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83-513: Most state plane zones are based on either a transverse Mercator projection or a Lambert conformal conic projection . The choice between the two map projections is based on the shape of the state and its zones. States that are long in the east–west direction are typically divided into zones that are also long east–west. These zones use the Lambert conformal conic projection , because it is good at maintaining accuracy along an east–west axis, due to

166-402: 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. Depending on

249-502: A common coordinate system. The National Geodetic Survey has announced a modernization of the National Spatial Reference System , and a replacement of the state plane coordinate system will be part of the modernization. The number of zones will be substantially higher than the 1983 system. One new feature will be a state-wide zone for projects that extend over more than one zone. The NGS expects to release

332-462: A course of constant bearing is always plotted as a straight line. A normal cylindrical projection is any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude (parallels) are mapped to horizontal 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. This cylinder

415-535: A cylinder tangential at the equator with axis along the polar axis of the sphere. The cylindrical projections are constructed so that all points on a meridian are projected to points with x = a λ {\displaystyle x=a\lambda } (where a {\displaystyle a} is the Earth radius ) and y {\displaystyle y} is a prescribed function of ϕ {\displaystyle \phi } . For

498-416: A cylindrical projection (for example) is one which: (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.) 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

581-399: A grid line of constant x , defining grid north. Therefore, γ is positive in the quadrant north of the equator and east of the central meridian and also in the quadrant south of the equator and west of the central meridian. The convergence must be added to a grid bearing to obtain a bearing from true north. For the secant transverse Mercator the convergence may be expressed either in terms of

664-499: A grid: the grid is an independent construct which could be defined arbitrarily. In practice the national implementations, and UTM, do use grids aligned with the Cartesian axes of the projection, but they are of finite extent, with origins which need not coincide with the intersection of the central meridian with the equator. The true grid origin is always taken on the central meridian so that grid coordinates will be negative west of

747-481: A modern English translation. ) Lambert did not name his projections; the name transverse Mercator dates from the second half of the nineteenth century. The principal properties of the transverse projection are here presented in comparison with the properties of the normal projection. The ellipsoidal form of the transverse Mercator projection was developed by Carl Friedrich Gauss in 1822 and further analysed by Johann Heinrich Louis Krüger in 1912. The projection

830-416: A parallel of origin (usually written φ 0 ) are often used to define the origin of the map projection. A globe is the only way to represent the Earth with constant scale 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. Some possible properties are: Projection construction

913-399: A plane without distortion. The same applies to other reference surfaces used as models for the Earth, such as oblate spheroids , ellipsoids , and geoids . Since any map projection is a representation of one of those surfaces on a plane, all map projections distort. The classical way of showing the distortion inherent in a projection is to use Tissot's indicatrix . For a given point, using

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996-406: A proxy for the combination of angular deformation and areal inflation; such methods arbitrarily choose what paths to measure and how to weight them in order to yield a single result. Many have been described. The creation of a map projection involves two steps: Some of the simplest map projections are literal projections, as obtained by placing a light source at some definite point relative to

1079-631: A regional area of interest—such as a metropolitan area covering several counties—crosses a state plane zone boundary. The Seattle metropolitan area in Washington is an example of this. King County , which includes the City of Seattle , uses the "Washington State Plane North" coordinate system, while Pierce County , which includes the City of Tacoma , uses "Washington State Plane South". Thus any regional agency that wants to combine regional data from local governments has to transform at least some data into

1162-480: A series expansion for the transformation between latitude and conformal latitude, Karney has implemented the series to thirtieth order. An exact solution by E. H. Thompson is described by L. P. Lee. It is constructed in terms of elliptic functions (defined in chapters 19 and 22 of the NIST handbook) which can be calculated to arbitrary accuracy using algebraic computing systems such as Maxima. Such an implementation of

1245-477: 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

1328-410: A tangent Normal Mercator projection the (unique) formulae which guarantee conformality are: Conformality implies that the point scale , k , is independent of direction: it is a function of latitude only: For the secant version of the projection there is a factor of k 0 on the right hand side of all these equations: this ensures that the scale is equal to k 0 on the equator. The figure on

1411-709: Is according to properties of the model they preserve. Some of the more common categories are: Because the sphere is not a developable surface , it is impossible to construct a map projection that is both equal-area and conformal. 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 thing, even most projections that do fall into those categories are not naturally attainable through physical projection. As L. P. Lee notes, No reference has been made in

1494-399: 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 sphere in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate ellipsoid . Whether spherical or ellipsoidal, the principles discussed hold without loss of generality. Selecting a model for a shape of

1577-405: Is applied to a narrow strip near the central meridians where the differences between the spherical and ellipsoidal versions are small, but nevertheless important in accurate mapping. Direct series for scale, convergence and distortion are functions of eccentricity and both latitude and longitude on the ellipsoid: inverse series are functions of eccentricity and both x and y on the projection. In

1660-567: Is centred on 42W and, at its broadest point, is no more than 750 km from that meridian while the span in longitude reaches almost 50 degrees. Krüger– n is accurate to within 1 mm but the Redfearn version of the Krüger– λ series has a maximum error of 1 kilometre. Karney's own 8th-order (in n ) series is accurate to 5 nm within 3900 km of the central meridian. The normal cylindrical projections are described in relation to

1743-405: Is evident from the global projections shown above. Near the central meridian the differences are small but measurable. The difference between the north-south grid lines and the true meridians is the angle of convergence . Map projection 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

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1826-445: Is given by φ): 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 secant lines —a pair of identical latitudes of opposite sign (or else the equator) at which the east-west scale matches the north-south-scale. Normal cylindrical projections map

1909-555: Is known by several names: the (ellipsoidal) transverse Mercator in the US; Gauss conformal or Gauss–Krüger in Europe; or Gauss–Krüger transverse Mercator more generally. Other than just a synonym for the ellipsoidal transverse Mercator map projection, the term Gauss–Krüger may be used in other slightly different ways: The projection is conformal with a constant scale on the central meridian. (There are other conformal generalisations of

1992-469: Is needed in order to accurately transform data from one coordinate system to another. The main problem with the state plane coordinate system is that each zone uses a different coordinate system. This is not a major problem as long as one's needs are within the boundaries of a given state plane zone, as is the case with most county and city governments. However, the need to transform spatial data from one coordinate system to another can be burdensome. Sometimes

2075-540: Is now the most widely used expression of coordinate information in local and regional surveying and mapping applications in the United States and its territories. It has been revised several times since then. When computers began to be used for mapping and GIS , the state plane system's cartesian grid system and simplified calculations made spatial processing faster and spatial data easier to work with. Even though computer processing power has improved radically since

2158-461: Is now the most widely used projection in accurate large-scale mapping. The projection, as developed by Gauss and Krüger, was expressed in terms of low order power series which were assumed to diverge in the east-west direction, exactly as in the spherical version. This was proved to be untrue by British cartographer E. H. Thompson, whose unpublished exact (closed form) version of the projection, reported by Laurence Patrick Lee in 1976, showed that

2241-493: Is the distance of the true grid origin north of the false origin. If the true origin of the grid is at latitude φ 0 on the central meridian and the scale factor the central meridian is k 0 then these definitions give eastings and northings by: The terms "eastings" and "northings" do not mean strict east and north directions. Grid lines of the transverse projection, other than the x and y axes, do not run north-south or east-west as defined by parallels and meridians. This

2324-587: Is the way a coordinate system is linked to the physical Earth). More recently there has been an effort to increase the accuracy of the NAD83 datum using technology that was not available in 1983. These efforts are known as "High Accuracy Reference Network" (HARN) or "High Precision GPS Network" (HPGN). In addition, the basic unit of distance used is sometimes feet and sometimes meters. Thus a fully described coordinate system often looks something like: "Washington State Plane North, NAD83 HARN, US Survey feet". This information

2407-401: 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 bivariate map . 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

2490-456: Is wrapped around the Earth, projected onto, and then unrolled. 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

2573-487: The Collignon projection in polar areas. The term "conic projection" is used to refer to any projection in which meridians are mapped to equally spaced lines radiating out from the apex and circles of latitude (parallels) are mapped to circular arcs centered on the apex. When making a conic map, the map maker arbitrarily picks two standard parallels. Those standard parallels may be visualized as secant lines where

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2656-722: 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 . Many properties can be measured on the Earth's surface independently of its geography: Map projections can be constructed to preserve some of these properties at

2739-454: 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 justify using the more complicated ellipsoid. The ellipsoidal model is commonly used to construct topographic maps and for other large- and medium-scale maps that need to accurately depict

2822-404: The Earth when the extent of the mapped region exceeds a few hundred kilometers in length in both dimensions. For maps of smaller regions, an ellipsoidal model must be chosen if greater accuracy is required; see next section. The spherical form of the transverse Mercator projection was one of the seven new projections presented, in 1772, by Johann Heinrich Lambert . (The text is also available in

2905-484: The Redfearn series used by GEOTRANS and the exact solution is less than 1 mm out to a longitude difference of 3 degrees, corresponding to a distance of 334 km from the central meridian at the equator but a mere 35 km at the northern limit of an UTM zone. Thus the Krüger– n series are very much better than the Redfearn λ series. The Redfearn series becomes much worse as the zone widens. Karney discusses Greenland as an instructive example. The long thin landmass

2988-494: 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 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

3071-427: The above equations gives In terms of the coordinates with respect to the rotated graticule the point scale factor is given by k  = sec  φ′ : this may be expressed either in terms of the geographical coordinates or in terms of the projection coordinates: The second expression shows that the scale factor is simply a function of the distance from the central meridian of the projection. A typical value of

3154-407: The central meridian. The true parallels and meridians (other than equator and central meridian) have no simple relation to the rotated graticule and they project to complicated curves. The angles of the two graticules are related by using spherical trigonometry on the spherical triangle NM′P defined by the true meridian through the origin, OM′N, the true meridian through an arbitrary point, MPN, and

3237-494: The central meridian. Therefore, meridians are equally spaced along a given parallel. On a pseudocylindrical map, any point 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: The HEALPix projection combines an equal-area cylindrical projection in equatorial regions with

3320-407: The central meridian. To avoid such negative grid coordinates, standard practice defines a false origin to the west (and possibly north or south) of the grid origin: the coordinates relative to the false origin define eastings and northings which will always be positive. The false easting , E 0 , is the distance of the true grid origin east of the false origin. The false northing , N 0 ,

3403-638: The combined error in the X and Y directions. In 1933, the North Carolina Department of Transportation asked the United States Coast and Geodetic Survey to assist in creating a comprehensive method for converting curvilinear coordinates (latitude and longitude) to a user-friendly, 2-dimensional Cartesian coordinate system. This request developed into the State Plane Coordinate System (SPCS), which

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3486-407: 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 both standard parallels or to the south of both standard parallels are stretched; distances along parallels between

3569-597: The distortion in projections. Like Tissot's indicatrix, the Goldberg-Gott indicatrix is based on infinitesimals, and depicts flexion and skewness (bending and lopsidedness) distortions. 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 small circle of fixed radius (e.g., 15 degrees angular radius ). Sometimes spherical triangles are used. In

3652-477: The early days of GIS, the size of spatial datasets and the complexity of geoprocessing tasks being demanded of computers have also increased. Thus the state plane coordinate system is still useful. Originally, the state plane coordinate systems were based on the North American Datum of 1927 (NAD27). Later, the more accurate North American Datum of 1983 (NAD83) became the standard (a geodetic datum

3735-467: The ellipsoidal projection is finite (below). This is the most striking difference between the spherical and ellipsoidal versions of the transverse Mercator projection: Gauss–Krüger gives a reasonable projection of the whole ellipsoid to the plane, although its principal application is to accurate large-scale mapping "close" to the central meridian. In most applications the Gauss–Krüger coordinate system

3818-445: The equator and not a meridian. Pseudocylindrical projections represent the central meridian as a straight line segment. Other meridians are longer than the central meridian and bow outward, away from the central meridian. Pseudocylindrical projections map parallels 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

3901-416: The exact solution is described by Karney (2011). The exact solution is a valuable tool in assessing the accuracy of the truncated n and λ series. For example, the original 1912 Krüger– n series compares very favourably with the exact values: they differ by less than 0.31 μm within 1000 km of the central meridian and by less than 1 mm out to 6000 km. On the other hand, the difference of

3984-437: The expansion parameter: The Krüger– λ series were the first to be implemented, possibly because they were much easier to evaluate on the hand calculators of the mid twentieth century. The Krüger– n series have been implemented (to fourth order in n ) by the following nations. Higher order versions of the Krüger– n series have been implemented to seventh order by Engsager and Poder and to tenth order by Kawase. Apart from

4067-458: 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 basic metric properties in different ways. The purpose of

4150-410: 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 how the projection distorts sizes and shapes according to position on the map. Another way to visualize local distortion

4233-436: The geographical coordinates or in terms of the projection coordinates: Details of actual implementations The projection coordinates resulting from the various developments of the ellipsoidal transverse Mercator are Cartesian coordinates such that the central meridian corresponds to the x axis and the equator corresponds to the y axis. Both x and y are defined for all values of λ and ϕ . The projection does not define

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4316-433: The geoid are used to project maps from. Other regular solids are sometimes used as generalizations for smaller bodies' geoidal equivalent. For example, Io is better modeled by triaxial ellipsoid or prolated spheroid with small eccentricities. Haumea 's shape is a Jacobi ellipsoid , with its major axis twice as long as its minor and with its middle axis one and half times as long as its minor. See map projection of

4399-400: 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 helpful in understanding the basic concept of a map projection. A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a developable surface . The cylinder , cone and

4482-429: The globe never preserves or optimizes metric properties, so that possibility is not discussed further here. Tangent and secant lines ( standard lines ) are represented undistorted. If these lines are a parallel of latitude, as in conical projections, it is called a standard parallel . The central meridian is the meridian to which the globe is rotated before projecting. The central meridian (usually written λ 0 ) and

4565-514: The great circle WM′PE. The results are: The direct formulae giving the Cartesian coordinates ( x , y ) follow immediately from the above. Setting x  =  y′ and y  = − x′ (and restoring factors of k 0 to accommodate secant versions) The above expressions are given in Lambert and also (without derivations) in Snyder, Maling and Osborne (with full details). Inverting

4648-498: The land surface. Auxiliary latitudes are often employed in projecting the ellipsoid. A third model is the geoid , a more complex and accurate representation of Earth's shape coincident with what mean sea level 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, conformality and equivalence . Therefore, in geoidal projections that preserve such properties,

4731-402: The left shows how a transverse cylinder is related to the conventional graticule on the sphere. It is tangential to some arbitrarily chosen meridian and its axis is perpendicular to that of the sphere. The x - and y -axes defined on the figure are related to the equator and central meridian exactly as they are for the normal projection. In the figure on the right a rotated graticule is related to

4814-466: 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. 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 datum (model) of the Earth. Different datums assign slightly different coordinates to

4897-427: The mapped graticule would deviate from a mapped ellipsoid's graticule. Normally the geoid is not used as an Earth model for projections, however, because Earth's shape is very regular, with the undulation of the geoid amounting to less than 100 m from the ellipsoidal model out of the 6.3 million m Earth radius . For irregular planetary bodies such as asteroids , however, sometimes models analogous to

4980-452: The new spatial reference system and new zones in 2025. As with earlier systems, the name of the new system, State Plane Coordinate System of 2022, indicates a year earlier than the actual year of release. Transverse Mercator projection The transverse Mercator map projection ( TM , TMP ) is an adaptation of the standard Mercator projection . The transverse version is widely used in national and international mapping systems around

5063-404: The normal Mercator: Since the central meridian of the transverse Mercator can be chosen at will, it may be used to construct highly accurate maps (of narrow width) anywhere on the globe. The secant, ellipsoidal form of the transverse Mercator is the most widely applied of all projections for accurate large-scale maps. In constructing a map on any projection, a sphere is normally chosen to model

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5146-403: 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.) 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

5229-431: 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

5312-460: The projection cone intersecting the Earth's surface along two lines of latitude. Zones that are long in the north–south direction use the transverse Mercator projection because it is better at maintaining accuracy along a north–south axis, due to the circumference of the projection cylinder being oriented along a meridian of longitude. The panhandle of Alaska, whose maximum dimension is on a diagonal, uses an Oblique Mercator projection, which minimizes

5395-468: The projection surface into a flat map. The most common projection surfaces are cylindrical (e.g., Mercator ), conic (e.g., Albers ), and planar (e.g., stereographic ). 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 polyconic . Another way to classify projections

5478-535: 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,

5561-405: The rotated graticule: φ′ (angle M′CP) is an effective latitude and − λ′ (angle M′CO) becomes an effective longitude. (The minus sign is necessary so that ( φ′ , λ′ ) are related to the rotated graticule in the same way that ( φ , λ ) are related to the standard graticule). The Cartesian ( x′ , y′ ) axes are related to the rotated graticule in the same way that the axes ( x , y ) axes are related to

5644-438: The same location, so in large scale maps, such as those from national mapping systems, it is important to match the datum to the projection. The slight differences in coordinate assignation between different datums is not a concern for world maps or those of large regions, where such differences are reduced to imperceptibility. Carl Friedrich Gauss 's Theorema Egregium proved that a sphere's surface cannot be represented on

5727-440: The scale factor h along the meridian, the scale factor k along the parallel, and the angle θ ′ between them, Nicolas Tissot 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. Many other ways have been described of showing

5810-419: The scale factor is k 0  = 0.9996 so that k  = 1 when x is approximately 180 km. When x is approximately 255 km and k 0  = 1.0004: the scale factor is within 0.04% of unity over a strip of about 510 km wide. The convergence angle γ at a point on the projection is defined by the angle measured from the projected meridian, which defines true north, to

5893-490: The secant version the lines of true scale on the projection are no longer parallel to central meridian; they curve slightly. The convergence angle between projected meridians and the x constant grid lines is no longer zero (except on the equator) so that a grid bearing must be corrected to obtain an azimuth from true north. The difference is small, but not negligible, particularly at high latitudes. In his 1912 paper, Krüger presented two distinct solutions, distinguished here by

5976-427: The sphere. In reality, cylinders and cones provide us with convenient descriptive terms, but little else. Lee's objection refers to the way the terms cylindrical , conic , and planar (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

6059-429: The standard graticule. The tangent transverse Mercator projection defines the coordinates ( x′ , y′ ) in terms of − λ′ and φ′ by the transformation formulae of the tangent Normal Mercator projection: This transformation projects the central meridian to a straight line of finite length and at the same time projects the great circles through E and W (which include the equator) to infinite straight lines perpendicular to

6142-423: The standard parallels are compressed. When a single standard parallel is used, distances along all other parallels are stretched. Conic projections that are commonly used are: Azimuthal projections have the property that directions from a central point are preserved and therefore great circles through the central point are represented by straight lines on the map. These projections also have radial symmetry in

6225-403: The surface into a plane. While the first step inevitably distorts some properties of the globe, the developable surface can then be unfolded without further distortion. Once a choice is made between projecting onto a cylinder, cone, or plane, the aspect of the shape must be specified. The aspect describes how the developable surface is placed relative to the globe: it may be normal (such that

6308-432: The surface's axis of symmetry coincides with the Earth's axis), transverse (at right angles to the Earth's axis) or oblique (any angle in between). The developable surface may also be either tangent or secant 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

6391-407: The term cylindrical 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; instead parallels are placed how they need to be in order to satisfy the property that

6474-423: 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

6557-540: The transverse Mercator from the sphere to the ellipsoid but only Gauss-Krüger has a constant scale on the central meridian.) Throughout the twentieth century the Gauss–Krüger transverse Mercator was adopted, in one form or another, by many nations (and international bodies); in addition it provides the basis for the Universal Transverse Mercator series of projections. The Gauss–Krüger projection

6640-455: The transverse cylinder in the same way that the normal cylinder is related to the standard graticule. The 'equator', 'poles' (E and W) and 'meridians' of the rotated graticule are identified with the chosen central meridian, points on the equator 90 degrees east and west of the central meridian, and great circles through those points. The position of an arbitrary point ( φ , λ ) on the standard graticule can also be identified in terms of angles on

6723-422: The triaxial ellipsoid for further information. 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 Earth's surface onto the projection surface, then unraveling and scaling

6806-411: The whole Earth as a finite rectangle, except in the first two cases, where the rectangle stretches infinitely tall while retaining constant width. 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. See: transverse Mercator . An oblique cylindrical projection aligns with a great circle, but not

6889-443: The world, including the Universal Transverse Mercator . When paired with a suitable geodetic datum , the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent. The transverse Mercator projection is the transverse aspect of the standard (or Normal ) Mercator projection. They share the same underlying mathematical construction and consequently the transverse Mercator inherits many traits from

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