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A map projection is any method used in cartography to represent the two-dimensional curved surface of the earth or other body on a plane. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001

 

The Mercator projection shows courses of constant bearing as straight lines. The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569

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The term "projection" here refers to any function defined on the earth's surface and with values on the plane, and not necessarily a geometric projection. Graphical projection is a Protocol by which an image of an imaginary three-dimensional object is projected onto a planar surface without the aid of mathematical calculation Planar projections are the subset of 3D graphical projections constructed by linearly mapping points in three dimensional space to points on a two-dimensional projection plane Perspective (from Latin perspicere to see through in the graphic arts such as drawing is an approximate representation on a flat surface (such as paper of an image as it is perceived Orthographic projection is a means of representing a three- Dimensional (3D object in two dimensions (2D Orthographic projection is a means of representing a three- Dimensional (3D object in two dimensions (2D A plan is an Orthographic projection of a 3-dimensional object from the position of a horizontal plane through the object A floor plan ( floorplan) in Architecture and Building engineering is a Diagram, usually to scale, of the relationships between rooms In Geometry, a cross section is the intersection of a body in 2-dimensional space with a line or of a body in 3-dimensional space with a plane etc An elevation is an Orthographic projection of a 3-dimensional object from the position of a horizontal plane beside an object An auxiliary view is an angle at which one can view an object that is not one of the primary views for an Orthographic projection. Axonometric projection ("to measure along axes" is a technique used in orthographic pictorials Isometric projection is a form of Graphical projection —more specifically an Axonometric projection. Dimetric projection is a form of Axonometric projection, in which its direction of viewing is such that two of the three axes of space appear equally foreshortened of which Trimetric projection is a form of Axonometric projection, where the direction of viewing is such that all of the three axes of space appear unequally foreshortened This article discusses imaging of three-dimensional objects For an abstract mathematical discussion see Projection (linear algebra. The cavalier perspective, also called cavalier projection or high view point, is a way to represent a three dimensional object on a flat drawing and more specifically Cabinet projection or sometimes cabinet perspective is a type of Oblique projection. 3D projection is any method of mapping three-dimensional points to a two-dimensional plane Owned by Atlassian Software Systems, FishEye is a Revision control browser and search engine Stereoscopy, stereoscopic imaging or 3-D (three-dimensional imaging is any technique capable of recording three-dimensional visual Anamorphosis is a distorted projection or perspective requiring the viewer to use special devices or occupy a specific vantage point to reconstitute the image A bird's-eye view is a View of an object from above as though the observer were a Bird, often used in the making of Blueprints, Floor plans Top-down perspective, also sometimes referred to as bird's-eye view, overhead view or helicopter view, A worm's-eye view is a View of an object from below as though the observer were a Worm. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

Flat maps could not exist without map projections, because a sphere cannot be laid flat over a plane without distortions. A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, Regions, and Themes One can see this mathematically as a consequence of Gauss's Theorema Egregium. Gauss's Theorema Egregium (Latin "Remarkable Theorem" is a foundational result in Differential geometry proved by Carl Friedrich Gauss that concerns the Flat maps can be more useful than globes in many situations: they are more compact and easier to store; they readily accommodate an enormous range of scales; they are viewed easily on computer displays; they can facilitate measuring properties of the terrain being mapped; they can show larger portions of the earth's surface at once; and they are cheaper to produce and transport. A globe is a three- Dimensional scale model of Earth ( terrestrial globe) or other spheroid celestial body such as a planet star or moon These useful traits of flat maps motivate the development of map projections.

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Metric properties of maps

Many properties can be measured on the earth's surface independently of its geography. Some of these properties are:

An Albers projection shows areas accurately, but distorts shapes.
An Albers projection shows areas accurately, but distorts shapes. The Albers equal-area conic projection, or Albers projection, is a conic, equal area Map projection that uses two standard parallels

Map projections can be constructed to preserve one or some of these properties, though not all of them simultaneously. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined Direction is the information contained in the relative position of one point with respect to another point without the Distance information In Navigation, a bearing is the direction one object is from another object Distance is a numerical description of how far apart objects are The concept of scale is applicable if a system is represented proportionally by another system Each projection preserves or compromises or approximates basic metric properties in different ways. The purpose of the map, then, determines which projection should form the base for the map. Since many purposes exist for maps, so do many projections exist upon which to construct them.

Another major concern that drives the choice of a projection is the compatibility of data sets. Data sets are geographic information. As such, their collection depends on the chosen model of the earth. Different models assign slightly different coordinates to the same location, so it is important that the model be known and that chosen projection be compatible with that model. On small areas (large scale) data compatibility issues are more important since metric distortions are minimal at this level. In very large areas (small scale), on the other hand, distortion is a more important factor to consider.

Construction of a map projection

The creation of a map projection involves three steps:

  1. Selection of a model for the shape of the earth or planetary body (usually choosing between a sphere or ellipsoid)
  2. Transformation of geographic coordinates (longitude and latitude) to plane coordinates (eastings and northings or x,y)
  3. Reduction of the scale (it does not matter in what order the second and third steps are performed)

Because the real earth's shape is irregular, information is lost in the first step, in which an approximating, regular model is chosen. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an Ellipse. Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the The terms easting and northing are geographic Cartesian coordinates for a point Reducing the scale may be considered to be part of transforming geographic coordinates to plane coordinates.

Most map projections, both practically and theoretically, are not "projections" in any physical sense. Rather, they depend on mathematical formulae that have no direct physical interpretation. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information However, in understanding the concept of a map projection it is helpful to think of a globe with a light source placed at some definite point with respect to it, projecting features of the globe onto a surface. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume The following discussion of developable surfaces is based on that concept.

Choosing a projection surface

A Miller cylindrical projection maps the globe onto a cylinder.
A Miller cylindrical projection maps the globe onto a cylinder. The Miller cylindrical projection is a modified Mercator projection, proposed by Osborn Maitland Miller ( 1897 - 1979) in 1942.

A surface that can be unfolded or unrolled into a flat plane or sheet without stretching, tearing or shrinking is called a 'developable surface'. In Mathematics, a developable surface is a Surface with zero Gaussian curvature. The cylinder, cone and of course the plane are all developable surfaces. A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex The sphere and ellipsoid are not developable surfaces. Any projection that attempts to project a sphere (or an ellipsoid) on a flat sheet will have to distort the image (similar to the impossibility of making a flat sheet from an orange peel).

One way of describing a projection is to project first from the earth's surface to a developable surface such as a cylinder or cone, followed by the simple second step of unrolling 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.


Orientation of the projection

This transverse Mercator projection is mathematically the same as a standard Mercator, but oriented around a different axis.
This transverse Mercator projection is mathematically the same as a standard Mercator, but oriented around a different axis. The transverse Mercator projection is an adaptation of the Mercator projection.

Once a choice is made between projecting onto a cylinder, cone, or plane, the orientation of the shape must be chosen. The orientation is how the shape is placed with respect to the globe. The orientation of the projection surface can be normal (inline with the earth's axis), transverse (at right angles to the earth's axis) or oblique (any angle in between). These surfaces may also be either tangent or secant to the spherical or ellipsoidal globe. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. Secant is a term in mathematics It comes from the Latin secare (to cut Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe. Insofar as preserving metric properties go, it is never advantageous to move the developable surface away from contact with the globe, so that practice is not discussed here.

Scale

A globe is the only way to represent the earth with constant scale throughout the entire map in all directions. A globe is a three- Dimensional scale model of Earth ( terrestrial globe) or other spheroid celestial body such as a planet star or moon The concept of scale is applicable if a system is represented proportionally by another system 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:

Choosing a model for the shape of the Earth

Projection construction is also affected by how the shape of the earth is approximated. In the following discussion on projection categories, a sphere is assumed. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe However, the Earth is not exactly spherical but is closer in shape to an oblate ellipsoid, a shape which bulges around the equator. An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an Ellipse. The equator (sometimes referred to colloquially as "the Line") is the intersection of the Earth 's surface with the plane perpendicular to the 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 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 the land surface. A topographic map is a type of Map characterized by large-scale detail and quantitative representation of relief, usually using Contour lines in modern

A third model of the shape of the earth is called a geoid, which is a complex and more or less accurate representation of the global mean sea level surface that is obtained through a combination of terrestrial and satellite gravity measurements. The geoid is that Equipotential surface which would coincide exactly with the mean ocean surface of the Earth if the oceans were in equilibrium at rest and extended through This model is not used for mapping due to its complexity but is instead used for control purposes in the construction of geographic datums. This article describes a concept from Surveying and Geodesy. For other meanings see Datum (disambiguation. (In geodesy, plural of "datum" is "datums," rather than "data". ) A geoid is used to construct a datum by adding irregularities to the ellipsoid in order to better match the Earth's actual shape (it takes into account the large scale features in the Earth's gravity field associated with mantle convection patterns, as well as the gravity signatures of very large geomorphic features such as mountain ranges, plateaus and plains). Mantle convection is the slow creeping motion of Earth's rocky mantle in response to perpetual gravitationally unstable variations in its density Historically, datums have been based on ellipsoids that best represent the geoid within the region the datum is intended to map. Each ellipsoid has a distinct major and minor axis. Different controls (modifications) are added to the ellipsoid in order to construct the datum, which is specialized for a specific geographic regions (such as the North American Datum). The North American Datum is the official datum used for the primary Geodetic network in North America A few modern datums, such as WGS84 (the one used in the Global Positioning System GPS), are optimized to represent the entire earth as well as possible with a single ellipsoid, at the expense of some accuracy in smaller regions. The World Geodetic System defines a reference frame for the earth for use in Geodesy and Navigation. Basic concept of GPS operation A GPS receiver calculates its position by carefully timing the signals sent by the constellation of GPS Satellites high above the Earth

Classification

A fundamental projection classification is based on type of projection surface onto which the globe is conceptually projected. The projections are described in terms of placing a gigantic surface in contact with the earth, followed by an implied scaling operation. These surfaces are cylindrical (e. g. , Mercator), conic (e. The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569 g. , Albers), and azimuthal or plane (e. The Albers equal-area conic projection, or Albers projection, is a conic, equal area Map projection that uses two standard parallels g. , stereographic). In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane Many mathematical projections, however, do not neatly fit into any of these three conceptual projection methods. Hence other peer categories have been described in the literature, such as pseudoconic (meridians are arcs of circles), pseudocylindrical (meridians are straight lines), pseudoazimuthal, retroazimuthal, and polyconic. A polyconic projection is a conical Map projection. The projection stems from "rolling" a cone tangent to the Earth at all parallels of latitude instead

Another way to classify projections is through the properties they preserve despite projection. Some of the more common categories are:

NOTE: Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal. A map projection is any method of representing the Surface of a sphere or other shape on a plane. The gnomonic Map projection displays all Great circles as straight lines

Projections by surface

Cylindrical

The space-oblique Mercator projection was developed by the USGS for use in Landsat images.
The space-oblique Mercator projection was developed by the USGS for use in Landsat images. Space-oblique Mercator projection is a Map projection. History The Space-oblique Mercator projection (SOM was developed by John P The United States Geological Survey ( USGS) is a scientific agency of the United States government. The Landsat program is the longest running enterprise for acquisition of imagery of Earth from space

The term "cylindrical projection" is used to refer to any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude (parallels) are mapped to horizontal lines (or, mutatis mutandis, more generally, radial lines from a fixed point are mapped to equally spaced parallel lines and concentric circles around it are mapped to perpendicular lines). This article is about the geographical concept For other uses of the word see Meridian. A circle of latitude, on the Earth, is an imaginary East - West circle connecting all locations (not taking into account elevation that share a given Mutatis mutandis, Latin literal meaning "with those things having been changed which need to be changed" or more simply "the necessary changes having

The mapping of meridians to vertical lines can be visualized by imagining a cylinder (of which the axis coincides with the Earth's axis of rotation) wrapped around the Earth and then projecting onto the cylinder, and subsequently unfolding the cylinder.

Unavoidably, all cylindrical projections have the same east-west stretching away from the equator by a factor equal to the secant of the latitude, compared with the scale at the equator. The equator (sometimes referred to colloquially as "the Line") is the intersection of the Earth 's surface with the plane perpendicular to the Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the The various cylindrical projections can be described in terms of the north-south stretching:

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 identical latitudes of opposite sign (or else the equator) at which the east-west scale matches the north-south-scale.

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.

Pseudocylindrical

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

Pseudocylindrical projections represent the central meridian and each parallel as a single straight line segment, but not the other meridians. This article is about the geographical concept For other uses of the word see Meridian. A circle of latitude, on the Earth, is an imaginary East - West circle connecting all locations (not taking into account elevation that share a given Each pseudocylindrical projection represents a point on the Earth along the straight line representing its parallel, at a distance which is a function of its difference in longitude from the central meridian.

The true distance between two points on the same meridian corresponds to the distance on the map between the two parallels, which is smaller than the distance between the two points on the map. The true distance between two points on the same parallel -- and the true area of shapes on the map -- are not distorted. The meridians drawn on the map help the user to realize the shape distortion and mentally compensate for it.

Hybrid

The HEALPix projection combines an equal-area cylindrical projection in equatorial regions with the Collignon projection in polar areas. The Mollweide projection is a pseudocylindrical map projection generally used for global maps of the world (or sky The Goode homolosine projection (or interrupted Goode homolosine projection) is an interrupted pseudocylindrical, equal-area, composite Map projection The Kavrayskiy VII is a Map projection invented by V V Kavrayskiy in 1939 for use as a general purpose pseudocylindrical projection The Tobler hyperelliptical projection is a family of pseudocylindrical projections used for mapping the Earth. HEALPix (sometimes written as Healpix an acronym for H ierarchical E qual A rea iso' L' atitude Pix elisation of a 2- Sphere The Collignon Projection is a pseudocylindrical Map projection first known to be published by Édouard Collignon in 1865 and subsequently cited

Conical

Pseudoconical

Azimuthal (projections onto a plane)

An azimuthal projection shows distances and directions accurately from the center point, but distorts shapes and sizes elsewhere.
An azimuthal projection shows distances and directions accurately from the center point, but distorts shapes and sizes elsewhere. A Lambert conformal conic projection ( LCC) is a conic Map projection, which is often used for Aeronautical charts In essence the projection The Albers equal-area conic projection, or Albers projection, is a conic, equal area Map projection that uses two standard parallels A Bonne projection is a pseudoconical equal-area Map projection, sometimes called a dépôt de la guerre or a Sylvanus projection The Werner projection is a pseudoconical equal-area Map projection, sometimes called the Stabius-Werner or the Stab-Werner projection A polyconic projection is a conical Map projection. The projection stems from "rolling" a cone tangent to the Earth at all parallels of latitude instead A map projection is any method of representing the Surface of a sphere or other shape on a plane.

Azimuthal projections have the property that directions from a central point are preserved (and hence, great circles through the central point are represented by straight lines on the map). Azimuth ( is a mathematical concept defined as the angle usually measured in degrees (° between a reference plane and a point. Usually 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 r(d) of the true distance d, 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.

The mapping of radial lines can be visualized by imagining a plane tangent to the Earth, with the central point as tangent point.

The radial scale is r'(d) and the transverse scale r(d)/(R sin(d/R)) where R is the radius of the Earth.

Some azimuthal projections are true perspective projections; that is, they can be constructed mechanically, projecting the surface of the Earth by extending lines from a points of perspective (along an infinite line through the tangent point and the tangent point's antipode) onto the plane:

Other azimuthal projections are not true perspective projections:

Projections by preservation of a metric property

A stereographic projection is conformal and perspective but not equal area or equidistant.
A stereographic projection is conformal and perspective but not equal area or equidistant. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane

Conformal

Conformal map projections preserve angles locally:

Equal-area

The equal-area Mollweide projection
The equal-area Mollweide projection

These projections preserve area:

Equidistant

These preserve distance from some standard point or line:

a two-point equidistant projection of Asia
a two-point equidistant projection of Asia

Gnomonic

The Gnomonic projection is thought to be the oldest map projection, developed by Thales in the 6th century BC
The Gnomonic projection is thought to be the oldest map projection, developed by Thales in the 6th century BC

Great circles are displayed as straight lines:

Retroazimuthal

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:

Compromise projections

The Robinson projection was adopted by National Geographic Magazine in 1988 but abandoned by them in about 1997 for the Winkel Tripel.
The Robinson projection was adopted by National Geographic Magazine in 1988 but abandoned by them in about 1997 for the Winkel Tripel. The gnomonic Map projection displays all Great circles as straight lines Thales of Miletus According to Bertrand Russell, "Philosophy begins with Thales The 6th century BC started the first day of 600 BC and ended the last day of 501 BC. A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. The gnomonic Map projection displays all Great circles as straight lines The Littrow projection is the only conformal retroazimuthal Map projection. The Craig retroazimuthal Map projection was created by James Ireland Craig in 1909 The Robinson projection is a Map projection popularly used since the 1960s to show the entire world at once The National Geographic Magazine is the official journal of the National Geographic Society. The Winkel tripel projection ( Winkel III) is a modified azimuthal Map projection, one of three projections proposed by Oswald Winkel in 1921

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:

Other noteworthy projections

See also

References

  1. ^ Sinusoidal Projection -- From MathWorld. Retrieved on November 18, 2005.
  2. ^ Gnomonic Projection -- From MathWorld. Retrieved on November 18, 2005.
  3. ^ The Gnomonic Projection. Retrieved on November 18, 2005.
  4. ^ Orthographic Projection -- From MathWorld. Retrieved on November 18, 2005.
  5. ^ Stereographic Projection -- From MathWorld. Retrieved on November 18, 2005.
  6. ^ Azimuthal Equidistant Projection -- From MathWorld. Retrieved on November 18, 2005.
  7. ^ Lambert Azimuthal Equal-Area Projection -- From MathWorld. Retrieved on November 18, 2005.
  8. ^ http://www.gis.psu.edu/projection/chap6figs.html. Retrieved on November 18, 2005.

External links

Dictionary

map projection

-noun

  1. any systematic method of transforming the spherical representation of parallels, meridians and geographic features of the Earth's surface to a nonspherical surface, usually a plane
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