Map projection - Knowledge (XXG)

1030: 515: 1106: 1730: 2027: 1142: 1198: 31: 1882: 1124: 1180: 1162: 5210: 1759: 882: 5222: 479: 288: 237:, resulting in shapes and bearings distorted in most places of the map. Each projection preserves, compromises, or approximates basic metric properties in different ways. The purpose of the map determines which projection should form the base for the map. Because maps have many different purposes, a diversity of projections have been created to suit those purposes. 820:

given rise to much misunderstanding. Particularly is this so with regard to the conic projections with two standard parallels: they may be regarded as developed on cones, but they are cones which bear no simple relationship to the sphere. In reality, cylinders and cones provide us with convenient descriptive terms, but little else.

2319:) designed to educate the public about map projections and distortion in maps. In 1989 and 1990, after some internal debate, seven North American geographic organizations adopted a resolution recommending against using any rectangular projection (including Mercator and Gall–Peters) for reference maps of the world. 469:

One way of describing a projection is first to project from the Earth's surface to a developable surface such as a cylinder or cone, and then to unroll the surface into a plane. While the first step inevitably distorts some properties of the globe, the developable surface can then be unfolded without

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All projections of a sphere on a plane necessarily distort the surface in some way. 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

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Some of the simplest map projections are literal projections, as obtained by placing a light source at some definite point relative to the globe and projecting its features onto a specified surface. Although most projections are not defined in this way, picturing the light source-globe model can be

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To measure distortion globally across areas instead of at just a single point necessarily involves choosing priorities to reach a compromise. Some schemes use distance distortion as a proxy for the combination of angular deformation and areal inflation; such methods arbitrarily choose what paths to

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Compromise projections give up the idea of perfectly preserving metric properties, seeking instead to strike a balance between distortions, or to simply make things look right. Most of these types of projections distort shape in the polar regions more than at the equator. These are some compromise

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One way to classify map projections is based on the type of surface onto which the globe is projected. In this scheme, the projection process is described as placing a hypothetical projection surface the size of the desired study area in contact with part of the Earth, transferring features of the

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are used. In the first half of the 20th century, projecting a human head onto different projections was common to show how distortion varies across one projection as compared to another. In dynamic media, shapes of familiar coastlines and boundaries can be dragged across an interactive map to show

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where the cone intersects the globe—or, if the map maker chooses the same parallel twice, as the tangent line where the cone is tangent to the globe. The resulting conic map has low distortion in scale, shape, and area near those standard parallels. Distances along the parallels to the north of

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as used in the field of map projections relaxes the last constraint entirely. Instead the parallels can be placed according to any algorithm the designer has decided suits the needs of the map. The famous Mercator projection is one in which the placement of parallels does not arise by projection;

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No reference has been made in the above definitions to cylinders, cones or planes. The projections are termed cylindric or conic because they can be regarded as developed on a cylinder or a cone, as the case may be, but it is as well to dispense with picturing cylinders and cones, since they have

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The three developable surfaces (plane, cylinder, cone) provide useful models for understanding, describing, and developing map projections. However, these models are limited in two fundamental ways. For one thing, most world projections in use do not fall into any of those categories. For another

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The time has come to discard for something that represents the continents and directions less deceptively ... Although its usage ... has diminished ... it is still highly popular as a wall map apparently in part because, as a rectangular map, it fills a rectangular wall space with more map, and

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Selecting a model for a shape of the Earth involves choosing between the advantages and disadvantages of a sphere versus an ellipsoid. Spherical models are useful for small-scale maps such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to

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as straight lines. Along parallels, each point from the surface is mapped at a distance from the central meridian that is proportional to its difference in longitude from the central meridian. Therefore, meridians are equally spaced along a given parallel. On a pseudocylindrical map, any point

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to the sphere or ellipsoid. Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe. Moving the developable surface away from contact with the globe never preserves or optimizes metric properties, so that possibility is not

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maps, such as those from national mapping systems, it is important to match the datum to the projection. The slight differences in coordinate assignation between different datums is not a concern for world maps or those of large regions, where such differences are reduced to imperceptibility.

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preserves distances from one or two special points to all other points. The special point or points may get stretched into a line or curve segment when projected. In that case, the point on the line or curve segment closest to the point being measured to must be used to measure the distance.

836:(azimuthal) have been abstracted in the field of map projections. If maps were projected as in light shining through a globe onto a developable surface, then the spacing of parallels would follow a very limited set of possibilities. Such a cylindrical projection (for example) is one which: 440: 1811:, or orthomorphic, map projections preserve angles locally, implying that they map infinitesimal circles of constant size anywhere on the Earth to infinitesimal circles of varying sizes on the map. In contrast, mappings that are not conformal distort most such small circles into 1075:

further from the equator than some other point has a higher latitude than the other point, preserving north-south relationships. This trait is useful when illustrating phenomena that depend on latitude, such as climate. Examples of pseudocylindrical projections include:

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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.

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If the length of the line segment connecting two projected points on the plane is proportional to the geodesic (shortest surface) distance between the two unprojected points on the globe, then we say that distance has been preserved between those two points. An

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operators to know the direction to point their antennas toward a point and see the distance to it. Distance from the tangent point on the map is proportional to surface distance on the Earth (; for the case where the tangent point is the North Pole, see the

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described how to construct an ellipse that illustrates the amount and orientation of the components of distortion. By spacing the ellipses regularly along the meridians and parallels, the network of indicatrices shows how distortion varies across the map.

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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

1815:. An important consequence of conformality is that relative angles at each point of the map are correct, and the local scale (although varying throughout the map) in every direction around any one point is constant. These are some conformal projections: 465:

and the plane are all developable surfaces. The sphere and ellipsoid do not have developable surfaces, so any projection of them onto a plane will have to distort the image. (To compare, one cannot flatten an orange peel without tearing and warping it.)

734:). Many mathematical projections, however, do not neatly fit into any of these three projection methods. Hence other peer categories have been described in the literature, such as pseudoconic, pseudocylindrical, pseudoazimuthal, retroazimuthal, and 1253:

both standard parallels or to the south of both standard parallels are stretched; distances along parallels between the standard parallels are compressed. When a single standard parallel is used, distances along all other parallels are stretched.

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In the first case (Mercator), the east-west scale always equals the north-south scale. In the second case (central cylindrical), the north-south scale exceeds the east-west scale everywhere away from the equator. Each remaining case has a pair of

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Another way to visualize local distortion is through grayscale or color gradations whose shade represents the magnitude of the angular deformation or areal inflation. Sometimes both are shown simultaneously by blending two colors to create a

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Has parallels constrained to where they fall when light shines through the globe onto the cylinder, with the light source someplace along the line formed by the intersection of the prime meridian with the equator, and the center of the

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The mathematics of projection do not permit any particular map projection to be best for everything. Something will always be distorted. Thus, many projections exist to serve the many uses of maps and their vast range of scales.

2292:, developed for navigational purposes, has often been used in world maps where other projections would have been more appropriate. This problem has long been recognized even outside professional circles. For example, a 1943 1334:

through the central point are represented by straight lines on the map. These projections also have radial symmetry in the scales and hence in the distortions: map distances from the central point are computed by a function

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The mapping of meridians to vertical lines can be visualized by imagining a cylinder whose axis coincides with the Earth's axis of rotation. This cylinder is wrapped around the Earth, projected onto, and then unrolled.

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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.

84:, 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. 908:

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

1082:, which was the first pseudocylindrical projection developed. On the map, as in reality, the length of each parallel is proportional to the cosine of the latitude. The area of any region is true. 240:

Another consideration in the configuration of a projection is its compatibility with data sets to be used on the map. Data sets are geographic information; their collection depends on the chosen

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Airy, G.B. (1861). "Explanation of a projection by balance of errors for maps applying to a very large extent of the Earth's surface; and comparison of this projection with other projections".

3794:—PDF versions of numerous projections, created and released into the Public Domain by Paul B. Anderson ... member of the International Cartographic Association's Commission on Map Projections 854:(If you rotate the globe before projecting then the parallels and meridians will not necessarily still be straight lines. Rotations are normally ignored for the purpose of classification.) 1263:, which keeps parallels evenly spaced along the meridians to preserve a constant distance scale along each meridian, typically the same or similar scale as along the standard parallels. 1006:

Normal cylindrical projections map the whole Earth as a finite rectangle, except in the first two cases, where the rectangle stretches infinitely tall while retaining constant width.

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so that phenomena per unit area are shown in correct proportion. However, representing area ratios correctly necessarily distorts shapes more than many maps that are not equal-area.

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would be if there were no winds, tides, or land. Compared to the best fitting ellipsoid, a geoidal model would change the characterization of important properties such as distance,

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Projection construction is also affected by how the shape of the Earth or planetary body is approximated. In the following section on projection categories, the earth is taken as a

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Where the light source emanates along the line described in this last constraint is what yields the differences between the various "natural" cylindrical projections. But the term

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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):

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throughout the entire map in all directions. A map cannot achieve that property for any area, no matter how small. It can, however, achieve constant scale along specific lines.

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proved that a sphere's surface cannot be represented on a plane without distortion. The same applies to other reference surfaces used as models for the Earth, such as oblate

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Earth's surface onto the projection surface, then unraveling and scaling the projection surface into a flat map. The most common projection surfaces are cylindrical (e.g.,

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Rather than the original (enlarged) infinitesimal circle as in Tissot's indicatrix, some visual methods project finite shapes that span a part of the map. For example, a

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Scale is constant along all straight lines radiating from a particular geographic location. This is the defining characteristic of an equidistant projection such as the

994:(undistorted at the equator). Since this projection scales north-south distances by the reciprocal of east-west stretching, it preserves area at the expense of shapes. 1288:, an equal-area projection on which most meridians and parallels appear as curved lines. It has a configurable standard parallel along which there is no distortion. 1070:

as a straight line segment. Other meridians are longer than the central meridian and bow outward, away from the central meridian. Pseudocylindrical projections map

2312: 1347:, independent of the angle; correspondingly, circles with the central point as center are mapped into circles which have as center the central point on the map. 1014:

A transverse cylindrical projection is a cylindrical projection that in the tangent case uses a great circle along a meridian as contact line for the cylinder.

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Near-sided perspective projection, which simulates the view from space at a finite distance and therefore shows less than a full hemisphere, such as used in

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maps each point on the Earth to the closest point on the plane. Can be constructed from a point of perspective an infinite distance from the tangent point;

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instead parallels are placed how they need to be in order to satisfy the property that a course of constant bearing is always plotted as a straight line.

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Scale is constant along any parallel in the direction of the parallel. This applies for any cylindrical or pseudocylindrical projection in normal aspect.

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maps, such as those spanning continents or the entire world, many projections are in common use according to their fitness for the purpose, such as

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Direction to a fixed location B (the bearing at the starting location A of the shortest route) corresponds to the direction on the map from A to B:

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to subdivide the globe into faces, and then projects each face to the globe. The most well-known polyhedral map projection is Buckminster Fuller's

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defined as a grid superimposed on the projection. In small-scale maps, eastings and northings are not meaningful, and grids are not superimposed.

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is constructed so that each point's distance from the center of the map is the logarithm of its distance from the tangent point on the Earth.

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An azimuthal equidistant projection shows distances and directions accurately from the center point, but distorts shapes and sizes elsewhere.

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Map projections can be constructed to preserve some of these properties at the expense of others. Because the Earth's curved surface is not

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Comparison of some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii.

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The scale depends on location, but not on direction. This is equivalent to preservation of angles, the defining characteristic of a

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Another way to classify projections is according to properties of the model they preserve. Some of the more common categories are:

1275:, which adjusts the north-south distance between non-standard parallels to equal the east-west stretching, giving a conformal map. 1003:—a pair of identical latitudes of opposite sign (or else the equator) at which the east-west scale matches the north-south-scale. 4975: 4772: 4699: 4655: 4351: 2132: 2076: 2054: 2031: 1612: 615: 611: 1248:

When making a conic map, the map maker arbitrarily picks two standard parallels. Those standard parallels may be visualized as

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thing, even most projections that do fall into those categories are not naturally attainable through physical projection. As

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of the shape must be specified. The aspect describes how the developable surface is placed relative to the globe: it may be

483: 3882: 1088:, which in its most common forms represents each meridian as two straight line segments, one from each pole to the equator. 151:. However, it has been criticized throughout the 20th century for enlarging regions further from the equator. To contrast, 5263: 5012: 4980: 4830: 4461: 4285: 4118: 4108: 3940: 3346: 2335: 2190: 1854: 946: 444: 2901:. FOSS4G Europe 2015. Geomatics Workbooks. Vol. 12. Como, Italy: Polytechnic University of Milan. pp. 697–700. 1600:). Can display nearly the entire sphere's surface on a finite circle. The sphere's full surface requires an infinite map. 4970: 4684: 4338: 4247: 3386: 2060: 1999: 1260: 671: 4960: 4910: 4873: 4640: 4333: 4182: 4032: 3818:, U.S. Geological Survey Professional Paper 1453, by John P. Snyder (USGS) and Philip M. Voxland (U. Minnesota), 1989. 2642: 2066: 1959: 1892:

Equal-area maps preserve area measure, generally distorting shapes in order to do so. Equal-area maps are also called

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Comparison of tangent and secant cylindrical, conic and azimuthal map projections with standard parallels shown in red

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A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a

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Gott, III, J. Richard; Mugnolo, Charles; Colley, Wesley N. (2006). "Map projections for minimizing distance errors".

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A sinusoidal projection shows relative sizes accurately, but grossly distorts shapes. Distortion can be reduced by "

542:) are represented undistorted. If these lines are a parallel of latitude, as in conical projections, it is called a 4845: 4689: 4280: 4113: 4103: 2397: 2278:. Due to distortions inherent in any map of the world, the choice of projection becomes largely one of aesthetics. 2185: 2048: 1910: 1633: 1523:

can be constructed by using a point of perspective outside the Earth. Photographs of Earth (such as those from the

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Combination of the above: the scale depends on latitude only, not on longitude or direction. This applies for the

4825: 4210: 2988: 2210: 1766: 1189: 284:. Since any map projection is a representation of one of those surfaces on a plane, all map projections distort. 57: 4559: 4065: 3565:

Snyder, John P. (2017). "Matching the Map Projection to the Need". In Lapaine, Miljenko; Usery, E. Lynn (eds.).

2727:. United States Geological Survey Professional Paper. Vol. 1395. United States Government Printing Office. 2079:: Two "control points" are arbitrarily chosen by the map maker; distances from each control point are preserved. 1969: 693:

Other regular solids are sometimes used as generalizations for smaller bodies' geoidal equivalent. For example,

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and, consequently, non-proportional presentation of areas. Similarly, an area-preserving projection can not be

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Hargitai, Henrik; Wang, Jue; Stooke, Philip J.; Karachevtseva, Irina; Kereszturi, Akos; Gede, Mátyás (2017),

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Many other ways have been described of showing the distortion in projections. Like Tissot's indicatrix, the

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Fran Evanisko, American River College, lectures for Geography 20: "Cartographic Design for GIS", Fall 2002

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Ghaderpour, E. (2016). "Some equal-area, conformal and conventional map projections: a tutorial review".

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as straight lines. A rhumb is a course of constant bearing. Bearing is the compass direction of movement.

244:(model) of the Earth. Different datums assign slightly different coordinates to the same location, so in 5258: 5137: 4950: 4740: 4694: 4521: 4498: 4481: 4192: 2833: 2670:, Lecture Notes in Geoinformation and Cartography, Springer International Publishing, pp. 177–202, 2231: 2026: 1989: 1876: 1079: 945:

North-south stretching grows with latitude, but less quickly than the east-west stretching: such as the

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and most atlases favor map projections that compromise between area and angular distortion, such as the

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is the meridian to which the globe is rotated before projecting. The central meridian (usually written

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An oblique cylindrical projection aligns with a great circle, but not the equator and not a meridian.

982:. This projection has many named specializations differing only in the scaling constant, such as the 5253: 4955: 4850: 4630: 4625: 4620: 4597: 4592: 4513: 4275: 4215: 4187: 4172: 4167: 4162: 4157: 3271: 2562: 2271: 2215: 2205: 1984: 1943: 1939: 1935: 1925: 1920: 1886: 1512: 1297: 1238: 1218: 1171: 1153: 1115: 1085: 1067: 894: 735: 423: 263: 214: 209: 3491: 1527:) give this perspective. It is a generalization of near-sided perspective projection, allowing tilt. 978:

North-south compression equals the cosine of the latitude (the reciprocal of east-west stretching):

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as straight lines. Can be constructed by using a point of perspective at the center of the Earth.

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justify using the more complicated ellipsoid. The ellipsoidal model is commonly used to construct

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twice as long as its minor and with its middle axis one and half times as long as its minor. See

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amounting to less than 100 m from the ellipsoidal model out of the 6.3 million m

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projections have the property that directions from a central point are preserved and therefore

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in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate

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Selection of a model for the shape of the Earth or planetary body (usually choosing between a

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Cheng, Y.; Lorre, J. J. (2000). "Equal Area Map Projection for Irregularly Shaped Objects".

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and for other large- and medium-scale maps that need to accurately depict the land surface.

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measure and how to weight them in order to yield a single result. Many have been described.

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projection combines an equal-area cylindrical projection in equatorial regions with the

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is better modeled by triaxial ellipsoid or prolated spheroid with small eccentricities.

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would deviate from a mapped ellipsoid's graticule. Normally the geoid is not used as an

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is mathematically the same as a standard Mercator, but oriented around a different axis.

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Bauer, H.A. (1942). "Globes, Maps, and Skyways (Air Education Series)". New York. p. 28

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Many properties can be measured on the Earth's surface independently of its geography:

116: 5242: 5214: 5180: 3824:, a visualization of distortion on a vast array of map projections in a single image. 3743: 3437: 3149: 2582: 2255: 2003: 1808: 1628: 594: 360: 234: 38:(1482, Johannes Schnitzer, engraver), constructed after the coordinates in Ptolemy's 3878:

Table of examples and properties of all common projections (RadicalCartography.net).

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show the correct sizes of countries relative to each other, but distort angles. The

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Most of this article assumes that the surface to be mapped is that of a sphere. The

5226: 5185: 5165: 4965: 3875: 2200: 2104: 1774: 1758: 1429: 1331: 690:, however, sometimes models analogous to the geoid are used to project maps from. 683: 658:, a more complex and accurate representation of Earth's shape coincident with what 387:). Because the Earth's actual shape is irregular, information is lost in this step. 343: 3840: 3549: 2947: 2699: 934:

North-south stretching grows with latitude faster than east-west stretching (sec

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MapRef: The Internet Collection of MapProjections and Reference Systems in Europe

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Peters, A. B. (1978). "Uber Weltkartenverzerrunngen und Weltkartenmittelpunkte".

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Slocum, Terry A.; Robert B. McMaster; Fritz C. Kessler; Hugh H. Howard (2005).

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Clark, P. E.; Clark, C. S. (2013). "CSNB Mapping Applied to Irregular Bodies".

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how the projection distorts sizes and shapes according to position on the map.

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The classical way of showing the distortion inherent in a projection is to use

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projections, such as those resulting from casting a shadow on a screen, or the

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Miller, Osborn Maitland (1942). "Notes on Cylindrical World Map Projections".

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Once a choice is made between projecting onto a cylinder, cone, or plane, the

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Systematic representation of the surface of a sphere or ellipsoid onto a plane

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Constant-Scale Natural Boundary Mapping to Reveal Global and Cosmic Processes

2906: 5155: 5094: 5022: 3626:(2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. p. 166. 3521: 3496: 3471: 3421: 3369: 3311: 2391: 812: 694: 687: 636: 498:(such that the surface's axis of symmetry coincides with the Earth's axis), 391: 384: 81: 3806: 3016: 2862: 2506: 2477:

Robinson, Arthur; Randall, Sale; Morrison, Joel; Muehrcke, Phillip (1985).

3218:"The Classification of Projections of Irregularly-shaped Celestial Bodies" 2574: 1416:(along an infinite line through the tangent point and the tangent point's 678:

for projections, however, because Earth's shape is very regular, with the

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American Cartographic Association's Committee on Map Projections, 1986.

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The term "conic projection" is used to refer to any projection in which

788:), a trait possible only between one or two points and every other point 4878: 2892:"Real-time projection visualisation with Indicatrix Mapper QGIS Plugin" 2429:

Notes and comments on the composition of terrestrial and celestial maps

1355: 1327: 1214: 524: 35: 3852: 3677: 2897:. In Brovelli, Maria Antonia; Minghini, Marco; Negreti, Marco (eds.). 1534:, which is conformal, can be constructed by using the tangent point's 4416: 2097: 886: 698: 632: 380: 749:), a trait possible only from one or two points to every other point 3669: 3225:

Proceedings of the 21st International Cartographic Conference (ICC)

2733: 2519: 2063:: Distances from the two poles are preserved, in equatorial aspect. 2051:: Distances from the two poles are preserved, in equatorial aspect. 1241:

are mapped to equally spaced lines radiating out from the apex and

5078: 3274:(1944). "The nomenclature and classification of map projections". 2557: 2152: 2087: 2025: 1784: 1757: 1318: 1228: 1050: 880: 655: 579: 286: 281: 229:

to a plane, preservation of shapes inevitably requires a variable

204: 124: 65: 29: 3925: 107:

Despite the name's literal meaning, projection is not limited to

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p. 12. Falls Church: American Congress on Surveying and Mapping.

3548:. Cartography and Geographic Information Society. Archived from 1504: 431:

helpful in understanding the basic concept of a map projection.

199: 5051: 5001: 4798: 4414: 3990: 3929: 3920: 2342:) – System to capture, manage, and present geographic data 1793:

is conformal and perspective but not equal area or equidistant.

1464:; so that even just a hemisphere is already infinite in extent. 5104: 3915: 3869: 3163:

Stooke, P. J. (1998). "Mapping Worlds with Irregular Shapes".

2773:(3). Cartography and Geographic Information Society: 167–182. 1245:(parallels) are mapped to circular arcs centered on the apex. 846:

Has straight parallels symmetrically placed about the equator;

100:. However, the term "map projection" refers specifically to a 2167:

magazine in 1988 but abandoned by them in about 1997 for the

1715:); locations closer than at a distance equal to the constant 1350:

The mapping of radial lines can be visualized by imagining a

564:) are often used to define the origin of the map projection. 3192:"Mathematical Basis for Non-spherical Celestial Bodies Maps" 971:

North-south distances neither stretched nor compressed (1):

2639:

Flattening the earth: two thousand years of map projections

2456:. New York, NY: American Elsevier Publishing Company, inc. 893:

A normal cylindrical projection is any projection in which

1033:

Cylindrical equal-area projection with oblique orientation

3864: 2311:

motivated the American Cartographic Association (now the

2096:

is thought to be the oldest map projection, developed by

986:

or Gall orthographic (undistorted at the 45° parallels),

791:

Preserving shortest route, a trait preserved only by the

5047: 3730:

Robinson, Arthur (1990). "Rectangular World Maps—No!".

3707:, second edition. New York: John Wiley and Sons. p. 82. 3190:

Shingareva, K.B.; Bugaevsky, L.M.; Nyrtsov, M. (2000).

2505:; Voxland, P.M. (1989). "An album of map projections". 2402:

Pages displaying short descriptions of redirect targets

2368: – drawings or diagrams used to describe an object 2356:

Pages displaying short descriptions of redirect targets

2303:

clearly because its familiarity breeds more popularity.

2834:"Flexion and Skewness in Map Projections of the Earth" 2760:"Symbolization of Map Projection Distortion: A Review" 2394: – Map of most or all of the surface of the Earth 1841:, great and small, maps to a circle or straight line. 582:

is the only way to represent the Earth with constant

375:

The creation of a map projection involves two steps:

3066:

London, Edinburgh, and Dublin Philosophical Magazine

2758:

Mulcahy, Karen A.; Clarke, Keith C. (January 2001).

2370:

Pages displaying wikidata descriptions as a fallback

931:; this distorts areas excessively in high latitudes. 920:

North-south stretching equals east-west stretching (

4941: 4896: 4887: 4864: 4811: 4754: 4731: 4713: 4673: 4583: 4535: 4512: 4489: 4480: 4427: 4369: 4319: 4306: 4261: 4233: 4150: 4141: 4041: 4012: 4003: 2246:Modern national mapping systems typically employ a 2057:: Distances from the center and edge are preserved. 3544:. In Robinson, Arthur H.; Snyder, John P. (eds.). 2629: 2627: 2625: 2623: 2621: 2619: 2617: 1956:(also known as Gall–Peters, or Peters, projection) 3624:Thematic Cartography and Geographic Visualization 3335:. Archived from the original on 12 December 2016. 2832:Goldberg, David M.; Gott III, J. Richard (2007). 2600:(3rd ed.). The University of Chicago Press. 2258:and low variation in scale over small areas. For 1900:. These are some projections that preserve area: 60:employed to represent the curved two-dimensional 2827: 2825: 2715: 2713: 1781:Projections by preservation of a metric property 1354:tangent to the Earth, with the central point as 1047:List of map projections § pseudocylindrical 897:are mapped to equally spaced vertical lines and 651:are often employed in projecting the ellipsoid. 147:. This map projection has the property of being 2300: 2274:. Reference maps of the world often appear on 2135:—also preserves distance from the central point 843:Has straight vertical meridians, spaced evenly; 817: 3841:Color images of map projections and distortion 3130:Cartography and Geographic Information Science 2767:Cartography and Geographic Information Science 2354: – Cartesian geographic coordinate system 2332: – Reference frame for measuring location 2313:Cartography and Geographic Information Society 1256:Conic projections that are commonly used are: 5063: 3941: 3830:, free software can render many projections ( 3694:. New York: McGraw–Hill. 2d ed., 1948. p. 87. 2315:) to produce a series of booklets (including 2129:—the only conformal retroazimuthal projection 8: 4888: 3847:Geometric aspects of mapping: map projection 3535: 3533: 1063:Pseudocylindrical projections represent the 901:(parallels) are mapped to horizontal lines. 824:Lee's objection refers to the way the terms 557:) and a parallel of origin (usually written 302:. For a given point, using the scale factor 191:shows areas accurately, but distorts shapes. 4812: 3992: 3242:. SpringerBriefs in Astronomy. p. 71. 2275: 522:The developable surface may also be either 5070: 5056: 5048: 4998: 4893: 4808: 4795: 4486: 4424: 4411: 4316: 4147: 4009: 4000: 3987: 3948: 3934: 3926: 2238:Suitability of projections for application 2144:Mecca or Qibla—also has vertical meridians 872:List of map projections § Cylindrical 627:Choosing a model for the shape of the body 390:Transformation of geographic coordinates ( 143:The most well-known map projection is the 5033:Map projection of the tri-axial ellipsoid 3870:PROJ.4 – Cartographic Projections Library 3517:"Lambert Azimuthal Equal-Area Projection" 3048: 3015: 2852: 2732: 2556: 2000:Snyder's equal-area polyhedral projection 1754:List of map projections § Polyhedral 1604:Other azimuthal projections are not true 686:. For irregular planetary bodies such as 502:(at right angles to the Earth's axis) or 2668:Map Projections in Planetary Cartography 1880: 1315:List of map projections § azimuthal 1028: 990:(undistorted at the 30° parallels), and 711:map projection of the triaxial ellipsoid 513: 477: 438: 339:(bending and lopsidedness) distortions. 331:is based on infinitesimals, and depicts 182: 3546:Matching the Map Projection to the Need 2538: 2536: 2427:Lambert, Johann; Tobler, Waldo (2011). 2419: 1865:Guyou hemisphere-in-a-square projection 1860:Adams hemisphere-in-a-square projection 614:. There are also projections (Maurer's 4142: 3338: 2890:Wirth, Ervin; Kun, Péter (July 2015). 2452:Richardus, Peter; Adler, Ron (1972). 306:along the meridian, the scale factor 7: 4370: 2307:A controversy in the 1980s over the 2073:are preserved, in equatorial aspect. 1826:are represented by straight segments 1408:Some azimuthal projections are true 1309:Azimuthal (projections onto a plane) 4800: 4755: 3327:Furuti, Carlos A. (11 April 2016). 2927:Jacobs, Frank (18 September 2013). 2700:"Which is the best map projection?" 618:, Close) where true distances from 44:and using his second map projection 4262: 3492:"Azimuthal Equidistant Projection" 3177:10.1111/j.1541-0064.1998.tb01553.x 2916:from the original on 23 July 2022. 2400: – Video projection technique 2281:Thematic maps normally require an 346:of fixed radius (e.g., 15 degrees 310:along the parallel, and the angle 25: 3703:Robinson, Arthur Howard. (1960). 3199:Journal of Geospatial Engineering 2969:"A cornucopia of map projections" 2802:Small-scale map projection design 2725:Map projections: A working manual 2698:Singh, Ishveena (25 April 2017). 2107:are displayed as straight lines: 1762:Buckminster Fuller's Dymaxion map 938:): The cylindric perspective (or 5220: 5209: 5208: 4976:Quadrilateralized spherical cube 4674: 4656:Quadrilateralized spherical cube 4004: 3908:based on work by Yu-Sung Chang ( 3744:10.1111/j.0033-0124.1990.00101.x 2032:two-point equidistant projection 1728: 1196: 1178: 1160: 1140: 1122: 1104: 616:two-point equidistant projection 612:azimuthal equidistant projection 135:, whereas small objects such as 131:are generally better modeled as 4942: 4732: 3883:"Understanding Map Projections" 3822:A Cornucopia of Map Projections 3690:Raisz, Erwin Josephus. (1938). 2282: 2206:B. J. S. Cahill's Butterfly Map 1507:, approximate this perspective. 667: 447:maps the globe onto a cylinder. 4565:Lambert cylindrical equal-area 4307: 3910:Wolfram Demonstrations Project 3855:, Henry Bottomley (SE16.info). 3542:"Enlarging the Heart of a Map" 3095:. City University of New York. 1980:Lambert cylindrical equal-area 1538:as the point of perspective. 1521:General Perspective projection 992:Lambert cylindrical equal-area 885:The Mercator projection shows 754: 663: 589:Some possible properties are: 484:transverse Mercator projection 1: 5013:Interruption (map projection) 4714: 3807:"An Album of Map Projections" 3216:Nyrtsov, M.V. (August 2003). 2336:Geographic information system 2201:Buckminster Fuller's Dymaxion 1855:Peirce quincuncial projection 1300:and other projections in the 947:Miller cylindrical projection 445:Miller cylindrical projection 435:Choosing a projection surface 291:Tissot's indicatrices on the 4651:Lambert azimuthal equal-area 4447:Guyou hemisphere-in-a-square 4437:Adams hemisphere-in-a-square 4234: 2929:"This is your brain on maps" 2431:. Redlands, CA: ESRI Press. 1975:Lambert azimuthal equal-area 1741:(click for detail) 1640:Lambert azimuthal equal-area 1405:is the radius of the Earth. 798:Because the sphere is not a 34:A medieval depiction of the 18:Pseudocylindrical projection 4865: 3575:10.1007/978-3-319-51835-0_3 3248:10.1007/978-1-4614-7762-4_6 3050:10.3138/Y51X-1590-PV21-136G 2989:Kartographische Nachrichten 2676:10.1007/978-3-319-51835-0_7 2643:University of Chicago Press 1525:International Space Station 1369:) and the transverse scale 165:National Geographic Society 5285: 3853:Java world map projections 3771:10.1559/152304089783814089 3467:"Stereographic Projection" 3142:10.1559/152304000783547957 2899:Open Innovation for Europe 2779:10.1559/152304001782153044 2545:Journal of Applied Geodesy 2398:Spherical image projection 1874: 1801: 1767:Polyhedral map projections 1751: 1634:flag of the United Nations 1312: 1302:polyconic projection class 1044: 973:equirectangular projection 869: 752:Preserving shape locally ( 571: 538:Tangent and secant lines ( 256: 5204: 5146: 5123: 5085: 5008: 4997: 4924: 4807: 4794: 4606: 4423: 4410: 4347: 4206: 4089: 3999: 3986: 3963: 3567:Choosing a Map Projection 3417:"Orthographic Projection" 3387:"The Gnomonic Projection" 3345:: CS1 maint: unfit URL ( 3288:10.1179/sre.1944.7.51.190 3078:10.1080/14786446108643179 2481:(fifth ed.). Wiley. 2211:Kavrayskiy VII projection 713:for further information. 179:Metric properties of maps 56:is any of a broad set of 3540:Snyder, John P. (1997). 3442:PROJ 7.1.1 documentation 3438:"Near-sided perspective" 2596:Monmonier, Mark (2018). 2508:Album of Map Projections 2380:South-up map orientation 1804:Conformal map projection 1791:stereographic projection 1532:stereographic projection 535:discussed further here. 506:(any angle in between). 474:Aspect of the projection 329:Goldberg-Gott indicatrix 323:Other distortion metrics 173:Winkel tripel projection 4452:Lambert conformal conic 3916:Compare Map Projections 3732:Professional Geographer 3705:Elements of Cartography 3307:"Sinusoidal Projection" 3165:The Canadian Geographer 3093:"Projection parameters" 2948:"Mercator Puzzle Redux" 2800:Canters, Frank (2002). 2479:Elements of Cartography 2382: – Map orientation 2361:List of map projections 1850:Lambert conformal conic 1469:orthographic projection 1410:perspective projections 1343:) of the true distance 1273:Lambert conformal conic 680:undulation of the geoid 371:Design and construction 72:. In a map projection, 5133:History of cartography 4585:Tobler hyperelliptical 4198:Tobler hyperelliptical 4124:Space-oblique Mercator 3031:Laskowski, P. (1997). 2863:10.3138/carto.42.4.297 2305: 2276:compromise projections 2172: 2149:Compromise projections 2101: 2041:equidistant projection 2035: 2010:Tobler hyperelliptical 1931:Cylindrical equal-area 1889: 1813:ellipses of distortion 1794: 1763: 1324: 1234: 1098:Tobler hyperelliptical 1060: 1034: 1010:Transverse cylindrical 980:equal-area cylindrical 890: 822: 806:Projections by surface 745:Preserving direction ( 519: 487: 448: 424:eastings and northings 295: 192: 161:Gall–Peters projection 153:equal-area projections 45: 5138:List of cartographers 3902:World Map Projections 3759:American Cartographer 3365:"Gnomonic Projection" 3111:ArcSDE Developer Help 2575:10.1515/jag-2015-0033 2283:equal area projection 2254:in order to preserve 2250:or close variant for 2232:AuthaGraph projection 2156: 2133:Hammer retroazimuthal 2100:in the 6th century BC 2091: 2077:Two-point equidistant 2055:Azimuthal equidistant 2029: 1884: 1877:Equal-area projection 1788: 1761: 1678:Logarithmic azimuthal 1613:Azimuthal equidistant 1322: 1232: 1054: 1032: 884: 784:Preserving distance ( 747:azimuthal or zenithal 730:), and planar (e.g., 654:A third model is the 622:points are preserved. 572:Further information: 517: 481: 442: 290: 186: 157:Sinusoidal projection 90:differential geometry 76:, often expressed as 33: 5264:Descriptive geometry 4961:Cahill–Keyes M-shape 4821:Chamberlin trimetric 3876:Projection Reference 3599:Choosing a World Map 3276:Empire Survey Review 3117:on 28 November 2018. 2598:How to lie with maps 2221:Chamberlin trimetric 2216:Wagner VI projection 2139:Craig retroazimuthal 1887:Mollweide projection 1513:The Blue Marble 2012 1414:point of perspective 1361:The radial scale is 1219:Collignon projection 1086:Collignon projection 470:further distortion. 264:Carl Friedrich Gauss 115:image produced by a 5028:Tissot's indicatrix 4929:Central cylindrical 4570:Smyth equal-surface 4472:Transverse Mercator 4321:General perspective 4076:Smyth equal-surface 4028:Transverse Mercator 3843:(Mapthematics.com). 3692:General Cartography 3658:Geographical Review 3329:"Conic Projections" 2946:Van Damme, Bramus. 2567:2016JAGeo..10..197G 2290:Mercator projection 2248:transverse Mercator 2164:National Geographic 2159:Robinson projection 2112:Gnomonic projection 2094:Gnomonic projection 2069:Distances from the 1830:Transverse Mercator 1426:gnomonic projection 1243:circles of latitude 1025:Oblique cylindrical 1019:transverse Mercator 940:central cylindrical 899:circles of latitude 800:developable surface 793:gnomonic projection 649:Auxiliary latitudes 605:Mercator projection 454:developable surface 352:spherical triangles 300:Tissot's indicatrix 293:Mercator projection 259:Tissot's indicatrix 169:Robinson projection 145:Mercator projection 94:projective geometry 4981:Waterman butterfly 4831:Miller cylindrical 4462:Peirce quincuncial 4357:Lambert equal-area 4109:Gall stereographic 3894:, Melita Kennedy ( 3849:(KartoWeb.itc.nl). 3514:Weisstein, Eric W. 3489:Weisstein, Eric W. 3464:Weisstein, Eric W. 3414:Weisstein, Eric W. 3362:Weisstein, Eric W. 3304:Weisstein, Eric W. 3091:Albrecht, Jochen. 3017:astro-ph/0608500v1 2298:editorial states: 2191:Miller cylindrical 2173: 2102: 2036: 1960:Goode's homolosine 1890: 1839:circle of a sphere 1795: 1764: 1420:) onto the plane: 1325: 1298:American polyconic 1235: 1061: 1035: 975:or "plate carrée". 891: 877:Normal cylindrical 520: 488: 449: 296: 193: 46: 5236: 5235: 5045: 5044: 5041: 5040: 4993: 4992: 4989: 4988: 4937: 4936: 4790: 4789: 4786: 4785: 4669: 4668: 4406: 4405: 4402: 4401: 4365: 4364: 4253:Lambert conformal 4229: 4228: 4143:Pseudocylindrical 4137: 4136: 3765:: 222–223. 1989. 3718:Which Map is Best 3584:978-3-319-51835-0 3257:978-1-4614-7761-7 3107:"Map projections" 2721:Snyder, John Parr 2685:978-3-319-51834-3 2607:978-0-226-43592-3 2438:978-1-58948-281-4 2317:Which Map Is Best 2061:Equidistant conic 1954:Gall orthographic 1261:Equidistant conic 1206: 1205: 1041:Pseudocylindrical 765:Preserving area ( 705:, with its major 607:in normal aspect. 544:standard parallel 269:Theorema Egregium 189:Albers projection 16:(Redirected from 5276: 5224: 5212: 5211: 5151:Animated mapping 5128:Early world maps 5100:Geovisualization 5072: 5065: 5058: 5049: 4999: 4956:Cahill Butterfly 4894: 4874:Goode homolosine 4809: 4796: 4761: 4760:(Mecca or Qibla) 4641:Goode homolosine 4487: 4425: 4412: 4317: 4312: 4183:Goode homolosine 4148: 4033:Oblique Mercator 4010: 4001: 3988: 3950: 3943: 3936: 3927: 3893: 3889: 3887: 3817: 3813: 3811: 3775: 3774: 3754: 3748: 3747: 3727: 3721: 3714: 3708: 3701: 3695: 3688: 3682: 3681: 3653: 3647: 3644: 3638: 3637: 3619: 3613: 3612: 3595: 3589: 3588: 3561: 3559: 3557: 3537: 3528: 3527: 3526: 3509: 3503: 3502: 3501: 3484: 3478: 3477: 3476: 3459: 3453: 3452: 3450: 3449: 3434: 3428: 3427: 3426: 3409: 3403: 3402: 3400: 3398: 3393:on 30 April 2016 3389:. Archived from 3382: 3376: 3375: 3374: 3357: 3351: 3350: 3344: 3336: 3324: 3318: 3317: 3316: 3299: 3293: 3291: 3268: 3262: 3261: 3235: 3229: 3228: 3222: 3213: 3207: 3206: 3196: 3187: 3181: 3180: 3160: 3154: 3153: 3125: 3119: 3118: 3113:. Archived from 3103: 3097: 3096: 3088: 3082: 3081: 3072:(149): 409–421. 3061: 3055: 3054: 3052: 3028: 3022: 3021: 3019: 3007: 3001: 3000: 2997: 2983: 2977: 2976: 2965: 2959: 2958: 2956: 2954: 2943: 2937: 2936: 2931:. Strange Maps. 2924: 2918: 2917: 2915: 2896: 2887: 2881: 2880: 2878: 2877: 2856: 2854:astro-ph/0608501 2838: 2829: 2820: 2819: 2797: 2791: 2790: 2764: 2755: 2749: 2748: 2736: 2717: 2708: 2707: 2695: 2689: 2688: 2663: 2657: 2656: 2631: 2612: 2611: 2593: 2587: 2586: 2560: 2540: 2531: 2530: 2528: 2526: 2513: 2499: 2493: 2492: 2474: 2468: 2467: 2449: 2443: 2442: 2424: 2403: 2371: 2357: 2341: 2252:large-scale maps 2067:Werner cordiform 1732: 1714: 1712: 1711: 1703: 1700: 1674: 1672: 1671: 1665: 1662: 1627:; it is used by 1599: 1597: 1596: 1590: 1587: 1570: 1568: 1567: 1561: 1558: 1502: 1500: 1499: 1494: 1491: 1463: 1461: 1460: 1455: 1452: 1400: 1398: 1397: 1392: 1389: 1292:Werner cordiform 1221:in polar areas. 1200: 1182: 1164: 1144: 1134:Goode homolosine 1126: 1108: 1092: 1091: 964: 962: 961: 958: 955: 726:), conic (e.g., 703:Jacobi ellipsoid 645:topographic maps 574:Map scale factor 548:central meridian 314:′ between them, 133:oblate spheroids 129:celestial bodies 127:and other large 21: 5284: 5283: 5279: 5278: 5277: 5275: 5274: 5273: 5249:Map projections 5239: 5238: 5237: 5232: 5200: 5191:Topographic map 5142: 5119: 5081: 5076: 5046: 5037: 5004: 4985: 4933: 4920: 4883: 4860: 4846:Van der Grinten 4803: 4801:By construction 4782: 4759: 4758: 4750: 4727: 4709: 4690:Equirectangular 4676: 4665: 4602: 4579: 4575:Trystan Edwards 4531: 4508: 4476: 4419: 4398: 4371:Pseudoazimuthal 4361: 4343: 4310: 4309: 4302: 4257: 4225: 4221:Winkel I and II 4202: 4133: 4114:Gall isographic 4104:Equirectangular 4085: 4081:Trystan Edwards 4037: 3995: 3982: 3959: 3954: 3906:Stephen Wolfram 3891: 3885: 3881: 3859:Map Projections 3815: 3809: 3805: 3802: 3797: 3792:Map Projections 3784: 3779: 3778: 3756: 3755: 3751: 3729: 3728: 3724: 3715: 3711: 3702: 3698: 3689: 3685: 3655: 3654: 3650: 3645: 3641: 3634: 3621: 3620: 3616: 3609: 3597: 3596: 3592: 3585: 3564: 3562: 3555: 3553: 3539: 3538: 3531: 3512: 3511: 3510: 3506: 3487: 3486: 3485: 3481: 3462: 3461: 3460: 3456: 3447: 3445: 3436: 3435: 3431: 3412: 3411: 3410: 3406: 3396: 3394: 3384: 3383: 3379: 3360: 3359: 3358: 3354: 3337: 3326: 3325: 3321: 3302: 3301: 3300: 3296: 3282:(51): 190–200. 3270: 3269: 3265: 3258: 3237: 3236: 3232: 3220: 3215: 3214: 3210: 3194: 3189: 3188: 3184: 3162: 3161: 3157: 3127: 3126: 3122: 3105: 3104: 3100: 3090: 3089: 3085: 3063: 3062: 3058: 3030: 3029: 3025: 3009: 3008: 3004: 2991: 2985: 2984: 2980: 2967: 2966: 2962: 2952: 2950: 2945: 2944: 2940: 2926: 2925: 2921: 2913: 2894: 2889: 2888: 2884: 2875: 2873: 2836: 2831: 2830: 2823: 2816: 2799: 2798: 2794: 2762: 2757: 2756: 2752: 2745: 2719: 2718: 2711: 2697: 2696: 2692: 2686: 2665: 2664: 2660: 2653: 2635:Snyder, John P. 2633: 2632: 2615: 2608: 2595: 2594: 2590: 2542: 2541: 2534: 2524: 2522: 2511: 2501: 2500: 2496: 2489: 2476: 2475: 2471: 2464: 2454:map projections 2451: 2450: 2446: 2439: 2426: 2425: 2421: 2416: 2411: 2406: 2401: 2369: 2355: 2339: 2325: 2240: 2186:van der Grinten 2161:was adopted by 2151: 2120: 2086: 2024: 2019: 1911:Boggs eumorphic 1885:The equal-area 1879: 1873: 1806: 1800: 1783: 1756: 1750: 1745: 1744: 1743: 1738: 1733: 1721: 1710: 1704: 1701: 1696: 1695: 1693: 1666: 1663: 1658: 1657: 1655: 1654: sin 1591: 1588: 1583: 1582: 1580: 1579: cos 1571:; the scale is 1562: 1559: 1554: 1553: 1551: 1550: tan 1495: 1492: 1487: 1486: 1484: 1483: sin 1456: 1453: 1448: 1447: 1445: 1444: tan 1393: 1390: 1385: 1384: 1382: 1381: sin 1317: 1311: 1282: 1227: 1211: 1049: 1043: 1027: 1012: 959: 956: 953: 952: 950: 879: 874: 868: 840:Is rectangular; 808: 719: 629: 576: 570: 563: 556: 512: 476: 437: 373: 325: 261: 255: 181: 58:transformations 28: 23: 22: 15: 12: 11: 5: 5282: 5280: 5272: 5271: 5266: 5261: 5256: 5251: 5241: 5240: 5234: 5233: 5231: 5230: 5218: 5205: 5202: 5201: 5199: 5198: 5193: 5188: 5183: 5178: 5176:Nautical chart 5173: 5171:Linguistic map 5168: 5163: 5161:Choropleth map 5158: 5153: 5147: 5144: 5143: 5141: 5140: 5135: 5130: 5124: 5121: 5120: 5118: 5117: 5112: 5110:Map projection 5107: 5102: 5097: 5092: 5086: 5083: 5082: 5077: 5075: 5074: 5067: 5060: 5052: 5043: 5042: 5039: 5038: 5036: 5035: 5030: 5025: 5020: 5015: 5009: 5006: 5005: 5002: 4995: 4994: 4991: 4990: 4987: 4986: 4984: 4983: 4978: 4973: 4968: 4963: 4958: 4953: 4947: 4945: 4939: 4938: 4935: 4934: 4932: 4931: 4925: 4922: 4921: 4919: 4918: 4913: 4908: 4902: 4900: 4891: 4885: 4884: 4882: 4881: 4876: 4870: 4868: 4862: 4861: 4859: 4858: 4853: 4848: 4843: 4838: 4833: 4828: 4826:Kavrayskiy VII 4823: 4817: 4815: 4805: 4804: 4799: 4792: 4791: 4788: 4787: 4784: 4783: 4781: 4780: 4775: 4770: 4764: 4762: 4756:Retroazimuthal 4752: 4751: 4749: 4748: 4743: 4737: 4735: 4729: 4728: 4726: 4725: 4719: 4717: 4711: 4710: 4708: 4707: 4702: 4697: 4692: 4687: 4681: 4679: 4675:Equidistant in 4671: 4670: 4667: 4666: 4664: 4663: 4658: 4653: 4648: 4643: 4638: 4633: 4628: 4623: 4618: 4613: 4607: 4604: 4603: 4601: 4600: 4595: 4589: 4587: 4581: 4580: 4578: 4577: 4572: 4567: 4562: 4557: 4552: 4547: 4541: 4539: 4533: 4532: 4530: 4529: 4524: 4518: 4516: 4510: 4509: 4507: 4506: 4501: 4495: 4493: 4484: 4478: 4477: 4475: 4474: 4469: 4464: 4459: 4454: 4449: 4444: 4439: 4433: 4431: 4421: 4420: 4415: 4408: 4407: 4404: 4403: 4400: 4399: 4397: 4396: 4391: 4386: 4381: 4375: 4373: 4367: 4366: 4363: 4362: 4360: 4359: 4354: 4348: 4345: 4344: 4342: 4341: 4336: 4331: 4325: 4323: 4314: 4304: 4303: 4301: 4300: 4295: 4294: 4293: 4288: 4278: 4273: 4267: 4265: 4259: 4258: 4256: 4255: 4250: 4245: 4239: 4237: 4231: 4230: 4227: 4226: 4224: 4223: 4218: 4213: 4211:Kavrayskiy VII 4207: 4204: 4203: 4201: 4200: 4195: 4190: 4185: 4180: 4175: 4170: 4165: 4160: 4154: 4152: 4145: 4139: 4138: 4135: 4134: 4132: 4131: 4126: 4121: 4116: 4111: 4106: 4101: 4096: 4090: 4087: 4086: 4084: 4083: 4078: 4073: 4068: 4063: 4058: 4053: 4047: 4045: 4039: 4038: 4036: 4035: 4030: 4025: 4019: 4017: 4007: 3997: 3996: 3991: 3984: 3983: 3981: 3980: 3975: 3970: 3964: 3961: 3960: 3957:Map projection 3955: 3953: 3952: 3945: 3938: 3930: 3924: 3923: 3918: 3913: 3899: 3892:(1.70 MB) 3879: 3873: 3867: 3862: 3856: 3850: 3844: 3838: 3825: 3819: 3816:(12.6 MB) 3801: 3800:External links 3798: 3796: 3795: 3789: 3785: 3783: 3780: 3777: 3776: 3749: 3738:(1): 101–104. 3722: 3709: 3696: 3683: 3670:10.2307/210384 3664:(3): 424–430. 3648: 3639: 3632: 3614: 3607: 3590: 3583: 3563:Reprinted in: 3552:on 2 July 2010 3529: 3504: 3479: 3454: 3429: 3404: 3385:Savard, John. 3377: 3352: 3319: 3294: 3263: 3256: 3230: 3208: 3182: 3155: 3120: 3098: 3083: 3056: 3023: 3002: 2978: 2960: 2938: 2919: 2882: 2847:(4): 297–318. 2821: 2814: 2792: 2750: 2743: 2734:10.3133/pp1395 2709: 2704:Geoawesomeness 2690: 2684: 2658: 2651: 2613: 2606: 2588: 2551:(3): 197–209. 2532: 2520:10.3133/pp1453 2494: 2487: 2469: 2462: 2444: 2437: 2418: 2417: 2415: 2412: 2410: 2407: 2405: 2404: 2395: 2389: 2383: 2377: 2375:Rubbersheeting 2372: 2366:Plan (drawing) 2363: 2358: 2352:Grid reference 2349: 2346:Geoinformatics 2343: 2333: 2330:Geodetic datum 2326: 2324: 2321: 2295:New York Times 2239: 2236: 2235: 2234: 2229: 2223: 2218: 2213: 2208: 2203: 2198: 2193: 2188: 2183: 2150: 2147: 2146: 2145: 2136: 2130: 2119: 2118:Retroazimuthal 2116: 2115: 2114: 2085: 2082: 2081: 2080: 2074: 2064: 2058: 2052: 2023: 2020: 2018: 2017: 2012: 2007: 2004:geodesic grids 1997: 1992: 1987: 1982: 1977: 1972: 1967: 1962: 1957: 1951: 1946: 1933: 1928: 1923: 1918: 1913: 1908: 1902: 1875:Main article: 1872: 1869: 1868: 1867: 1862: 1857: 1852: 1847: 1842: 1832: 1827: 1802:Main article: 1799: 1796: 1782: 1779: 1749: 1746: 1735: 1734: 1727: 1726: 1725: 1724: 1723: 1722:are not shown. 1719: 1708: 1692: ln 1675: 1637: 1602: 1601: 1528: 1517: 1508: 1465: 1310: 1307: 1306: 1305: 1295: 1289: 1281: 1278: 1277: 1276: 1270: 1264: 1226: 1223: 1210: 1207: 1204: 1203: 1202: 1201: 1193: 1192: 1190:Kavrayskiy VII 1185: 1184: 1183: 1175: 1174: 1167: 1166: 1165: 1157: 1156: 1148: 1147: 1146: 1145: 1137: 1136: 1129: 1128: 1127: 1119: 1118: 1111: 1110: 1109: 1101: 1100: 1090: 1089: 1083: 1042: 1039: 1026: 1023: 1011: 1008: 996: 995: 976: 969: 943: 932: 878: 875: 867: 864: 852: 851: 847: 844: 841: 807: 804: 796: 795: 789: 782: 763: 750: 718: 717:Classification 715: 701:'s shape is a 660:mean sea level 628: 625: 624: 623: 608: 601: 598: 569: 566: 561: 554: 540:standard lines 511: 508: 475: 472: 436: 433: 428: 427: 388: 372: 369: 348:angular radius 324: 321: 316:Nicolas Tissot 257:Main article: 254: 251: 223: 222: 217: 212: 207: 202: 180: 177: 117:pinhole camera 54:map projection 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 5281: 5270: 5267: 5265: 5262: 5260: 5257: 5255: 5252: 5250: 5247: 5246: 5244: 5229: 5228: 5223: 5219: 5217: 5216: 5215:Category:Maps 5207: 5206: 5203: 5197: 5194: 5192: 5189: 5187: 5184: 5182: 5181:Pictorial map 5179: 5177: 5174: 5172: 5169: 5167: 5164: 5162: 5159: 5157: 5154: 5152: 5149: 5148: 5145: 5139: 5136: 5134: 5131: 5129: 5126: 5125: 5122: 5116: 5113: 5111: 5108: 5106: 5103: 5101: 5098: 5096: 5093: 5091: 5088: 5087: 5084: 5080: 5073: 5068: 5066: 5061: 5059: 5054: 5053: 5050: 5034: 5031: 5029: 5026: 5024: 5021: 5019: 5016: 5014: 5011: 5010: 5007: 5000: 4996: 4982: 4979: 4977: 4974: 4972: 4969: 4967: 4964: 4962: 4959: 4957: 4954: 4952: 4949: 4948: 4946: 4944: 4940: 4930: 4927: 4926: 4923: 4917: 4916:Stereographic 4914: 4912: 4909: 4907: 4904: 4903: 4901: 4899: 4895: 4892: 4890: 4886: 4880: 4877: 4875: 4872: 4871: 4869: 4867: 4863: 4857: 4856:Winkel tripel 4854: 4852: 4849: 4847: 4844: 4842: 4839: 4837: 4836:Natural Earth 4834: 4832: 4829: 4827: 4824: 4822: 4819: 4818: 4816: 4814: 4810: 4806: 4802: 4797: 4793: 4779: 4776: 4774: 4771: 4769: 4766: 4765: 4763: 4757: 4753: 4747: 4744: 4742: 4739: 4738: 4736: 4734: 4730: 4724: 4721: 4720: 4718: 4716: 4712: 4706: 4703: 4701: 4698: 4696: 4693: 4691: 4688: 4686: 4683: 4682: 4680: 4678: 4672: 4662: 4659: 4657: 4654: 4652: 4649: 4647: 4644: 4642: 4639: 4637: 4634: 4632: 4629: 4627: 4624: 4622: 4619: 4617: 4616:Briesemeister 4614: 4612: 4609: 4608: 4605: 4599: 4596: 4594: 4591: 4590: 4588: 4586: 4582: 4576: 4573: 4571: 4568: 4566: 4563: 4561: 4558: 4556: 4553: 4551: 4548: 4546: 4543: 4542: 4540: 4538: 4534: 4528: 4525: 4523: 4520: 4519: 4517: 4515: 4511: 4505: 4502: 4500: 4497: 4496: 4494: 4492: 4488: 4485: 4483: 4479: 4473: 4470: 4468: 4467:Stereographic 4465: 4463: 4460: 4458: 4455: 4453: 4450: 4448: 4445: 4443: 4440: 4438: 4435: 4434: 4432: 4430: 4426: 4422: 4418: 4413: 4409: 4395: 4394:Winkel tripel 4392: 4390: 4387: 4385: 4382: 4380: 4377: 4376: 4374: 4372: 4368: 4358: 4355: 4353: 4350: 4349: 4346: 4340: 4339:Stereographic 4337: 4335: 4332: 4330: 4327: 4326: 4324: 4322: 4318: 4315: 4313: 4305: 4299: 4296: 4292: 4289: 4287: 4284: 4283: 4282: 4279: 4277: 4274: 4272: 4269: 4268: 4266: 4264: 4263:Pseudoconical 4260: 4254: 4251: 4249: 4246: 4244: 4241: 4240: 4238: 4236: 4232: 4222: 4219: 4217: 4214: 4212: 4209: 4208: 4205: 4199: 4196: 4194: 4191: 4189: 4186: 4184: 4181: 4179: 4176: 4174: 4171: 4169: 4166: 4164: 4161: 4159: 4156: 4155: 4153: 4149: 4146: 4144: 4140: 4130: 4127: 4125: 4122: 4120: 4117: 4115: 4112: 4110: 4107: 4105: 4102: 4100: 4097: 4095: 4092: 4091: 4088: 4082: 4079: 4077: 4074: 4072: 4069: 4067: 4064: 4062: 4059: 4057: 4054: 4052: 4049: 4048: 4046: 4044: 4040: 4034: 4031: 4029: 4026: 4024: 4021: 4020: 4018: 4015: 4011: 4008: 4006: 4002: 3998: 3994: 3989: 3985: 3979: 3976: 3974: 3971: 3969: 3966: 3965: 3962: 3958: 3951: 3946: 3944: 3939: 3937: 3932: 3931: 3928: 3922: 3919: 3917: 3914: 3911: 3907: 3903: 3900: 3897: 3884: 3880: 3877: 3874: 3871: 3868: 3866: 3863: 3860: 3857: 3854: 3851: 3848: 3845: 3842: 3839: 3836: 3833: 3829: 3826: 3823: 3820: 3808: 3804: 3803: 3799: 3793: 3790: 3787: 3786: 3781: 3772: 3768: 3764: 3760: 3753: 3750: 3745: 3741: 3737: 3733: 3726: 3723: 3719: 3713: 3710: 3706: 3700: 3697: 3693: 3687: 3684: 3679: 3675: 3671: 3667: 3663: 3659: 3652: 3649: 3643: 3640: 3635: 3633:0-13-035123-7 3629: 3625: 3618: 3615: 3610: 3608:0-9613459-2-6 3604: 3600: 3594: 3591: 3586: 3580: 3576: 3572: 3568: 3551: 3547: 3543: 3536: 3534: 3530: 3524: 3523: 3518: 3515: 3508: 3505: 3499: 3498: 3493: 3490: 3483: 3480: 3474: 3473: 3468: 3465: 3458: 3455: 3443: 3439: 3433: 3430: 3424: 3423: 3418: 3415: 3408: 3405: 3392: 3388: 3381: 3378: 3372: 3371: 3366: 3363: 3356: 3353: 3348: 3342: 3334: 3330: 3323: 3320: 3314: 3313: 3308: 3305: 3298: 3295: 3289: 3285: 3281: 3277: 3273: 3267: 3264: 3259: 3253: 3249: 3245: 3241: 3234: 3231: 3226: 3219: 3212: 3209: 3204: 3200: 3193: 3186: 3183: 3178: 3174: 3170: 3166: 3159: 3156: 3151: 3147: 3143: 3139: 3135: 3131: 3124: 3121: 3116: 3112: 3108: 3102: 3099: 3094: 3087: 3084: 3079: 3075: 3071: 3067: 3060: 3057: 3051: 3046: 3042: 3038: 3037:Cartographica 3034: 3027: 3024: 3018: 3013: 3006: 3003: 2998: 2995: 2990: 2982: 2979: 2974: 2970: 2964: 2961: 2949: 2942: 2939: 2934: 2930: 2923: 2920: 2912: 2908: 2904: 2900: 2893: 2886: 2883: 2872: 2868: 2864: 2860: 2855: 2850: 2846: 2842: 2841:Cartographica 2835: 2828: 2826: 2822: 2817: 2815:9780203472095 2811: 2807: 2803: 2796: 2793: 2788: 2784: 2780: 2776: 2772: 2768: 2761: 2754: 2751: 2746: 2744:9780318235622 2740: 2735: 2730: 2726: 2722: 2716: 2714: 2710: 2705: 2701: 2694: 2691: 2687: 2681: 2677: 2673: 2669: 2662: 2659: 2654: 2652:0-226-76746-9 2648: 2644: 2640: 2636: 2630: 2628: 2626: 2624: 2622: 2620: 2618: 2614: 2609: 2603: 2599: 2592: 2589: 2584: 2580: 2576: 2572: 2568: 2564: 2559: 2554: 2550: 2546: 2539: 2537: 2533: 2521: 2517: 2510: 2509: 2504: 2498: 2495: 2490: 2488:0-471-09877-9 2484: 2480: 2473: 2470: 2465: 2463:0-444-10362-7 2459: 2455: 2448: 2445: 2440: 2434: 2430: 2423: 2420: 2413: 2408: 2399: 2396: 2393: 2390: 2387: 2384: 2381: 2378: 2376: 2373: 2367: 2364: 2362: 2359: 2353: 2350: 2347: 2344: 2337: 2334: 2331: 2328: 2327: 2322: 2320: 2318: 2314: 2310: 2304: 2299: 2297: 2296: 2291: 2286: 2284: 2279: 2277: 2273: 2269: 2265: 2264:Winkel tripel 2261: 2260:smaller-scale 2257: 2253: 2249: 2244: 2237: 2233: 2230: 2227: 2224: 2222: 2219: 2217: 2214: 2212: 2209: 2207: 2204: 2202: 2199: 2197: 2196:Winkel Tripel 2194: 2192: 2189: 2187: 2184: 2182: 2179: 2178: 2177: 2176:projections: 2170: 2169:Winkel tripel 2166: 2165: 2160: 2155: 2148: 2143: 2140: 2137: 2134: 2131: 2128: 2125: 2124: 2123: 2117: 2113: 2110: 2109: 2108: 2106: 2105:Great circles 2099: 2095: 2090: 2083: 2078: 2075: 2072: 2068: 2065: 2062: 2059: 2056: 2053: 2050: 2047: 2046: 2045: 2042: 2033: 2028: 2021: 2016: 2013: 2011: 2008: 2005: 2001: 1998: 1996: 1993: 1991: 1988: 1986: 1983: 1981: 1978: 1976: 1973: 1971: 1968: 1966: 1963: 1961: 1958: 1955: 1952: 1950: 1947: 1945: 1941: 1937: 1934: 1932: 1929: 1927: 1924: 1922: 1919: 1917: 1914: 1912: 1909: 1907: 1904: 1903: 1901: 1899: 1895: 1888: 1883: 1878: 1870: 1866: 1863: 1861: 1858: 1856: 1853: 1851: 1848: 1846: 1843: 1840: 1836: 1835:Stereographic 1833: 1831: 1828: 1825: 1821: 1818: 1817: 1816: 1814: 1810: 1805: 1797: 1792: 1787: 1780: 1778: 1776: 1772: 1768: 1760: 1755: 1747: 1742: 1737: 1731: 1718: 1707: 1699: 1691: 1687: 1683: 1679: 1676: 1670: 1661: 1653: 1649: 1645: 1641: 1638: 1635: 1630: 1629:amateur radio 1626: 1622: 1618: 1614: 1611: 1610: 1609: 1608:projections: 1607: 1595: 1586: 1578: 1574: 1566: 1557: 1549: 1545: 1541: 1537: 1533: 1529: 1526: 1522: 1518: 1515: 1514: 1509: 1506: 1498: 1490: 1482: 1478: 1474: 1470: 1466: 1459: 1451: 1443: 1439: 1435: 1431: 1430:great circles 1427: 1423: 1422: 1421: 1419: 1415: 1411: 1406: 1404: 1396: 1388: 1380: 1376: 1372: 1368: 1364: 1359: 1357: 1353: 1348: 1346: 1342: 1338: 1333: 1332:great circles 1329: 1321: 1316: 1308: 1303: 1299: 1296: 1293: 1290: 1287: 1284: 1283: 1279: 1274: 1271: 1268: 1265: 1262: 1259: 1258: 1257: 1254: 1251: 1246: 1244: 1240: 1231: 1224: 1222: 1220: 1216: 1208: 1199: 1195: 1194: 1191: 1188: 1187: 1186: 1181: 1177: 1176: 1173: 1170: 1169: 1168: 1163: 1159: 1158: 1155: 1152: 1151: 1150: 1149: 1143: 1139: 1138: 1135: 1132: 1131: 1130: 1125: 1121: 1120: 1117: 1114: 1113: 1112: 1107: 1103: 1102: 1099: 1096: 1095: 1094: 1093: 1087: 1084: 1081: 1078: 1077: 1076: 1073: 1069: 1066: 1058: 1053: 1048: 1040: 1038: 1031: 1024: 1022: 1020: 1015: 1009: 1007: 1004: 1002: 993: 989: 985: 981: 977: 974: 970: 967: 948: 944: 941: 937: 933: 930: 926: 923: 919: 918: 917: 915: 911: 906: 902: 900: 896: 888: 883: 876: 873: 865: 863: 860: 855: 848: 845: 842: 839: 838: 837: 835: 831: 827: 821: 816: 814: 805: 803: 801: 794: 790: 787: 783: 780: 776: 772: 768: 764: 761: 757: 756: 751: 748: 744: 743: 742: 739: 737: 733: 732:stereographic 729: 725: 716: 714: 712: 708: 704: 700: 696: 691: 689: 685: 681: 677: 673: 669: 665: 661: 657: 652: 650: 646: 640: 638: 634: 626: 621: 617: 613: 609: 606: 602: 599: 596: 595:conformal map 592: 591: 590: 587: 585: 581: 575: 567: 565: 560: 553: 549: 545: 541: 536: 533: 532: 527: 526: 516: 510:Notable lines 509: 507: 505: 501: 497: 493: 485: 480: 473: 471: 467: 464: 460: 456: 455: 446: 441: 434: 432: 425: 421: 417: 413: 409: 405: 401: 397: 393: 389: 386: 382: 378: 377: 376: 370: 368: 364: 362: 361:bivariate map 356: 353: 350:). Sometimes 349: 345: 340: 338: 334: 330: 322: 320: 317: 313: 309: 305: 301: 294: 289: 285: 283: 279: 275: 271: 270: 265: 260: 252: 250: 247: 243: 238: 236: 232: 228: 221: 218: 216: 213: 211: 208: 206: 203: 201: 198: 197: 196: 190: 185: 178: 176: 174: 170: 166: 162: 158: 154: 150: 146: 141: 138: 134: 130: 126: 121: 118: 114: 110: 105: 103: 99: 95: 91: 85: 83: 79: 75: 71: 67: 63: 59: 55: 51: 43: 42: 37: 32: 19: 5259:Infographics 5227:Portal:Atlas 5225: 5213: 5186:Thematic map 5166:Geologic map 5109: 4911:Orthographic 4442:Gauss–Krüger 4334:Orthographic 4129:Web Mercator 4023:Gauss–Krüger 3956: 3861:(MathWorld). 3762: 3758: 3752: 3735: 3731: 3725: 3717: 3712: 3704: 3699: 3691: 3686: 3661: 3657: 3651: 3642: 3623: 3617: 3598: 3593: 3566: 3554:. Retrieved 3550:the original 3545: 3520: 3507: 3495: 3482: 3470: 3457: 3446:. Retrieved 3444:. 2020-09-17 3441: 3432: 3420: 3407: 3397:November 18, 3395:. Retrieved 3391:the original 3380: 3368: 3355: 3332: 3322: 3310: 3297: 3279: 3275: 3266: 3239: 3233: 3227:: 1158–1164. 3224: 3211: 3202: 3198: 3185: 3168: 3164: 3158: 3133: 3129: 3123: 3115:the original 3110: 3101: 3086: 3069: 3065: 3059: 3040: 3036: 3026: 3005: 2987: 2981: 2973:Mapthematics 2972: 2963: 2951:. Retrieved 2941: 2932: 2922: 2898: 2885: 2874:. Retrieved 2844: 2840: 2801: 2795: 2770: 2766: 2753: 2724: 2703: 2693: 2667: 2661: 2638: 2597: 2591: 2548: 2544: 2523:. 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