Misplaced Pages

#641358

57-701: The Cassini Grid system was introduced in 1927 for maps of the United Kingdom for the British military. It is so called from the use of Cassini map projection . It modified and replaced a grid coordinate system first deployed in 1919 known as the British System . The Cassini Grid system is thus more properly called the Modified British System . It may also be referred to as Military Grid , War Office Grid , or Purple Grid (due to

114-401: A map projection is any of a broad set of transformations employed to represent the curved two-dimensional surface of a globe on a plane . In a map projection, coordinates , often expressed as latitude and longitude , of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of

171-462: 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

228-416: 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

285-412: A grid square is the 500 km letter in lowercase followed by the 100 km letter in uppercase. For instance, the squares covering London are wL and wQ. Each 100 km square is subdivided into 10 km squares, and each of these is further subdivided into 1 km squares. These squares are numbered according to eastings and northings , that is, counting from the south-west corner of the 100 km square. For instance,

342-416: 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

399-399: 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

456-406: 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

513-646: A shadow on a screen, or the rectilinear image produced by a pinhole camera on a flat film plate. Rather, any mathematical function that transforms coordinates from the curved surface distinctly and smoothly to the plane is a projection. Few projections in practical use are perspective. Most of this article assumes that the surface to be mapped is that of a sphere. The Earth and other large celestial bodies are generally better modeled as oblate spheroids , whereas small objects such as asteroids often have irregular shapes. The surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with

570-477: 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

627-709: 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

SECTION 10

#1732776040642

684-399: 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

741-477: Is because square qY is three 100 km squares East of square vV and five 100 km squares North of vV. However, it is frequently not necessary to explicitly give the grid square at all, this can be implied from context. The Ordnance Survey National Grid was introduced in 1938 and the intention was to replace the Cassini Grid system with this. However, it had not become established when World War II broke out and

798-517: Is composed of the operations: In practice, the projection has always been applied to models of the earth as an ellipsoid , which greatly complicates the mathematical development but is suitable for surveying. Nevertheless, the use of the Cassini projection has largely been superseded by the transverse Mercator projection, at least with central mapping agencies. Areas along the central meridian, and at right angles to it, are not distorted. Elsewhere,

855-445: 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

912-401: 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

969-456: 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

1026-479: The Cassini–Soldner projection or Soldner projection ) is a map projection first described in an approximate form by César-François Cassini de Thury in 1745. Its precise formulas were found through later analysis by Johann Georg von Soldner around 1810. It is the transverse aspect of the equirectangular projection , in that the globe is first rotated so the central meridian becomes the "equator", and then

1083-487: 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

1140-722: 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

1197-647: The Cassini Grid continued in use throughout the war. Work on the National Grid maps stopped for the duration of the war. In fact, progress went backwards due to the Southampton offices of the Ordnance Survey being bombed by the Germans in 1940 destroying some of their data. National Grid maps began to be issued after the war, but it was some time before there was full coverage. In the interim,

SECTION 20

#1732776040642

1254-454: 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

1311-528: The National Grid resulted in National Grid and Cassini Grid maps being used side by side for an extended time. Converting between the two systems is problematic. Although the two grids are superficially similar (both use 100 km squares), different projections, baselines and origins make the conversion non-trivial. Conversion of Cassini Grid references is of interest to modern archaeologists attempting to locate wartime sites and artefacts. Cassini projection The Cassini projection (also sometimes known as

1368-494: 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

1425-494: 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

1482-480: The characterization of their distortions. There is no limit to the number of possible map projections. More generally, projections are considered in several fields of pure mathematics, including differential geometry , projective geometry , and manifolds . However, the term "map projection" refers specifically to a cartographic projection. Despite the name's literal meaning, projection is not limited to perspective projections, such as those resulting from casting

1539-407: 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

1596-597: 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

1653-418: The distortion is largely in a north–south direction, and varies by the square of the distance from the central meridian. As such, the greater the longitudinal extent of the area, the worse the distortion becomes. Due to this, the Cassini projection works best for areas with greater north–south extent than east–west. For example, Ordnance Survey maps of Great Britain used the Cassini projection from 1924 until

1710-445: 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

1767-402: The essential elements of cartography. 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 properties. The study of map projections is primarily about

Cassini Grid - Misplaced Pages Continue

1824-458: 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

1881-410: 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

1938-433: 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

1995-400: 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

2052-429: 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

2109-517: The grid colour on some issues of maps). It continued in use until some time after World War II . After the war, the Cassini Grid system was replaced with the Ordnance Survey National Grid (see § Transition to National Grid ). However, the Cassini Grid system continued to be used to map colonial possessions into the 1960s. Hong Kong , for instance, was surveyed in 1962 for a series of 1:10,000 and 1:25,000 maps of

2166-411: The grid reference 25 76 refers to a 1 km square within the 10 km square 2 7. That is, it is two squares across and seven squares up (counting from zero) from the reference corner. Greater precision can be obtained by using six, eight, or even more digits, but always an even number with the same precision on both coordinates. The complete grid reference includes both the grid square letters and

2223-883: The introduction of the National Grid . Cassini is known as a spherical projection, but can be generalised as an elliptical form. Considering the earth as an ellipse, the projection is composed of these operations: and M is the meridional distance function . The reverse operation is composed of the operations: If φ ′ = π 2 {\displaystyle \varphi '={\frac {\pi }{2}}} then φ = φ ′ {\displaystyle \varphi =\varphi '} and λ = 0. {\displaystyle \lambda =0.} Otherwise calculate T and N as above with φ ′ {\displaystyle \varphi '} , and Map projection In cartography ,

2280-498: 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,

2337-466: 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

Cassini Grid - Misplaced Pages Continue

2394-427: 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

2451-409: The normal equirectangular projection is applied. Considering the earth as a sphere, the projection is composed of the operations: where λ is the longitude from the central meridian and φ is the latitude. When programming these equations, the inverse tangent function used is actually the atan2 function, with the first argument sin  φ and the second cos φ cos λ . The reverse operation

2508-646: The numerical reference giving the location within the grid square. For instance, wQ 683 825 is a location in South London. Grid references can also be given in an all numeral notation. To do this, square vV (the South-West corner of the grid out in the Atlantic) is taken as square zero and then the grid squares are counted from there to the location being referenced. For instance qY 985 613 (somewhere around Carlisle ) can also be expressed as 3985 5613. This

2565-403: 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

2622-468: 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

2679-622: The public were allowed to purchase War Office Cassini Grid maps. National Grid maps at 1:25,000 scale (often referred to as 2 1 ⁄ 2 inches to the mile, or just 2 1 ⁄ 2 -inch maps) were not introduced until as late as 1950. Cassini Grid maps at this scale had been available, but apparently the public were discouraged from using them by the Ordnance Survey. They were poor quality, rushed into production for wartime needs, and had been produced by merely photographically reducing six-inch maps. The long period of introduction of

2736-438: 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

2793-440: 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

2850-427: 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

2907-423: 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

SECTION 50

#1732776040642

2964-403: 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

3021-432: 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

3078-407: 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

3135-640: The territory. These were published between 1965 and 1972. The map is divided into a grid of 500 km squares. The squares covering England are Q, R, V and W with an intersection somewhere between York and Thirsk . Ireland is covered by square I, but the Irish grid is tilted slightly to the East relative to the English grid. Each 500 km square is further divided into 25 100 km squares. The 100 km squares are also given letters (with I omitted). The full designation for

3192-422: 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

3249-411: 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

#641358