to the center of earth. It is positive in the direction
General Perspective projection
14738183
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 geoscience. Construction In the figure, the area-preserving property of the projection can be seen by comparing a grid sector near the center of the net with one at the far right of the net. The two sectors have the same area on the sphere and the same area on the disk. The angle-distorting property can be seen by examining the grid lines; most of them do not intersect at right angles on the Schmidt net. A single Schmidt net can only represent one hemisphere of the earth; typically a pair of Schmidt nets is used to represent both sides of the globe. It is relatively simple to re-plot a gridded map of the world onto a Schmidt net if the azimuth is chosen to be the junction of the equator with any particular meridian from the world-map's grid. Each grid square surrounding this chosen longitude is simply re-plotted into the corresponding distorted grid-square in the Schmidt net. Points of latitude-longitude can be plotted relative to the azimuth's longitude, interpolating between grid lines in the Schmidt net. For greater accuracy, it is helpful to have a net with finer spacing than 10°; spacings of 2° are common. 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. The Schmidt net's horizontal axis can then be used as a scalar measuring device to convert the point's latitude (relative to the pole) into a radial distance from the centre of the circle. Alternatively, the Schmidt net could be replaced entirely with a correctly projected polar grid, and grid squares from a map re-drawn into this disc. Use 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. Conversely, the Wulff net ("equal-angle projection") is used to plot crystallographic axes and faces. Sources References Technical drawing Map projections
Schmidt net
12518245
The pinhole camera model describes the mathematical relationship between the coordinates of a point in three-dimensional space and its projection onto the image plane of an ideal pinhole camera, where the camera aperture is described as a point and no lenses are used to focus light. The model does not include, for example, geometric distortions or blurring of unfocused objects caused by lenses and finite sized apertures. It also does not take into account that most practical cameras have only discrete image coordinates. This means that the pinhole camera model can only be used as a first order approximation of the mapping from a 3D scene to a 2D image. Its validity depends on the quality of the camera and, in general, decreases from the center of the image to the edges as lens distortion effects increase. Some of the effects that the pinhole camera model does not take into account can be compensated, for example by applying suitable coordinate transformations on the image coordinates; other effects are sufficiently small to be neglected if a high quality camera is used. This means that the pinhole camera model often can be used as a reasonable description of how a camera depicts a 3D scene, for example in computer vision and computer graphics. Geometry The geometry related to the mapping of a pinhole camera is illustrated in the figure. The figure contains the following basic objects: A 3D orthogonal coordinate system with its origin at O. This is also where the camera aperture is located. The three axes of the coordinate system are referred to as X1, X2, X3. Axis X3 is pointing in the viewing direction of the camera and is referred to as the optical axis, principal axis, or principal ray. The plane which is spanned by axes X1 and X2 is the front side of the camera, or principal plane. An image plane, where the 3D world is projected through the aperture of the camera. The image plane is parallel to axes X1 and X2 and is located at distance from the origin O in the negative direction of the X3 axis, where f is the focal length of the pinhole camera. A practical implementation of a pinhole camera implies that the image plane is located such that it intersects the X3 axis at coordinate -f where f > 0. A point R at the intersection of the optical axis and the image plane. This point is referred to as the principal point or image center. A point P somewhere in the world at coordinate relative to the axes X1, X2, and X3. The projection line of point P into the camera. This is the green line which passes through point P and the point O. The projection of point P onto the image plane, denoted Q. This point is given by the intersection of the projection line (green) and the image plane. In any practical situation we can assume that > 0 which means that the intersection point
Pinhole camera model
3663444
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. The poles accrue infinite distortion, becoming lines instead of points. History 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. Formulae where φ is the latitude, λ is the longitude and λ is the central meridian. See also List of map projections Lambert azimuthal equal-area projection Lambert conformal conic projection References External links Table of examples and properties of all common projections, from radicalcartography.net Map projections Cylindrical equal-area projections
Lambert cylindrical equal-area projection
1968518
In geometric optics, distortion is a deviation from rectilinear projection; a projection in which straight lines in a scene remain straight in an image. It is a form of optical aberration. Radial distortion Although distortion can be irregular or follow many patterns, the most commonly encountered distortions are radially symmetric, or approximately so, arising from the symmetry of a photographic lens. These radial distortions can usually be classified as either barrel distortions or pincushion distortions. Mathematically, barrel and pincushion distortion are quadratic, meaning they increase as the square of distance from the center. In mustache distortion the quartic (degree 4) term is significant: in the center, the degree 2 barrel distortion is dominant, while at the edge the degree 4 distortion in the pincushion direction dominates. Other distortions are in principle possible – pincushion in center and barrel at the edge, or higher order distortions (degree 6, degree 8) – but do not generally occur in practical lenses, and higher order distortions are small relative to the main barrel and pincushion effects. Occurrence In photography, distortion is particularly associated with zoom lenses, particularly large-range zooms, but may also be found in prime lenses, and depends on focal distance – for example, the Canon EF 50mm 1.4 exhibits barrel distortion at extremely short focal distances. Barrel distortion may be found in wide-angle lenses, and is often seen at the wide-angle end of zoom lenses, while pincushion distortion is often seen in older or low-end telephoto lenses. Mustache distortion is observed particularly on the wide end of zooms, with certain retrofocus lenses, and more recently on large-range zooms such as the Nikon 18–200 mm. A certain amount of pincushion distortion is often found with visual optical instruments, e.g., binoculars, where it serves to counteract the globe effect. In order to understand these distortions, it should be remembered that these are radial defects; the optical systems in question have rotational symmetry (omitting non-radial defects), so the didactically correct test image would be a set of concentric circles having even separation – like a shooter's target. It will then be observed that these common distortions actually imply a nonlinear radius mapping from the object to the image: What is seemingly pincushion distortion, is actually simply an exaggerated radius mapping for large radii in comparison with small radii. A graph showing radius transformations (from object to image) will be steeper in the upper (rightmost) end. Conversely, barrel distortion is actually a diminished radius mapping for large radii in comparison with small radii. A graph showing radius transformations (from object to image) will be less steep in the upper (rightmost) end. Chromatic aberration Radial distortion that depends on wavelength is called "lateral chromatic aberration" – "lateral" because radial, "chromatic" because dependent on color (wavelength). This can cause colored fringes in high-contrast areas in the outer parts of the image. This should not be confused with axial (longitudinal) chromatic aberration, which causes aberrations throughout the field, particularly purple fringing. Origin of terms The names for these distortions
Distortion (optics)
3909097
In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection is also called a projection, even if the idempotence property is lost. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: The projection from a point onto a plane or central projection: If C is a point, called the center of projection, then the projection of a point P different from C onto a plane that does not contain C is the intersection of the line CP with the plane. The points P such that the line CP is parallel to the plane does not have any image by the projection, but one often says that they project to a point at infinity of the plane (see Projective geometry for a formalization of this terminology). The projection of the point C itself is not defined. The projection parallel to a direction D, onto a plane or parallel projection: The image of a point P is the intersection with the plane of the line parallel to D passing through P. See for an accurate definition, generalized to any dimension. The concept of projection in mathematics is a very old one, and most likely has its roots in the phenomenon of the shadows cast by real-world objects on the ground. This rudimentary idea was refined and abstracted, first in a geometric context and later in other branches of mathematics. Over time different versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations. In cartography, a map projection is a map of a part of the surface of the Earth onto a plane, which, in some cases, but not always, is the restriction of a projection in the above meaning. The 3D projections are also at the basis of the theory of perspective. The need for unifying the two kinds of projections and of defining the image by a central projection of any point different of the center of projection are at the origin of projective geometry. However, a projective transformation is a bijection of a projective space, a property not shared with the projections of this article. Definition Generally, a mapping where the domain and codomain are the same set (or mathematical structure) is a projection if the mapping is idempotent, which means that a projection is equal to
Projection (mathematics)
180583
The Dymaxion map or Fuller map is a projection of a world map onto the surface of an icosahedron, which can be unfolded and flattened to two dimensions. The flat map is heavily interrupted in order to preserve shapes and sizes. The projection was invented by Buckminster Fuller. The March 1, 1943, edition of Life magazine included a photographic essay titled "Life Presents R. Buckminster Fuller's Dymaxion World". The article included several examples of its use together with a pull-out section that could be assembled as a "three-dimensional approximation of a globe or laid out as a flat map, with which the world may be fitted together and rearranged to illuminate special aspects of its geography". Fuller applied for a patent in the United States in February 1944, showing a projection onto a cuboctahedron, which he called "dymaxion". The patent was issued in January 1946. In 1954, Fuller and cartographer Shoji Sadao produced the Airocean World Map, a version of the Dymaxion map that used a modified but mostly regular icosahedron as the base for the projection. The version most commonly referred to today, it depicts Earth's continents as "one island", or nearly contiguous land masses. The Dymaxion projection is intended only for representations of the entire globe. It is not a gnomonic projection, whereby global data expands from the center point of a tangent facet outward to the edges. Instead, each triangle edge of the Dymaxion map matches the scale of a partial great circle on a corresponding globe, and other points within each facet shrink toward its middle, rather than enlarging to the peripheries. Fuller's 1980 version of the Dymaxion map was the first definition and use of a mathematical transformation process to make the map. It is a polyhedral map projection. The name Dymaxion was applied by Fuller to several of his inventions. Properties Though neither conformal nor equal-area, Fuller claimed that his map had several advantages over other projections for world maps. It has less distortion of relative size of areas, most notably when compared to the Mercator projection; and less distortion of shapes of areas, notably when compared to the Gall–Peters projection. Other compromise projections attempt a similar trade-off. More unusually, the Dymaxion map does not have any "right way up". Fuller argued that in the universe there is no "up" and "down", or "north" and "south": only "in" and "out". Gravitational forces of the stars and planets created "in", meaning "towards the gravitational center", and "out", meaning "away from the gravitational center". He attributed the north-up-superior/south-down-inferior presentation of most other world maps to cultural bias. Fuller intended the map to be unfolded in different ways to emphasize different aspects of the world. Peeling the triangular faces of the icosahedron apart in one way results in an icosahedral net that shows an almost contiguous land mass comprising all of Earth's continents – not groups of continents divided by oceans. Peeling the solid apart in a different way presents a view of the world dominated by
Dymaxion map
1032887
The Mollweide projection is an equal-area, pseudocylindrical map projection generally used for maps of the world or celestial sphere. It is also known as the Babinet projection, homalographic projection, homolographic projection, and elliptical projection. The projection trades accuracy of angle and shape for accuracy of proportions in area, and as such is used where that property is needed, such as maps depicting global distributions. The projection was first published by mathematician and astronomer Karl (or Carl) Brandan Mollweide (1774–1825) of Leipzig in 1805. It was reinvented and popularized in 1857 by Jacques Babinet, who gave it the name homalographic projection. The variation homolographic arose from frequent nineteenth-century usage in star atlases. Properties The Mollweide is a pseudocylindrical projection in which the equator is represented as a straight horizontal line perpendicular to a central meridian that is one-half the equator's length. The other parallels compress near the poles, while the other meridians are equally spaced at the equator. The meridians at 90 degrees east and west form a perfect circle, and the whole earth is depicted in a proportional 2:1 ellipse. The proportion of the area of the ellipse between any given parallel and the equator is the same as the proportion of the area on the globe between that parallel and the equator, but at the expense of shape distortion, which is significant at the perimeter of the ellipse, although not as severe as in the sinusoidal projection. Shape distortion may be diminished by using an interrupted version. A sinusoidal interrupted Mollweide projection discards the central meridian in favor of alternating half-meridians which terminate at right angles to the equator. This has the effect of dividing the globe into lobes. In contrast, a parallel interrupted Mollweide projection uses multiple disjoint central meridians, giving the effect of multiple ellipses joined at the equator. More rarely, the projection can be drawn obliquely to shift the areas of distortion to the oceans, allowing the continents to remain truer to form. The Mollweide, or its properties, has inspired the creation of several other projections, including the Goode's homolosine, van der Grinten and the Boggs eumorphic. Mathematical formulation The projection transforms from latitude and longitude to map coordinates x and y via the following equations: where θ is an auxiliary angle defined by and λ is the longitude, λ is the central meridian, φ is the latitude, and R is the radius of the globe to be projected. The map has area 4R, conforming to the surface area of the generating globe. The x-coordinate has a range of [−2R, 2R], and the y-coordinate has a range of [−R, R]. Equation (1) may be solved with rapid convergence (but slow near the poles) using Newton–Raphson iteration: If φ = ±, then also θ = ±. In that case the iteration should be bypassed; otherwise, division by zero may result. There exists a closed-form inverse transformation: where θ can be found by the relation The inverse transformations allow one to find the latitude and longitude corresponding to
Mollweide projection
7711317
The Chamberlin trimetric projection is a map projection where three points are fixed on the globe and the points on the sphere are mapped onto a plane by triangulation. It was developed in 1946 by Wellman Chamberlin for the National Geographic Society. Chamberlin was chief cartographer for the Society from 1964 to 1971. The projection's principal feature is that it compromises between distortions of area, direction, and distance. A Chamberlin trimetric map therefore gives an excellent overall sense of the region being mapped. Many National Geographic Society maps of single continents use this projection. As originally implemented, the projection algorithm begins with the selection of three base points to form a spherical triangle minimally enclosing the area to be mapped. These points are mapped at the correct distance from each other according to the map’s chosen scale; aside from arbitrary rotation and translation, the position of the three points on the plane are unambiguous because a triangle is determined by the lengths of its sides. To map any point P, the spherical distances from each of the base points to P are calculated. Using each of the three mapped base points as center, a circle is drawn with radius equal to the scale spherical distance of P from the base point. The three circles will always intersect at one, two, or three points. Intersecting at one point happens only at the base points, which are already mapped and therefore need no further processing. Intersecting at two points happens only along the straight line between two mapped base points. Chamberlin did not specify how to handle this case, but it would be determined by which definition of triangle center is chosen, as noted next. In the remaining case, which is most of the map, connecting the three points of intersection of the circles by line segments creates a small triangle. The position of P′ is determined by the center of the triangle. Chamberlin did not specify which definition of center to use. A Chamberlin trimetric projection map was originally obtained by graphically mapping points at regular intervals of latitude and longitude, with shorelines and other features then mapped by interpolation. 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. In some cases, the Chamberlin trimetric projection is difficult to distinguish visually from the Lambert azimuthal equal-area projection centered on the same area. See also Two-point equidistant projection References External links The Chamberlin Trimetric Projection - Implementations of the projection using Matlab scripts. The Chamberlin Trimetric Projection - Notes on the projection from a cartography class at Colorado State University. Map projections
Chamberlin trimetric projection
26410981
In cartography, the normal cylindrical equal-area projection is a family of normal cylindrical, equal-area map projections. History 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. Description The projection: is cylindrical, that means it has a cylindrical projection surface is normal, that means it has a normal aspect is an equal-area projection, that means any two areas in the map have the same relative size compared to their size on the sphere. The term "normal cylindrical projection" is used to refer to any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude are mapped to horizontal lines (or, mutatis mutandis, more generally, radial lines from a fixed point are mapped to equally spaced parallel lines and concentric circles around it are mapped to perpendicular lines). The mapping of meridians to vertical lines can be visualized by imagining a cylinder whose axis coincides with the Earth's axis of rotation, then projecting onto the cylinder, and subsequently unfolding the cylinder. By the geometry of their construction, cylindrical projections stretch distances east-west. The amount of stretch is the same at any chosen latitude on all cylindrical projections, and is given by the secant of the latitude as a multiple of the equator's scale. The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude is given by φ): The only normal cylindrical projections that preserve area have a north-south compression precisely the reciprocal of east-west stretching (cos φ). This divides north-south distances by a factor equal to the secant of the latitude, preserving area but distorting shapes. East–west scale matching the north–south scale Depending on the stretch factor S, any particular cylindrical equal-area projection either has zero, one or two latitudes for which the east–west scale matches the north–south scale. S>1 : zero S=1 : one, that latitude is the equator S<1 : a pair of identical latitudes of opposite sign Formulae The formulae presume a spherical model and use these definitions: λ is the longitude λ is the central meridian φ is the latitude φ is the standard latitude S is the stretch factor x is the horizontal coordinate of the projected location on the map y is the vertical coordinate of the projected location on the map Relationship between and : Specializations The specializations differ only in the ratio of the vertical to horizontal axis. Some specializations have been described, promoted, or otherwise named. Derivatives The Tobler hyperelliptical projection, first described by Tobler in 1973, is a further generalization of the cylindrical equal-area family. 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. References External links Table of examples and properties of all common projections,
Cylindrical equal-area projection
7688641
A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron. History The earliest use of the (icosahedral) geodesic grid in geophysical modeling dates back to 1968 and the work by Sadourny, Arakawa, and Mintz and Williamson. Later work expanded on this base. Construction A geodesic grid is a global Earth reference that uses triangular tiles based on the subdivision of a polyhedron (usually the icosahedron, and usually a Class I subdivision) to subdivide the surface of the Earth. Such a grid does not have a straightforward relationship to latitude and longitude, but conforms to many of the main criteria for a statistically valid discrete global grid. Primarily, the cells' area and shape are generally similar, especially near the poles where many other spatial grids have singularities or heavy distortion. The popular Quaternary Triangular Mesh (QTM) falls into this category. 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. Applications In biodiversity science, geodesic grids are a global extension of local discrete grids that are staked out in field studies to ensure appropriate statistical sampling and larger multi-use grids deployed at regional and national levels to develop an aggregated understanding of biodiversity. These grids translate environmental and ecological monitoring data from multiple spatial and temporal scales into assessments of current ecological condition and forecasts of risks to our natural resources. A geodesic grid allows local to global assimilation of ecologically significant information at its own level of granularity. When modeling the weather, ocean circulation, or the climate, partial differential equations are used to describe the evolution of these systems over time. Because computer programs are used to build and work with these complex models, approximations need to be formulated into easily computable forms. Some of these numerical analysis techniques (such as finite differences) require the area of interest to be subdivided into a grid — in this case, over the shape of the Earth. Geodesic grids can be used in video game development to model fictional worlds instead of the Earth. They are a natural analog of the hex map to a spherical surface. Pros and cons Pros: Largely isotropic. Resolution can be easily increased by binary division. Does not suffer from over sampling near the poles like more traditional rectangular longitude–latitude square grids. Does not result in dense linear systems like spectral methods do (see also Gaussian grid). No single points of contact between neighboring grid cells. Square grids and isometric grids suffer from the ambiguous problem of how to handle neighbors that only touch at a single point. Cells can be both minimally distorted and near-equal-area. In contrast, square grids are not equal area, while equal-area rectangular grids vary in shape from equator to poles.
Geodesic grid
25187803
In mathematics, the distortion is a measure of the amount by which a function from the Euclidean plane to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it is conformal; if the distortion is bounded and the function is a homeomorphism, then it is quasiconformal. The distortion of a function ƒ of the plane is given by which is the limiting eccentricity of the ellipse produced by applying ƒ to small circles centered at z. This geometrical definition is often very difficult to work with, and the necessary analytical features can be extrapolated to the following definition. A mapping ƒ : Ω → R2 from an open domain in the plane to the plane has finite distortion at a point x ∈ Ω if ƒ is in the Sobolev space W(Ω, R2), the Jacobian determinant J(x,ƒ) is locally integrable and does not change sign in Ω, and there is a measurable function K(x) ≥ 1 such that almost everywhere. Here Df is the weak derivative of ƒ, and |Df| is the Hilbert–Schmidt norm. For functions on a higher-dimensional Euclidean space Rn, there are more measures of distortion because there are more than two principal axes of a symmetric tensor. The pointwise information is contained in the distortion tensor The outer distortion KO and inner distortion KI are defined via the Rayleigh quotients The outer distortion can also be characterized by means of an inequality similar to that given in the two-dimensional case. If Ω is an open set in Rn, then a function has finite distortion if its Jacobian is locally integrable and does not change sign, and there is a measurable function KO (the outer distortion) such that almost everywhere. See also Deformation (mechanics) References . . Conformal mappings Real analysis Complex analysis Topology Measure theory Euclidean geometry
Distortion (mathematics)
2721825
In cartography, a Tissot's indicatrix (Tissot indicatrix, Tissot's ellipse, Tissot ellipse, ellipse of distortion) (plural: "Tissot's indicatrices") 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, generally 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 meridians and 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 precisely at each point. Because all ellipses on the map occupy the same area, the distortion imposed by the map projection is evident. There is a one-to-one correspondence between the Tissot indicatrix and the metric tensor of the map projection coordinate conversion. Description Tissot's theory was developed in the context of cartographic analysis. Generally the geometric model represents the Earth, and comes in the form of a sphere or ellipsoid. Tissot's indicatrices illustrate linear, angular, and areal distortions of maps: A map distorts distances (linear distortion) wherever the quotient between the lengths of an infinitesimally short line as projected onto the projection surface, and as it originally is on the Earth model, deviates from 1. 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 which is 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 whose areas vary across the map. 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, possibly also with varying orientation (given the four circle quadrants split by meridians and parallels). 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. Mathematics In the diagram below, the circle has unit area as defined on the surface of a sphere. The circle is the Tissot's indicatrix that results from some projection of onto a plane. Linear scale has not been preserved in this projection, as and . Because , we know that
Tissot's indicatrix
16390404
ImageAmerica Aviation, Inc. is an aerial photography company that was acquired by Google in July 2007. The company specialized in creating aerial photos with "accuracy, quick delivery and low cost". It previously sold its services primarily to city, county, state, and federal governments and to corporate customers. ImageAmerica also made money by selling low-cost imagery to county appraisers and assessors. The company's clients include the Texas Department of Transportation, the U.S. Geological Survey, and the Lucas County office that covers Toledo, Ohio. For satellite imaging, the company charges US$99 per , compared to other companies which could charge from $500 to $700 and for a lower quality than what ImageAmerica offers. It also developed its own DDP-2 (Direct Digital Panoramic) camera system. The system is housed in an aircraft. It has the ability to capture details as small as to . ImageAmerica's patented processing system has the ability to produce orthorectified imagery corrected for perspective distortions. The company's technology also uses sensors which are based on a unique design from Sarnoff Research Labs in Princeton, New Jersey. The company was founded in and is based in the Spirit of St. Louis Airport, Clayton, Missouri, United States. ImageAmerica builds high resolution cameras for aerial photographs. The company provided high resolution black and white images of New Orleans following the events of Hurricane Katrina. The company's products provide images for Google Maps and Google Earth. When the company was acquired by Google, the acquisition price was not disclosed. After the acquisition, the project manager of Google Maps and Earth said in a statement that Google was excited "about how ImageAmerica’s technology will contribute to [Google's] mapping services down the road." He went on to say, "Since we're in the research and development phase right now it may be some time before you see any of this imagery in Google Maps or Earth." In the same month, Google was on an acquisition-spree and had already acquired six other companies: Panoramio, PeakStream, Zenter, Feedburner, GrandCentral, and Postini. References External links Imaging America Companies established in 1998 Photography companies of the United States Google acquisitions 1998 establishments in Missouri
ImageAmerica
20771
The Mercator projection () is a conformal cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation due to its ability to represent north as 'up' and south as 'down' everywhere while preserving local directions and shapes. However, as a result, the Mercator projection inflates the size of objects the further they are from the equator. In a Mercator projection, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator. Despite these drawbacks, the Mercator projection is well-suited to marine navigation and internet web maps and continues to be widely used today. History There is some controversy over the origins of the Mercator. German polymath Erhard Etzlaub engraved miniature "compass maps" (about 10×8 cm) of Europe and parts of Africa that spanned latitudes 0°–67° to allow adjustment of his portable pocket-size sundials. The projection found on these maps, dating to 1511, was stated by John Snyder in 1987 to be the same projection as Mercator's. However, given the geometry of a sundial, these maps may well have been based on the similar central cylindrical projection, a limiting case of the gnomonic projection, which is the basis for a sundial. Snyder amended his assessment to "a similar projection" in 1993. Joseph Needham, a historian of China, speculated that some star charts of the Chinese Song Dynasty may have been based on the Mercator projection; however, this claim was presented without evidence, and astronomical historian Kazuhiko Miyajima concluded using cartometric analysis that these charts used an equirectangular projection instead. Portuguese mathematician and cosmographer Pedro Nunes first described the mathematical principle of the loxodrome and its use in marine navigation. In 1537, he proposed constructing a nautical atlas composed of several large-scale sheets in the equirectangular projection as a way to minimize distortion of directions. If these sheets were brought to the same scale and assembled, they would approximate the Mercator projection. In 1569, Gerhard Kremer, known by his trade name Gerardus Mercator, announced a new projection by publishing a large planispheric map measuring and printed in eighteen separate sheets. Mercator titled the map : "A new and augmented description of Earth corrected for the use of sailors". This title, along with an elaborate explanation for using the projection that appears as a section of text on the map, shows that Mercator understood exactly what he had achieved and that he intended the projection to aid navigation. Mercator never explained the method of construction or how he arrived at it. Various hypotheses have been tendered over the years, but in any case Mercator's friendship with Pedro Nunes and his access to the loxodromic tables Nunes created likely aided his efforts. English mathematician Edward Wright published the first accurate tables for constructing the projection in 1599 and, in more detail, in 1610, calling his treatise "Certaine Errors in Navigation". The first mathematical formulation was publicized around 1645 by a mathematician named Henry Bond (–1678).
Mercator projection
61469969
In map projections, an interruption is any place where the globe has been split. All map projections are interrupted at at least one point. Typical world maps are interrupted along an entire meridian. In that typical case, the interruption forms an east/west boundary, even though the globe has no boundaries. Most map projections can be interrupted beyond what is required by the projection mathematics. The reason for doing so is to improve distortion within the map by sacrificing proximity—that is, by separating places on the globe that ought to be adjacent. Effectively, this means that the resulting map is actually an amalgam of several partial map projections of smaller regions. Because the regions are smaller, they cover less of the globe, are closer to flat, and therefore accrue less inevitable distortion. These extra interruptions do not create a new projection. Rather, the result is an "arrangement" of an existing projection. In casual parlance, interrupted projection usually means a projection that has been interrupted beyond mathematical necessity. In this casual sense, the usual east/west interruption of a pseudocylindric map is ignored as an interruption to focus on the elective interruptions. An archetypical example is the Goode homolosine projection. In 1916, John Paul Goode experimented by interrupting the Mollweide projection. Satisfied with the interruption scheme, he then devised a new projection as a composite of the Mollweide and the sinusoidal projection and applied the same interruption scheme to the new projection, which he dubbed "homolosine". Because pseudocylindric projections map parallels as straight lines, and meridians to have constant spacing, they are easy to interrupt. This is normally done to optimize either for continental areas or for oceanic areas, as explored by Goode. Many interruption schemes that are much more elaborate have been developed. Since antiquity, for example, globe gores have been developed in order to paste map sections onto model globes. These are regular interruption either along the equator, or in polar form as "rosettes". The Cahill butterfly projection divides the world into octahedral sections. More generally, any mapping onto polyhedral faces becomes an interrupted map when laid flat. Buckminster Fuller proposed his "dymaxion" map in 1943, using a modified icosahedral interruption scheme to divide the oceans up in a way that shows the continents in a nearly continuous mass as "one island". The most elaborate interruptions schemes include those of Athelstan Spilhaus along continental boundaries, and JJ Wijk's myriahedral projections. References Map projections
Interruption (map projection)
1775417
The equirectangular projection (also called the equidistant cylindrical projection or la carte parallélogrammatique projection), and which includes the special case of the plate carrée projection (also called the geographic projection, lat/lon projection, or plane chart), is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. 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, NASA World Wind, the USGS Astrogeology Research Program, and Natural Earth, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth or other spherical solar system bodies. In addition it is frequently used in panoramic photography to represent a spherical panoramic image. Definition The forward projection transforms spherical coordinates into planar coordinates. The reverse projection transforms from the plane back onto the sphere. The formulae presume a spherical model and use these definitions: is the longitude of the location to project; is the latitude of the location to project; are the standard parallels (north and south of the equator) where the scale of the projection is true; is the central parallel of the map; is the central meridian of the map; is the horizontal coordinate of the projected location on the map; is the vertical coordinate of the projected location on the map; is the radius of the globe. Longitude and latitude variables are defined here in terms of radians. Forward The (French, for flat square), is the special case where is zero. This projection maps x to be the value of the longitude and y to be the value of the latitude, and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. Despite sometimes being called "unprojected", it is actually projected. When the is not zero, such as Marinus's , or Ronald Miller's , the projection can portray particular latitudes of interest at true scale. While a projection with equally spaced parallels is possible for an ellipsoidal model, it would no longer be equidistant because the distance between parallels on an ellipsoid is not constant. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing. Reverse Alternative names In spherical panorama viewers, usually: is called "yaw"; is called "pitch"; where both are defined in degrees. See also Cartography Cassini projection Gall–Peters projection (mentions a resolution rejecting the use of all rectangular world maps) List of map projections Mercator projection 360 video projection Wikimedia Gallery of Equirectangular World Maps References External links Global MODIS based satellite map The blue marble: land surface, ocean color,
Equirectangular projection
7151375
Space-oblique Mercator projection is a map projection devised in the 1970s for preparing maps from Earth-survey satellite data. It is a generalization of the oblique Mercator projection that incorporates the time evolution of a given satellite ground track to optimize its representation on the map. The oblique Mercator projection, on the other hand, optimizes for a given geodesic. History The space-oblique Mercator projection (SOM) was developed by John P. Snyder, Alden Partridge Colvocoresses and John L. Junkins in 1976. Snyder had an interest in maps, originating back to his childhood and he regularly attended cartography conferences while on vacation. In 1972, the United States Geological Survey (USGS) needed to develop a system for reducing the amount of distortion caused when satellite pictures of the ellipsoidal Earth were printed on a flat page. Colvocoresses, the head of the USGS's national mapping program, asked attendees of a geodetic sciences conferences for help solving the projection problem in 1976. Snyder attended the conference and became motivated to work on the problem armed with his newly purchased pocket calculator and devised the mathematical formulas needed to solve the problem. After submitting his calculations to Waldo Tobler for review, Snyder submitted these to the USGS at no charge. Impressed with his work, USGS officials offered Snyder a job with the organization, which he accepted. His formulas were used to produce maps from Landsat 4 images launched in the summer of 1978. Projection description The space-oblique Mercator projection provides continual, nearly conformal mapping of the swath sensed by a satellite. Scale is true along the ground track, varying 0.01 percent within the normal sensing range of the satellite. Conformality is correct within a few parts per million for the sensing range. Distortion is essentially constant along lines of constant distance parallel to the ground track. The space-oblique Mercator is the only projection presented that takes the rotation of Earth into account. Equations The forward equations for the Space-oblique Mercator projection for the sphere are as follows: References John Hessler, Projecting Time: John Parr Snyder and the Development of the Space Oblique Mercator Projection, Library of Congress, 2003 Snyder's 1981 Paper Detailing the Projection's Derivation Map projections
Space-oblique Mercator projection
3663509
The Behrmann projection is a cylindrical equal-area map projection described by Walter Behrmann in 1910. Cylindrical equal-area projections 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. While equal-area, distortion of shape increases in the Behrmann projection according to distance from the standard parallels. The Behrmann projection has the property that half of the Earth's surface is stretched horizontally and the other half is stretched vertically. This projection is not equidistant. See also List of map projections References External links Table of examples and properties of all common projections, from radicalcartography.net Map projections Cylindrical equal-area projections
Behrmann projection
33429851
Spatial cognition is the acquisition, organization, utilization, and revision of knowledge about spatial environments. It is most about how animals including humans behave within space and the knowledge they built around it, rather than space itself. These capabilities enable individuals to manage basic and high-level cognitive tasks in everyday life. Numerous disciplines (such as cognitive psychology, neuroscience, artificial intelligence, geographic information science, cartography, etc.) work together to understand spatial cognition in different species, especially in humans. Thereby, spatial cognition studies also have helped to link cognitive psychology and neuroscience. Scientists in both fields work together to figure out what role spatial cognition plays in the brain as well as to determine the surrounding neurobiological infrastructure. In humans, spatial cognition is closely related to how people talk about their environment, find their way in new surroundings, and plan routes. Thus a wide range of studies is based on participants reports, performance measures and similar, for example in order to determine cognitive reference frames that allow subjects to perform. In this context the implementation of virtual reality becomes more and more widespread among researchers, since it offers the opportunity to confront participants with unknown environments in a highly controlled manner. Spatial cognition can be seen from a psychological point of view, meaning that people's behaviour within that space is key. When people behave in space, they use cognitive maps, the most evolved form of spatial cognition. When using cognitive maps, information about landmarks and the routes between landmarks are stored and used. This knowledge can be built from various sources; from a tightly coordinated vision and locomotion (movement), but also from map symbols, verbal descriptions, and computer-based pointing systems. According to Montello, space is implicitly referring to a person's body and their associated actions. He mentions different kinds of space; figural space which is a space smaller than the body, vista space which the space is more extended than the human body, environmental space which is learned by locomotion, and geographical space which is the biggest space and can only be learned through cartographic representation. However, since space is represented in the human brain, this can also lead to distortions. When perceiving space and distance, a distortion can occur. Distances are perceived differently on whether they are considered between a given location and a location that has a high cognitive saliency, meaning that it stands out. Different perceived locations and distances can have a "reference point", which are better known than others, more frequently visited and more visible. There are other kinds of distortions as well. Furthermore, there the distortion in distance estimation and the distortion in angle alignment. Distortion in angle alignment means that your personal north will be viewed as "the north". The map is mentally represented according to the orientation of the personal point of view of learning. Since perceived distortion is "subjective" and not necessarily correlated with "objective distance", distortions can happen in this phenomenon too. There can be an overestimation in downtown routes, routes with turns, curved
Spatial cognition
14740464
A stereoplotter uses stereo photographs to determine elevations. It has been the primary method to plot contour lines on topographic maps since the 1930s. Although the specific devices have advanced technologically, they are all based on the apparent change in position of a feature in the two stereo photographs. Stereoplotters have changed as technology has improved. The first stereoplotters where projection stereoplotters they used only the light rays and optics to adjust the image. The Kelsh Plotter is an example of the projection stereoplotters. The analog stereoplotters came next and were more sophisticated in that they used more sophisticated optics to view the image. The analytical stereoplotter is used today. It incorporates a computer which does the work of mathematically aligning the images so that they line up properly. The analytic stereoplotter also allows for storing the data and redrawing at any desired scale. Analogical The stereoplotter requires two photographs that have considerable overlap (60%) and are corrected for distortion due to angle of photo. The photos are put onto transparent media and projected with a light source. Each image will be projected with overlap on the other. The operator, using a special set of optics, would then see the image as three-dimensional due to the differing perspective of each photo. The optics of the stereoplotter is what allows the operator to plot the contours and features. The light source used to project the photo is what begins the process. One photo is projected using cyan/blue filter and the other photo is projected with a red filter. The operator wears a special set of glasses that have the same color filter for lenses. Seeing the left photo in blue light while the left eye has the blue filter and the right photo projected with red light and the right eye seeing through the red filter, the overlapping image becomes three-dimensional. The images will have control points that detail how the overlap of the photos should occur. The resultant overlapping image is called an anaglyph and is a three-dimensional model of the terrain. Once the two photos are projected and the desired control points aligned the operator will then start to record the desired elevations on the terrain by "flying" a light spot along the contours. If the light spot appears to hover above the terrain or appears to dive into the terrain, the operator knows that he has moved it too far away from a slope or too far toward a slope, respectively. Originally, stereoplotters recorded the path of the flying light spot by directly scribing the path on a sheet of acetate or polyester coated with an opaque lacquer, which could be photographed to make the topographic map printing plates. If mistakes were made during contour tracing, the operator would daub some lacquer on the incorrect trace, allow it to dry and then try flying the light spot again. Current systems that use digital capture techniques allow simple erasure of part of the faulty data vector in computer memory,
Stereoplotter
59758307
A Valeriepieris circle is a figure drawn on the Earth's surface such that the majority of the human population lives within its interior. The concept was originally popularized by a map posted on Reddit in 2013, made by a Texas ESL teacher named Ken Myers, whose username on the site gave the figure its name. Myers's original circle covers only about 6.7% of the Earth's total surface area, with a radius of around , centered in the South China Sea. The map became a popular meme, and was featured in numerous internet media outlets. Myers's original map uses the Winkel tripel projection, which means that his circle, not having been adjusted to the projection, does not correspond to a circle on the surface of a sphere. In 2015, Singaporean professor Danny Quah—with the aid of an intern named Ken Teoh—verified Myers's original claim, as well as presenting a new, considerably smaller circle centered on the township of Mong Khet in Myanmar, with a radius of . In fact, Quah claimed this circle to be the smallest one possible, having been produced from more rigorous calculations and updated data, as well as being a proper circle on the Earth's surface. In 2022, Myers's original circle was again tested by Riaz Shah, a professor at Hult International Business School. Shah used recently published data from the United Nations' World Population Prospects to estimate that 4.2 billion people lived inside the circle as of 2022, out of a total human population of 8 billion. Myers's idea has been formalized and a Valeriepieris circle can be defined for any spatial area, like a single country. These generalised Valeriepieris circles can be used for studying population changes over time, dimensional reduction and measuring population centralization. A python package to compute Valeriepieris circles is available. References Geographical regions World population Population density World maps Internet memes
Valeriepieris circle
651822
Orthographic projection in cartography has been used since antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective (or azimuthal) projection in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges. History The orthographic projection has been known since antiquity, with its cartographic uses being well documented. Hipparchus used the projection in the 2nd century BC to determine the places of star-rise and star-set. In about 14 BC, Roman engineer Marcus Vitruvius Pollio used the projection to construct sundials and to compute sun positions. Vitruvius also seems to have devised the term orthographic (from the Greek orthos (= “straight”) and graphē (= “drawing”)) for the projection. However, the name analemma, which also meant a sundial showing latitude and longitude, was the common name until François d'Aguilon of Antwerp promoted its present name in 1613. The earliest surviving maps on the projection appear as crude woodcut drawings of terrestrial globes of 1509 (anonymous), 1533 and 1551 (Johannes Schöner), and 1524 and 1551 (Apian). A highly-refined map, designed by Renaissance polymath Albrecht Dürer and executed by Johannes Stabius, appeared in 1515. Photographs of the Earth and other planets from spacecraft have inspired renewed interest in the orthographic projection in astronomy and planetary science. Mathematics The formulas for the spherical orthographic projection are derived using trigonometry. They are written in terms of longitude (λ) and latitude (φ) on the sphere. Define the radius of the sphere R and the center point (and origin) of the projection (λ0, φ0). The equations for the orthographic projection onto the (x, y) tangent plane reduce to the following: Latitudes beyond the range of the map should be clipped by calculating the angular distance c from the center of the orthographic projection. This ensures that points on the opposite hemisphere are not plotted: . The point should be clipped from the map if cos(c) is negative. That is, all points that are included in the mapping satisfy: . The inverse formulas are given by: where For computation of the inverse formulas the use of the two-argument atan2 form of the inverse tangent function (as opposed to atan) is recommended. This ensures that the sign of the orthographic projection as written is correct in all quadrants. The inverse formulas are particularly useful when trying to project a variable defined on a (λ, φ) grid onto a rectilinear grid in (x, y). Direct application of the orthographic projection yields scattered points in (x, y), which creates problems for plotting and numerical integration. One solution is to start from the (x, y) projection plane and construct the image from the values defined in (λ, φ) by using the inverse formulas of the orthographic projection. See References for an ellipsoidal version of the orthographic
Orthographic map projection
2515660
The universal polar stereographic (UPS) coordinate system is used in conjunction with the universal transverse Mercator (UTM) coordinate system to locate positions on the surface of the Earth. Like the UTM coordinate system, the UPS coordinate system uses a metric-based cartesian grid laid out on a conformally projected surface. UPS covers the Earth's polar regions, specifically the areas north of 84°N and south of 80°S, which are not covered by the UTM grids, plus an additional 30 minutes of latitude extending into UTM grid to provide some overlap between the two systems. In the polar regions, directions can become complicated, with all geographic north–south lines converging at the poles. The difference between UPS grid north and true north can therefore be anything up to 180°—in some places, grid north is true south, and vice versa. UPS grid north is arbitrarily defined as being along the prime meridian in the Antarctic and the 180th meridian in the Arctic; thus, east and west on the grids when moving directly away from the pole are along the 90°E and 90°W meridians respectively. Projection system As the name indicates, the UPS system uses a stereographic projection. Specifically, the projection used in the system is a secant version based on an elliptical model of the earth. The scale factor at each pole is adjusted to 0.994 so that the latitude of true scale is 81.11451786859362545° (about 81° 06' 52.3") North and South. The scale factor inside the regions at latitudes higher than this parallel is too small, whereas the regions at latitudes below this line have scale factors that are too large, reaching 1.0016 at 80° latitude. The scale factor at the origin (the poles) is adjusted to minimize the overall distortion of scale within the mapped region. As with the Mercator projection, the region near the tangent (or secant) point on a Stereographic map remains very close to true scale for an angular distance of a few degrees. In the ellipsoidal model, a stereographic projection tangent to the pole has a scale factor of less than 1.003 at 84° latitude and 1.008 at 80° latitude. The adjustment of the scale factor in the UPS projection reduces the average scale distortion over the entire zone. References External links National Geospatial-Intelligence Agency, Geospatial Sciences Publications GeographicLib provides a utility GeoConvert (with source code) for conversions between geographic, UTM, UPS, and MGRS. Here is an online version of GeoConvert. Geographic coordinate systems
Universal polar stereographic coordinate system
12484243
Wagner VI is a pseudocylindrical whole Earth map projection. Like the Robinson projection, it is a compromise projection, not having any special attributes other than a pleasing, low distortion appearance. Wagner VI is equivalent to the Kavrayskiy VII horizontally elongated by a factor of . This elongation results in proper preservation of shapes near the equator but slightly more distortion overall. The aspect ratio of this projection is 2:1, as formed by the ratio of the equator to the central meridian. This matches the ratio of Earth’s equator to any meridian. The Wagner VI is defined by: where is the longitude and is the latitude. Inverse formula: See also List of map projections Cartography Kavrayskiy VII projection Robinson projection References Map projections
Wagner VI projection
20250761
The Joint Dark Energy Mission (JDEM) was an Einstein probe that planned to focus on investigating dark energy. JDEM was a partnership between NASA and the U.S. Department of Energy (DOE). In August 2010, the Board on Physics and Astronomy of the National Science Foundation (NSF) recommended the Wide Field Infrared Survey Telescope (WFIRST) mission, a renamed JDEM-Omega proposal which has superseded SNAP, Destiny, and Advanced Dark Energy Physics Telescope (ADEPT), as the highest priority for development in the decade around 2020. This would be a 1.5-meter telescope with a 144-megapixel HgCdTe focal plane array, located at the Sun-Earth L2 Lagrange point. The expected cost is around US$1.6 billion. Earlier proposals Dark Energy Space Telescope (Destiny) The Dark Energy Space Telescope (Destiny), was a planned project by NASA and DOE, designed to perform precision measurements of the universe to provide an understanding of dark energy. The space telescope will derive the expansion of the universe by measuring up to 3,000 distant supernovae each year of its three-year mission lifetime, and will additionally study the structure of matter in the universe by measuring millions of galaxies in a weak gravitational lensing survey. The Destiny spacecraft features an optical telescope with a 1.8 metre primary mirror. The telescope images infrared light onto an array of solid-state detectors. The mission is designed to be deployed in a halo orbit about the Sun-Earth Lagrange point. The Destiny proposal has been superseded by the Wide Field Infrared Survey Telescope (WFIRST). SuperNova Acceleration Probe (SNAP) The SuperNova Acceleration Probe (SNAP) mission was proposed to provide an understanding of the mechanism driving the acceleration of the universe and determine the nature of dark energy. To achieve these goals, the spacecraft needed to be able to detect these supernova when they are at their brightest moment. The mission was proposed as an experiment for the JDEM. The satellite observatory would be capable of measuring up to 2,000 distant supernovae each year of its three-year mission lifetime. SNAP was also planned to observe the small distortions of light from distant galaxies to reveal more about the expansion history of the universe. SNAP was initially planned to launch in 2013. To understand what is driving the acceleration of the universe, scientists need to see greater redshifts from supernovas than what is seen from Earth. The SNAP would detect redshifts of 1.7 from distant supernovas up to 10 billion light years away. At this distance, the acceleration of the universe is easily seen. To measure the presence of dark energy, a process called weak lensing can be used. The SNAP would have used an optical setup called the three-mirror anastigmat. This consists of a main mirror with a diameter of 2 meters to take in light. It reflects this light to a second mirror. Then this light is transferred to two additional smaller mirrors which direct the light to the spacecraft's instruments. It will also contain 72 different cameras. 36 of them are able to detect visible light and the other
Joint Dark Energy Mission
44885218
A rhumbline network (or windrose network) is a navigational aid consisting in multiple lines in a web-like grid drawn on portolan charts (and other early maps). These nautical charts were used in the medieval age and age of exploration in marine navigation. Since the invention of the Mercator projection , the term rhumb line (or loxodrome) has been redefined to mean a mathematically precise curve of constant bearing on the Earth's surface. To avoid confusion, the lines on earlier sailing charts can be unambiguously called windrose lines, since they are not true rhumb lines by the modern definition. A rhumb line in the modern sense is only straight on a chart drawn with the Mercator projection, but not on charts from the 13th–16th centuries. Older windrose lines were a close approximation on charts of the Mediterranean Sea and surrounding areas, but the rhumb lines on small-scale maps such as the Teixeira planisphere were highly inaccurate. The grid can be easily spotted (as parchment is quite translucent) by observing a chart from its rear face, with a light source illuminating the other side. The hole in the center of the circle, origin of the whole network, is also clearly visible from the rear. Use of windrose lines on a rhumbline network To calculate on a portolan chart the course to follow from a point of origin to a point of destination, one should transfer — using a parallel rule — the "line of course" drawn from the point of origin to the point of destination, on top of the windrose line on the compass rose closest to the ship's position, obtaining on it the theoretical course to be followed when sailing towards the destination. This theoretical course may have to be modified (as many times as needed) when tacking if the wind is right ahead of you, or to correct the effects of leeway, currents, etc. that a sailor with experience should be able to calculate empirically. Rhumblines vs. windrose lines Before modern accurate surveying, there was no method for measuring longitude at sea so maps used to have many distortions, especially in the east west direction. There was also distortion due to the curvature of the Earth's surface. The multitude of compass roses with straight lines extending outwards across the map derived from how the maps were then made by compiling empirical observations from navigators who attempted to follow a constant bearing at sea. All portolan maps share these characteristic "windrose networks", which emanate from compass roses located at various points on the map (or mapamundi). These better called "windrose lines" are generated "by observation and the compass", and are designated today as "lines of course" or "lines of rhumb" ("rhumb lines" in the fourteenth century, traced on portolan's particular projection, though not to be confused with modern rhumb lines, meridians or isoazimuthals). To understand that those lines should be better called "windrose lines", one has to know that portolan maps are characterized by the lack of map projection, for
Rhumbline network
25001172
This is a summary of map projections that have articles of their own on Wikipedia or that are otherwise notable. Because there is no limit to the number of possible map projections, there can be no comprehensive list. Table of projections *The first known popularizer/user and not necessarily the creator. Key Type of projection surface Cylindrical In normal aspect, these map regularly-spaced meridians to equally spaced vertical lines, and parallels to horizontal lines. Pseudocylindrical In normal aspect, these map the central meridian and parallels as straight lines. Other meridians are curves (or possibly straight from pole to equator), regularly spaced along parallels. Conic In normal aspect, conic (or conical) projections map meridians as straight lines, and parallels as arcs of circles. Pseudoconical In normal aspect, pseudoconical projections represent the central meridian as a straight line, other meridians as complex curves, and parallels as circular arcs. Azimuthal In standard presentation, azimuthal projections map meridians as straight lines and parallels as complete, concentric circles. They are radially symmetrical. In any presentation (or aspect), they preserve directions from the center point. This means great circles through the central point are represented by straight lines on the map. Pseudoazimuthal In normal aspect, pseudoazimuthal projections map the equator and central meridian to perpendicular, intersecting straight lines. They map parallels to complex curves bowing away from the equator, and meridians to complex curves bowing in toward the central meridian. Listed here after pseudocylindrical as generally similar to them in shape and purpose. Other Typically calculated from formula, and not based on a particular projection Polyhedral maps Polyhedral maps can be folded up into a polyhedral approximation to the sphere, using particular projection to map each face with low distortion. Properties Conformal Preserves angles locally, implying that local shapes are not distorted and that local scale is constant in all directions from any chosen point. Equal-area Area measure is conserved everywhere. Compromise Neither conformal nor equal-area, but a balance intended to reduce overall distortion. Equidistant All distances from one (or two) points are correct. Other equidistant properties are mentioned in the notes. Gnomonic All great circles are straight lines. Retroazimuthal Direction to a fixed location B (by the shortest route) corresponds to the direction on the map from A to B. Perspective Can be constructed by light shining through a globe onto a developable surface. See also 360 video projection List of national coordinate reference systems Snake Projection Notes Further reading
List of map projections
1334311
Nicolas Auguste Tissot (; March 16, 1824 – July 14, 1907) was a French cartographer, who in 1859 and 1881 published an analysis of the distortion that occurs on map projections. He devised Tissot's indicatrix, or distortion circle, which when plotted on a map will appear as an ellipse whose elongation depends on the amount of distortion by the map at that point. The angle and extent of the elongation represents the amount of angular distortion of the map. The size of the ellipse indicates the amount that the area is distorted. Biography Born in Nancy, Meurthe, France, Tissot was trained as an engineer in the French Army, from which he graduated as capitaine du génie. In the early 1860s he became an instructor in geodesy at the well-reputed Ecole Polytechnique in Paris. Around the same time, he indulged a research program meant to determine the best way of cartographic projection for a particular region and presented his findings to the French Académie des Sciences. In the eighteenth century, the German cartographer Johann H. Lambert had enunciated a mathematical theory of map projections and of the attendant characteristics of distortions that any given projection involved. Carl Friedrich Gauss had also studied the subject before Tissot's contributions later in the nineteenth century. Tissot's research in the mid-1850s on methods for finding good projections for particular regions led him to develop a projection that he saw as optimal. While not quite equal-area or conformal, his projection resulted in "negligible distortion for a very small region." Subsequently, his optimal projection was adopted by the geographic service of the French Army. While his first concepts regarding cartographic distortions developed in mid-century, it was only with the publication of Mémoire sur la représentation des surfaces et les projections des cartes géographiques in 1881 that the Tissot's Indicatrix became popular. In the book, Tissot argued for his method, reportedly demonstrating that "whatever the system of transformation, there is at each point on the spherical surface at least one pair of orthogonal directions which will also be orthogonal on the projection." Tissot employed a graphical device he called the ellipse indicatrice or distortion circle. When plotted on a map it reveals the amount of distortion by the map at the particular point where the ellipse is plotted. He suggested that the angle and extent of the elongation of the distortion circle represented the amount of angular distortion of the map, while the size of the ellipse corresponded to the amount of distortion in area. In a nineteenth-century cartographic context, in which professionals looked for ways to apply mathematical principles to the science of mapping and the map projection, Tissot's theory was favorably received, at least in continental Europe. Even in the more restrained Anglo-American academic world, a columnist of Science, a publication sponsored by the American Association for the Advancement of Science, hailed Tissot's method and encouraged his readers to study his work in the hope that such a study "will lead to the adoption of better
Nicolas Auguste Tissot
2514694
The Universal Transverse Mercator (UTM) is a map projection system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means it ignores altitude and treats the earth surface as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the x, y coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system. Most zones in UTM span 6 degrees of longitude, and each has a designated central meridian. The scale factor at the central meridian is specified to be 0.9996 of true scale for most UTM systems in use. History The National Oceanic and Atmospheric Administration (NOAA) website states that the system was developed by the United States Army Corps of Engineers, starting in the early 1940s. However, a series of aerial photos found in the Bundesarchiv-Militärarchiv (the military section of the German Federal Archives) apparently dating from 1943–1944 bear the inscription UTMREF followed by grid letters and digits, and projected according to the transverse Mercator, a finding that would indicate that something called the UTM Reference system was developed in the 1942–43 time frame by the Wehrmacht. It was probably carried out by the Abteilung für Luftbildwesen (Department for Aerial Photography). From 1947 onward the US Army employed a very similar system, but with the now-standard 0.9996 scale factor at the central meridian as opposed to the German 1.0. For areas within the contiguous United States the Clarke Ellipsoid of 1866 was used. For the remaining areas of Earth, including Hawaii, the International Ellipsoid was used. The World Geodetic System WGS84 ellipsoid is now generally used to model the Earth in the UTM coordinate system, which means current UTM northing at a given point can differ up to 200 meters from the old. For different geographic regions, other datum systems can be used. Prior to the development of the Universal Transverse Mercator coordinate system, several European nations demonstrated the utility of grid-based conformal maps by mapping their territory during the interwar period. Calculating the distance between two points on these maps could be performed more easily in the field (using the Pythagorean theorem) than was possible using the trigonometric formulas required under the graticule-based system of latitude and longitude. In the post-war years, these concepts were extended into the Universal Transverse Mercator/Universal Polar Stereographic (UTM/UPS) coordinate system, which is a global (or universal) system of grid-based maps. The transverse Mercator projection is a variant of the Mercator projection, which was originally developed by the Flemish geographer and cartographer Gerardus Mercator, in 1570. This projection is conformal, which means it preserves angles and therefore shapes across small regions. However, it distorts distance and area. Definitions UTM
Universal Transverse Mercator coordinate system
36814009
The central cylindrical projection is a perspective cylindrical map projection. It corresponds to projecting the Earth's surface onto a cylinder tangent to the equator as if from a light source at Earth's center. The cylinder is then cut along one of the projected meridians and unrolled into a flat map. The projection is neither conformal nor equal-area. Distortion increases so rapidly away from the equator that the central cylindrical is only used as an easily understood illustration of projection, rather than for practical maps. Its vertical stretching is even greater than that of the Mercator projection, whose construction method is sometimes erroneously described equivalently to the central cylindrical's. The scale becomes infinite at the poles. It is not known who first developed the projection, but it appeared with other new cylindrical projections in the 19th century, and regularly finds its way into textbooks, chiefly to illustrate that this is not the way the Mercator is constructed. As with any cylindrical projection, the construction can be generalized by positioning the cylinder to be tangent to a great circle of the globe that is not the equator. This projection has prominent use in panoramic photography, where it is usually called the "cylindrical projection". It can present a full 360° panorama and preserves vertical lines. Unlike other cylindrical projections, it gives correct perspective for tall objects, an important trait for architectural scenes. Formulae R denotes the radius of the generating globe; φ is the latitude; λ is the longitude; λ0 is the longitude of the central meridian; and x and y are the mapped coordinates. See also Gnomonic projection List of map projections References Map projections Cylindrical projections
Central cylindrical projection
51784
In cartography, a map projection is any of a broad set of transformations employed to represent the curved two-dimensional surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. The study of map projections is primarily about the characterization of their distortions. There is no limit to the number of possible map projections. More generally, projections are considered in several fields of pure mathematics, including differential geometry, projective geometry, and manifolds. However, the term "map projection" refers specifically to a cartographic projection. Despite the name's literal meaning, 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 that transforms coordinates from the curved surface distinctly and smoothly to the plane is a projection. Few projections in practical use are perspective. Most of this article assumes that the surface to be mapped is that of a sphere. The Earth and other large celestial bodies are generally better modeled as oblate spheroids, whereas small objects such as asteroids often have irregular shapes. The surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid. Therefore, more generally, a map projection is any method of flattening a continuous curved surface onto a plane. The most well-known map projection is the Mercator projection. This map projection has the property of being conformal. However, it has been criticized throughout the 20th century for enlarging regions further from the equator. To contrast, equal-area projections such as the Sinusoidal projection and the Gall–Peters projection show the correct sizes of countries relative to each other, but distort angles. The National Geographic Society and most atlases favor map projections that compromise between area and angular distortion, such as the Robinson projection and the Winkel tripel projection. Metric properties of maps Many properties can be measured on the Earth's surface independently of its geography: Area Shape Direction Bearing Distance Map projections can be constructed to preserve some of these properties at the expense of others. Because the Earth's curved surface is not isometric to a plane, preservation of shapes inevitably requires a variable scale and, consequently, non-proportional presentation of areas. Similarly, an area-preserving projection can not be conformal, resulting in shapes and bearings distorted in most places of the map. Each projection preserves, compromises, or approximates
Map projection
28955607
The equidistant conic projection is a conic map projection commonly used for maps of small countries as well as for larger regions such as the continental United States that are elongated east-to-west. Also known as the simple conic projection, a rudimentary version was described during the 2nd century CE by the Greek astronomer and geographer Ptolemy in his work Geography. The projection has the useful property that distances along the meridians are proportionately correct, and distances are also correct along two standard parallels that the mapmaker has chosen. The two standard parallels are also free of distortion. For maps of regions elongated east-to-west (such as the continental United States) the standard parallels are chosen to be about a sixth of the way inside the northern and southern limits of interest. This way distortion is minimized throughout the region of interest. Transformation Coordinates from a spherical datum can be transformed to an equidistant conic projection with rectangular coordinates by using the following formulas, where λ is the longitude, λ the reference longitude, φ the latitude, φ the reference latitude, and φ and φ the standard parallels: where Constants n, G, and ρ need only be determined once for the entire map. If one standard parallel is used (i.e. φ = φ), the formula for n above is indeterminate, but then The reference point (λ, φ) with longitude λ and latitude φ, transforms to the x,y origin at (0,0) in the rectangular coordinate system. The Y axis maps the central meridian λ, with y increasing northwards, which is orthogonal to the X axis mapping the central parallel φ, with x increasing eastwards. Other versions of these transformation formulae include parameters to offset the map coordinates so that all x,y values are positive, as well as a scaling parameter relating the radius of the sphere (Earth) to the units used on the map. The formulae used for ellipsoidal datums are more involved. See also List of map projections References Sources External links Table of examples and properties of all common projections, from radicalcartography.net Map projections Equidistant projections
Equidistant conic projection
7710044
The Cassini projection (also sometimes known as the Cassini–Soldner projection or Soldner projection) is a map projection first described in an approximate form by César-François Cassini de Thury in 1745. Its precise formulas were found through later analysis by Johann Georg von Soldner around 1810. It is the transverse aspect of the equirectangular projection, in that the globe is first rotated so the central meridian becomes the "equator", and then the normal equirectangular projection is applied. Considering the earth as a sphere, the projection is composed of the operations: where λ is the longitude from the central meridian and φ is the latitude. When programming these equations, the inverse tangent function used is actually the atan2 function, with the first argument sin φ and the second . The reverse operation is composed of the operations: In practice, the projection has always been applied to models of the earth as an ellipsoid, which greatly complicates the mathematical development but is suitable for surveying. Nevertheless, the use of the Cassini projection has largely been superseded by the transverse Mercator projection, at least with central mapping agencies. Distortions Areas along the central meridian, and at right angles to it, are not distorted. Elsewhere, the distortion is largely in a north–south direction, and varies by the square of the distance from the central meridian. As such, the greater the longitudinal extent of the area, the worse the distortion becomes. Due to this, the Cassini projection works best for areas with greater north–south extent than east–west. For example, Ordnance Survey maps of Great Britain used the Cassini projection from 1924 until the introduction of the National Grid. Elliptical form Cassini is known as a spherical projection, but can be generalised as an elliptical form. Considering the earth as an ellipse, the projection is composed of these operations: and M is the meridional distance function. The reverse operation is composed of the operations: If then and Otherwise calculate T and N as above with , and See also Cassini Grid References External links Table of examples and properties of all common projections, from radicalcartography.net Ordnance Survey GeoFacts on the Cassini Projection Cassini dans proj4 Map projections Cylindrical projections
Cassini projection
6990113
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. Polar regions are subject to extreme distortion. Lines of longitude converge to points at the poles. History Alphons J. van der Grinten invented the projection in 1898 and received US patent #751,226 for it and three others in 1904. The National Geographic Society adopted the projection for their reference maps of the world in 1922, raising its visibility and stimulating its adoption elsewhere. In 1988, National Geographic replaced the van der Grinten projection with the Robinson projection. Geometric construction The geometric construction given by van der Grinten can be written algebraically: where x takes the sign of , y takes the sign of φ, and If φ = 0, then Similarly, if λ = λ or φ = ±/2, then In all cases, φ is the latitude, λ is the longitude, and λ is the central meridian of the projection. Van der Grinten IV projection The van der Grinten IV projection is a later polyconic map projection developed by Alphons J. van der Grinten. The central meridian and equator are straight lines. All other meridians and parallels are arcs of circles. See also List of map projections Robinson projection (successor) References Bibliography Map projections
Van der Grinten projection
2961833
A spatial reference system (SRS) or coordinate reference system (CRS) is a framework used to precisely measure locations on the surface of Earth as coordinates. It is thus the application of the abstract mathematics of coordinate systems and analytic geometry to geographic space. A particular SRS specification (for example, "Universal Transverse Mercator WGS 84 Zone 16N") comprises a choice of Earth ellipsoid, horizontal datum, map projection (except in the geographic coordinate system), origin point, and unit of measure. Thousands of coordinate systems have been specified for use around the world or in specific regions and for various purposes, necessitating transformations between different SRS. Although they date to the Hellenic Period, spatial reference systems are now a crucial basis for the sciences and technologies of Geoinformatics, including cartography, geographic information systems, surveying, remote sensing, and civil engineering. This has led to their standardization in international specifications such as the EPSG codes and ISO 19111:2019 Geographic information—Spatial referencing by coordinates, prepared by ISO/TC 211, also published by the Open Geospatial Consortium as Abstract Specification, Topic 2: Spatial referencing by coordinate. Types of systems The thousands of spatial reference systems used today are based on a few general strategies, which have been defined in the EPSG, ISO, and OGC standards: Geographic coordinate system (or geodetic) A spherical coordinate system measuring locations directly on the Earth (modeled as a sphere or ellipsoid) using latitude (degrees north or south of the equator) and longitude (degrees west or east of a prime meridian). Geocentric coordinate system (or Earth-centered Earth-fixed) A three-dimensional cartesian coordinate system that models the Earth as a three-dimensional object, measuring locations from a center point, usually the center of mass of the Earth, along x, y, and z axes aligned with the equator and the prime meridian. This system is commonly used to track the orbits of satellites, because they are based on the center of mass. Thus, this is the internal coordinate system used by Satellite navigation systems such as GPS to compute locations using multilateration. Projected coordinate system (or planar, grid) A standardized cartesian coordinate system that models the Earth (or more commonly, a large region thereof) as a plane, measuring locations from an arbitrary origin point along x and y axes more or less aligned with the cardinal directions. Each of these systems is based on a particular Map projection to create a planar surface from the curved Earth surface. These are generally defined and used strategically to minimize the distortions inherent to projections. Common examples include the Universal transverse mercator (UTM) and national systems such as the British National Grid, and State Plane Coordinate System (SPCS). Engineering coordinate system (or local, custom) A cartesian coordinate system (2-D or 3-D) that is created bespoke for a small area, often a single engineering project, over which the curvature of the Earth can be safely approximated as flat without significant distortion. Locations are typically measured directly from an arbitrary origin point using surveying techniques. These may or may not be aligned with
Spatial reference system
37257405
The Gall stereographic projection, presented by James Gall in 1855, is a cylindrical projection. It is neither equal-area nor conformal but instead tries to balance the distortion inherent in any projection. Formulae The projection is conventionally defined as: where λ is the longitude from the central meridian in degrees, φ is the latitude, and R is the radius of the globe used as the model of the earth for projection. It is a perspective projection if the point of projection is allowed to vary with longitude: the point of projection being on the equator on the opposite side of the earth from the point being mapped and with the projective surface being a cylinder secant to the sphere at 45°N and 45°S. Gall called the projection "stereographic" because the spacing of the parallels is the same as the spacing of the parallels along the central meridian of the equatorial stereographic projection. The reverse projection is defined as : Braun stereographic projection This later (1867) cylindrical projection by Carl Braun is similar, differing only in the asymmetric scaling horizontally and vertically. This yields a projection tangent to the sphere. Its formula is: See also List of map projections References Bibliography Further reading External links gall in proj4 Progonos.com Map projections Cylindrical projections
Gall stereographic projection
58246920
The Equal Earth map projection is an equal-area pseudocylindrical projection for world maps, invented by Bojan Šavrič, Bernhard Jenny, and Tom Patterson in 2018. It is inspired by the widely used Robinson projection, but unlike the Robinson projection, retains the relative size of areas. The projection equations are simple to implement and fast to evaluate. 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 north or 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. According to the creators, the projection was created in response to the decision of the Boston public schools to adopt the Gall-Peters projection for world maps in March 2017, to accurately show the relative sizes of equatorial and non-equatorial regions. The decision generated controversy in the world of cartography due to this projection’s extreme distortion in the polar regions. At that time Šavrič, Jenny, and Patterson sought alternative map projections of equal areas for world maps, but could not find any that met their aesthetic criteria. Therefore, they created a new projection that had more visual appeal compared to existing projections of equal areas. As with the Natural Earth projection (2012) by Tom Patterson, a visual method was used to choose the parameters of the projection. A combination of Putniņš P4ʹ and Eckert IV projections was used as the basis. Formulation The projection is formulated as the equations where and refers to latitude and to longitude. Use 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. References External links Map projections
Equal Earth projection
2216927
In computer graphics, cube mapping is a method of environment mapping that uses the six faces of a cube as the map shape. The environment is projected onto the sides of a cube and stored as six square textures, or unfolded into six regions of a single texture. The cube map is generated by first rendering the scene six times from a viewpoint, with the views defined by a 90 degree view frustum representing each cube face. Or if the environment is first considered to be projected onto a sphere, then each face of the cube is its gnomonic projection. In the majority of cases, cube mapping is preferred over the older method of sphere mapping because it eliminates many of the problems that are inherent in sphere mapping such as image distortion, viewpoint dependency, and computational inefficiency. Also, cube mapping provides a much larger capacity to support real-time rendering of reflections relative to sphere mapping because the combination of inefficiency and viewpoint dependency severely limits the ability of sphere mapping to be applied when there is a consistently changing viewpoint. Variants of cube mapping are also commonly used in 360 video projection. History Cube mapping was first proposed in 1986 by Ned Greene in his paper “Environment Mapping and Other Applications of World Projections”, ten years after environment mapping was first put forward by Jim Blinn and Martin Newell. However, hardware limitations on the ability to access six texture images simultaneously made it infeasible to implement cube mapping without further technological developments. This problem was remedied in 1999 with the release of the Nvidia GeForce 256. Nvidia touted cube mapping in hardware as “a breakthrough image quality feature of GeForce 256 that ... will allow developers to create accurate, real-time reflections. Accelerated in hardware, cube environment mapping will free up the creativity of developers to use reflections and specular lighting effects to create interesting, immersive environments.” Today, cube mapping is still used in a variety of graphical applications as a favored method of environment mapping. Advantages Cube mapping is preferred over other methods of environment mapping because of its relative simplicity. Also, cube mapping produces results that are similar to those obtained by ray tracing, but is much more computationally efficient – the moderate reduction in quality is compensated for by large gains in efficiency. Pre-dating cube mapping, sphere mapping has many inherent flaws that made it impractical for most applications. Sphere mapping is view-dependent, meaning that a different texture is necessary for each viewpoint. Therefore, in applications where the viewpoint is mobile, it would be necessary to dynamically generate a new sphere mapping for each new viewpoint (or, to pre-generate a mapping for every viewpoint). Also, a texture mapped onto a sphere's surface must be stretched and compressed, and warping and distortion (particularly along the edge of the sphere) are a direct consequence of this. Although these image flaws can be reduced using certain tricks and techniques like “pre-stretching”, this just adds another layer of complexity to sphere
Cube mapping