The Mercator projection shows courses of constant bearing as straight lines. While common, scholars advise against using it for reference maps of the world because it drastically inflates the high latitudes.

A map projection is any method used in cartography (mapmaking) to represent the two-dimensional curved surface of the earth or other body on a plane. 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. Rio de Janeiro birds-eye view. ... A worms-eye view is a view of an object from below, as though the observer were a worm. ... Grand Theft Auto Top-down perspective, also sometimes referred to as birds-eye view or helicopter view, is a view used in computer and video games that shows the player and the area around him or her from above. ... Image File history File links Usgs_map_mercator. ... Mercator world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigatium Emendate (1569) The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569. ... Cartography or mapmaking (in Greek chartis = map and graphein = write) is the study and practice of making maps or globes. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... An open surface with X-, Y-, and Z-contours shown. ... This article is about Earth as a planet. ... Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ... The word projection can mean more than one thing. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...

Flat maps could not exist without map projections, because a sphere cannot be laid flat over a plane without distortions. One can see this mathematically as a consequence of Gauss's Theorema Egregium. 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. These useful traits of flat maps motivate the development of map projections. For other uses, see Map (disambiguation). ... The Theorema Egregium (Remarkable Theorem) is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces. ... World globe A Baroque era celestial globe A globe is a three-dimensional scale model of a spheroid celestial body such as a planet, star or moon, in particular Earth, or, alternatively, a spherical representation of the sky with the stars (but without the Sun, Moon, or planets, because their...

Metric properties of maps

An Albers projection shows areas accurately, but distorts shapes.

Many properties can be measured on the earth's surface independently of its geography. Some of these properties are: Image File history File links World map projection (source) File links The following pages link to this file: Map projection ... Map of the Earth using an Albers projection The Albers equal-area conic projection, or Albers projection, is a conic, equal area map projection that uses two standard parallels. ...

• Area
• Shape
• Direction
• Bearing
• Distance
• Scale

Map projections can be constructed to preserve one or some of these properties, though not all of them simultaneously. 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. Area is a physical quantity expressing the size of a part of a surface. ... Look up shape in Wiktionary, the free dictionary. ... The shape of each panel of this road sign, and the broken lines at the ends, represents an arrow; a space-consuming central bar of the arrow sign is dispensed with. ... In navigation, a bearing is the clockwise angle between a reference direction (or a datum line) and the direction to an object. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... A scale is either a device used for measurement of weights, or a series of ratios against which different measurements can be compared. ...

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. Reducing the scale may be considered to be part of transforming geographic coordinates to plane coordinates. A sphere is a symmetrical geometrical object. ... 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... Longitude is the east-west geographic coordinate measurement most commonly utilized in cartography and global navigation. ... Latitude,usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ... The terms easting and northing are geographic Cartesian coordinates for a point. ...

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. 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. The following discussion of developable surfaces is based on that concept. In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ...

Choosing a projection surface

A Miller cylindrical projection maps the globe onto a cylinder.

A surface that can be unfolded or unrolled into a flat plane or sheet without stretching, tearing or shrinking is called a 'developable surface'. The cylinder, cone and of course the plane are all developable surfaces. 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). Image File history File links World map projection (source) File links The following pages link to this file: Map projection ... A Miller cylindrical projection maps the globe onto a cylinder. ... A developable surface is a surface that can be flattened onto a plane without distortion (i. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... This article is about the geometric object, for other uses see Cone. ...

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.

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. 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. Image File history File links World map projection (source) File links The following pages link to this file: Map projection Transverse Mercator projection ... // Transverse Mercator Projection A Transverse Mercator projection A Transverse Mercator projection is an adaptation of the Mercator projection. ... The term transverse means side-to-side, as opposed to longitudinal, which means front-to-back. In automotive engineering, the term transverse refers to an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle. ... Oblique can mean one of several things: In linguistics, oblique case. ... For other uses, see tangent (disambiguation). ... Secant is a term in mathematics. ...

Scale

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. World globe A Baroque era celestial globe A globe is a three-dimensional scale model of a spheroid celestial body such as a planet, star or moon, in particular Earth, or, alternatively, a spherical representation of the sky with the stars (but without the Sun, Moon, or planets, because their... A scale is either a device used for measurement of weights, or a series of ratios against which different measurements can be compared. ...

Possible properties are:

• The scale depends on location, but not on direction; this is equivalent with preservation of angles: conformal map
• For a given latitude and direction, the scale is the same everywhere; this applies for any cylindrical projection
• Combination of the two: the scale depends on latitude only, not on longitude or direction; this applies for the Mercator projection

In mathematics, a conformal map is a function which preserves angles. ... Mercator world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigatium Emendate (1569) The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569. ...

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. However, the Earth is not exactly spherical but is closer in shape to an oblate ellipsoid, a shape which bulges around the equator. 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 sphere is a symmetrical geometrical object. ... 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses Ilhéu das Rolas, in São Tomé and Príncipe. ... // Topographic maps are a variety of maps characterized by large-scale detail and quantitative representation of relief, usually using contour lines in modern mapping, but historically using a variety of methods. ...

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. This model is not used for mapping due to its complexity but is instead used for control purposes in the construction of geographic datums. (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). 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). 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 GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ... This article describes a concept from cartography and geodesy. ... Mantle convection is the slow creeping motion of Earths rocky mantle in response to perpetual gravitationally unstable variations in its density. ... The North American Datum is the official reference ellipsoid used for the primary geodetic network in North America. ... WGS 84 is the 1984 revision of the World Geodetic System. ... Over fifty GPS satellites such as this NAVSTAR have been launched since 1978. ...

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.g., Albers), and azimuthal or plane (e.g., stereographic). 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. Mercator world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigatium Emendate (1569) The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569. ... Map of the Earth using an Albers projection The Albers equal-area conic projection, or Albers projection, is a conic, equal area map projection that uses two standard parallels. ... Stereographic projection of a circle of radius R onto the x axis. ... Map of the Earth using a polyconic projection A polyconic projection is a conical map projection. ...

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

• Direction preserving, called azimuthal (but only possible from the central point)
• Locally shape-preserving, called conformal or orthomorphic
• Area-preserving, called equal-area or equiareal or equivalent or authalic
• Distance preserving - equidistant (preserving distances between one or two points and every other point)
• Shortest-route preserving - gnomonic projection

NOTE: Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal. The Mercator projection shows courses of constant bearing as straight lines. ... Gnomonic projections are used in seismic work because seismic waves tend to travel along great circles. ...

Projections by surface

Cylindrical

The space-oblique Mercator projection was developed by the USGS for use in Landsat images.

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). Image File history File links File links The following pages link to this file: Map projection ... Space-oblique Mercator projection is a map projection. ... The United States Geological Survey (USGS) is a scientific agency of the United States government. ... The Landsat program is the longest running enterprise for acqusition of imagery of Earth from space. ... On the earth, a meridian is a north-south line between the North Pole and the South Pole. ... The 4 main circles of latitude on Earth A circle of latitude is an imaginary east-west circle on the Earth, that connects all locations with a given latitude. ... In Latin, mutatis mutandis means upon changing what needs to be changed, where what needs to be changed is usually implied by a prior statement assumed to be understood by the reader. ...

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 various cylindrical projections can be described in terms of the north-south stretching: World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses Ilhéu das Rolas, in São Tomé and Príncipe. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... Latitude,usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ...

• North-south stretching is equal to the east-west stretching (secant(L)): The east-west scale matches the north-south-scale: conformal cylindrical or Mercator; this distorts areas excessively in high latitudes (see also transverse Mercator).
• North-south stretching growing rapidly with latitude, even faster than east-west stretching (secant(L))²: The cylindric perspective (= central cylindrical) projection; unsuitable because distortion is even worse than in the Mercator projection.
• North-south stretching grows with latitude, but less quickly than the east-west stretching: such as the Miller cylindrical projection (secant(L*4/5)).
• North-south distances neither stretched nor compressed (1): equidistant cylindrical or plate carrée.
• North-south compression precisely the reciprocal of east-west stretching (cos(L)): equal-area cylindrical (with many named specializations such as Gall-Peters or Gall orthographic, Behrmann, and Lambert cylindrical equal-area). This divides north-south distances by a factor equal to the secant of the latitude, preserving area but heavily distorting shapes.

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. Mercator world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigatium Emendate (1569) The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569. ... // Transverse Mercator Projection A Transverse Mercator projection A Transverse Mercator projection is an adaptation of the Mercator projection. ... A Miller cylindrical projection maps the globe onto a cylinder. ... Equirectangular projection of the Globe Equirectangular projection of a composite satellite image (NASA) The plate carrée projection or geographic projection or equirectangular projection, is a very simple map projection that has been in use since the earliest days of spherical cartography. ... Peters map The Gall-Peters projection is one specialization of a configurable equal-area map projection known as the equal-area cylindric or cylindrical equal-area projection. ... Map of the Earth using a Behrmann projection The Behrmann Projection is a cylindrical map projection. ... Map of the Earth using a Lambert Cylindrical projection The Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is a cylindrical, equal area map projection. ...

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.

Pseudocylindrical projections represent the central meridian and each parallel as a straight line segment, but not the other meridians, except for the Collignon projection, which in its most common forms represents all meridians as straight lines from the poles to the equators as straight line segments. 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. Image File history File links Please see the file description page for further information. ... On the earth, a meridian is a north-south line between the North Pole and the South Pole. ... On the Earth, a circle of latitude is an imaginary east-west circle that connects all locations with a given latitude. ... The Collignon Projection is a pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881. ...

• Sinusoidal: the north-south scale is the same everywhere at the central meridian, and the east-west scale is throughout the map the same as that; correspondingly, on the map, as in reality, the length of each parallel is proportional to the cosine of the latitude. Thus the shape of the map for the whole earth is the area between two symmetric rotated cosine curves ^[1].

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 meridians drawn on the map help the user to realize the distortion and mentally compensate for it. Sinusoidal projection A sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson-Flamsteed or the Mercator equal-area projection. ...

• Mollweide
• Goode homolosine
• Eckert IV

• Eckert VI

• Collignon
• Kavrayskiy VII

Example of a Mollweide projection. ... Goode homolosine projection The Goode homolosine projection is an interrupted, pseudocylindrical, equal-area, composite map projection. ... Image File history File links Map_projection-Eckert_IV.png Created by User:Reisio with GMT. File links The following pages link to this file: Map projection ... Image File history File links Map_projection-Eckert_VI.png Created by User:Reisio with GMT. File links The following pages link to this file: Map projection ... The Collignon Projection is a pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881. ... Kavrayskiy VII projection of the Earth. ...

Hybrid

The HEALPix projection combines an equal-area cylindrical projection in equatorial regions with the Collignon projection in polar areas. Healpix (sometimes written as HEALPix) is a scheme for partitioning (or pixelizing) the sphere into a set of equal area pixels. ... The Collignon Projection is a pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881. ...

Conical

• Equidistant conic
• Lambert conformal conic
• Albers conic

Hello ... Map of the Earth using an Albers projection The Albers equal-area conic projection, or Albers projection, is a conic, equal area map projection that uses two standard parallels. ...

Pseudoconical

• Bonne
• Werner cordiform designates a pole and a meridian; distances from the pole are preserved, as are distances from the meridian (which is straight) along the parallels
• Continuous American polyconic

A Bonne projection A Bonne projection is a pseudoconical equal-area map projection, sometimes called a dépôt de la guerre or a Sylvanus projection. ... Werner projection The Werner projection is a pseudoconical equal-area map projection, sometimes called the Stabius-Werner or the Stab-Werner projection. ... Map of the Earth using a polyconic projection A polyconic projection is a conical map projection. ...

Azimuthal (projections onto a plane)

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

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). 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. Image File history File links World map projection (source) File links The following pages link to this file: Map projection Azimuthal equidistant projection ... A map projection is any of many methods used in cartography (mapmaking) to represent the two-dimensional curved surface of the earth or other body on a plane. ... Azimuth is the horizontal component of a direction (compass direction), measured around the horizon, from the north toward the east (i. ...

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: Perspective projection is a type of drawing that graphically approximates on a planar (two-dimensional) surface (e. ... In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. ...

• The gnomonic projection displays great circles as straight lines. Can be constructed by using a point of perspective at the center of the Earth. r(d) = c tan(d/R); a hemisphere already requires an infinite map ^[2], ^[3]
• The General Perspective Projection can be constructed by using a point of perspective outside the earth. Photographs of Earth (such as those from the International Space Station) give this perspective.
• The orthographic projection 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; r(d) = c sin(d/R) ^[4]. Can display up to a hemisphere on a finite circle. Photographs of Earth from far enough away, such as the Moon, give this perspective.
• The azimuthal conformal projection, also known as the stereographic projection, can be constructed by using the tangent point's antipode as the point of perspective. r(d) = c tan(d/2R); the scale is c/(2R cos²(d/2R))^[5]. Can display nearly the entire sphere on a finite circle. The full sphere requires an infinite map.

Other azimuthal projections are not true perspective projections: Gnomonic projections are used in seismic work because seismic waves tend to travel along great circles. ... For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the... General Vertical Perspective; origin at 90W, 0N General Perspective Projection is a map projection of cartography. ... “ISS” redirects here. ... Orthographic projection (equatorial aspect) of the hemisphere 30W–150E Orthographic projection is a map projection of cartography. ... This article is about Earths moon. ... Stereographic projection of a circle of radius R onto the x axis. ... In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. ... A cube in two-point perspective. ...

• Azimuthal equidistant: r(d) = cd; it is used by amateur radio 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 (^[6]; for the case where the tangent point is the North Pole, see the flag of the United Nations)
• Lambert azimuthal equal-area. Distance from the tangent point on the map is proportional to straight-line distance through the earth: r(d) = c sin(d/2R) ^[7]
• Logarithmic azimuthal 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. Works well with cognitive maps^{[citation needed]}. r(d) = c ln(d/d₀); locations closer than at a distance equal to the constant d₀ are not shown (^[8], figure 6-5)

This projection shows all distances and directions correctly from a single point. ... Amateur radio station with modern solid-state transceiver featuring LCD display and DSP capabilities Amateur radio, often called ham radio, is a hobby that uses various types of radio broadcasting equipment to communicate with other radio amateurs for public service, recreation and self-training. ... The olive branches symbolise peace. ... Map of the Earth using a Lambert azimuthal equal-area projection The Lambert azimuthal equal-area projection, or Lambert azimuthal projection, is an equal-area map projection. ... Cognitive Maps, Mental Maps, Mind Maps, Cognitive Models, or mental models are a type of mental processing, or cognition, composed of a series of psychological transformations by which an individual can acquire, code, store, recall, and decode information about the relative locations and attributes of phenomena in their everyday or...

Projections by preservation of a metric property

A stereographic projection is conformal and perspective but not equal area or equidistant.

Image File history File links World map projection (source) File links The following pages link to this file: Map projection Stereographic projection ... Stereographic projection of a circle of radius R onto the x axis. ...

Conformal

Conformal map projections preserve angles locally: In mathematics, a conformal map is a function which preserves angles. ...

• Mercator - rhumb lines are represented by straight segments
• Stereographic - shape of circles is conserved
• Roussilhe
• Lambert conformal conic
• Quincuncial map
• Adams hemisphere-in-a-square projection
• Guyou hemisphere-in-a-square projection

Mercator world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigatium Emendate (1569) The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569. ... Line crossing all meridians at the same angle. ... Stereographic projection of a circle of radius R onto the x axis. ... The Roussilhe oblique stereographic projection was developed by a Hydrographer of the French Navy in the late 19th century. ... Hello ... A quincuncial map is a conformal map projection that is conformal everywhere except at the corners of the inner hemisphere. ... The Adams-hemisphere-in-a-square is a conformal map map projection for a hemisphere (except for four points where the conformality fails). ... The Guyou hemisphere-in-a-square projection is a conformal map projection for the hemisphere (except for four points where the conformality fails). ...

Equal-area

The equal-area Mollweide projection

These projections preserve area: Image File history File links Size of this preview: 800 × 400 pixelsFull resolution (2048 × 1025 pixel, file size: 766 KB, MIME type: image/jpeg) A Mollweide projection of a Visible Earth image collected by the Earth Observatory experiment of the U.S. Governments NASA space agency. ... Image File history File links Size of this preview: 800 × 400 pixelsFull resolution (2048 × 1025 pixel, file size: 766 KB, MIME type: image/jpeg) A Mollweide projection of a Visible Earth image collected by the Earth Observatory experiment of the U.S. Governments NASA space agency. ... Example of a Mollweide projection. ...

Peters map The Gall-Peters projection is one specialization of a configurable equal-area map projection known as the equal-area cylindric or cylindrical equal-area projection. ... Map of the Earth using an Albers projection The Albers equal-area conic projection, or Albers projection, is a conic, equal area map projection that uses two standard parallels. ... Map of the Earth using a Lambert azimuthal equal-area projection The Lambert azimuthal equal-area projection, or Lambert azimuthal projection, is an equal-area map projection. ... Example of a Mollweide projection. ... A Hammer projection of the Earth The Hammer projection is an equal-area map projection, described by Ernst Hammer in 1892. ... Sinusoidal projection A sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson-Flamsteed or the Mercator equal-area projection. ... Werner projection The Werner projection is a pseudoconical equal-area map projection, sometimes called the Stabius-Werner or the Stab-Werner projection. ... A Bonne projection A Bonne projection is a pseudoconical equal-area map projection, sometimes called a dépôt de la guerre or a Sylvanus projection. ... A Bottomley projection A Bottomley projection is an equal-area map projection. ... Goode homolosine projection The Goode homolosine projection is an interrupted, pseudocylindrical, equal-area, composite map projection. ... The Hobo-Dyer map projection is an equal area map projection. ... The Collignon Projection is a pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881. ... Healpix (sometimes written as HEALPix) is a scheme for partitioning (or pixelizing) the sphere into a set of equal area pixels. ... The Tobler hyperelliptical projection is a family of pseudocylindrical projections used for mapping the earth. ...

Equidistant

These preserve distance from some standard point or line:

• Plate carrée - north-south scale is constant
• Equirectangular - equal distance between all latitudes and longitudes.
• Azimuthal equidistant - radial scale with respect to the central point is constant
• Equidistant conic
• sinusoidal - east-west scale is constant and corresponds to distances between parallels (but the north-south scale away from the central meridian is larger due to the obliqueness of the meridians)
• Werner cordiform distances from the North Pole are correct as are the curved distance on parallels
• Soldner

a two-point equidistant projection of Asia

• Two-point equidistant: two "control points" are arbitrarily chosen by the map maker. Distance from any point on the map to each control point is proportional to surface distance on the earth.

Equirectangular projection of the Globe Equirectangular projection of a composite satellite image (NASA) The plate carrée projection or geographic projection or equirectangular projection, is a very simple map projection that has been in use since the earliest days of spherical cartography. ... In cartography, the equirectangular projection is a modification of the plate carrée projection, with the longitude lines (meridians) spaced closer together, forming rectangles with the latitude lines (parallels) instead of squares. ... This projection shows all distances and directions correctly from a single point. ... Sinusoidal projection A sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson-Flamsteed or the Mercator equal-area projection. ... Werner projection The Werner projection is a pseudoconical equal-area map projection, sometimes called the Stabius-Werner or the Stab-Werner projection. ... For other uses, see North Pole (disambiguation). ... Image File history File links Download high-resolution version (2048x1529, 1404 KB) A Two Point Equidistant projection of Asia. ... Image File history File links Download high-resolution version (2048x1529, 1404 KB) A Two Point Equidistant projection of Asia. ... A Two Point Equidistant projection of Asia The two-point equidistant projection is a map projection first described by Hans Maurer in 1919[1]. Distances from any point on the map to two control points scale to the geodesic distances of the same points on the sphere. ... A Two Point Equidistant projection of Asia The two-point equidistant projection is a map projection first described by Hans Maurer in 1919[1]. Distances from any point on the map to two control points scale to the geodesic distances of the same points on the sphere. ...

Gnomonic

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: Image File history File links World map projection (source) File links The following pages link to this file: Map projection Gnomonic projection ... Gnomonic projections are used in seismic work because seismic waves tend to travel along great circles. ... Thales of Miletos (, ca. ... (2nd millennium BC - 1st millennium BC - 1st millennium) The 6th century BC started on January 1, 600 BC and ended on December 31, 501 BC. // Monument 1, an Olmec colossal head at La Venta The 5th and 6th centuries BC were a time of empires, but more importantly, a time... For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the...

• Gnomonic projection

Gnomonic projections are used in seismic work because seismic waves tend to travel along great circles. ...

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:

• Littrow - the only conformal retroazimuthal projection
• Hammer retroazimuthal - also preserves distance from the central point
• Craig retroazimuthal aka Mecca or Qibla - also has vertical meridians

The Littrow projection is the only conformal retroazimuthal map projection. ... The Craig retroazimuthal map projection was created by James Ireland Craig in 1909. ...

Compromise projections

The Robinson projection was adopted by National Geographic Magazine in 1988 but abandoned by them in about 1997 for the Winkel Tripel.

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: Image File history File links World map projection (source) File links The following pages link to this file: Map projection Robinson projection ... Robinson projection map (Large 2 MB). ... The National Geographic Magazine, later shortened to National Geographic, is the official journal of the National Geographic Society. ... A Winkel Tripel projection of the Earth Winkels Tripel projection was developed to show a map of the round Earth on flat paper with minimal distortion. ...

• Robinson
• van der Grinten
• Miller cylindrical
• Winkel Tripel
• Buckminster Fuller's Dymaxion
• B.J.S. Cahill's Butterfly Map
• Steve Waterman's Butterfly Map
• Kavrayskiy VII
• Wagner VI

Robinson projection map (Large 2 MB). ... A van der Grinten projection of the Earth The van der Grinten projection is neither equal-area nor conformal. ... A Miller cylindrical projection maps the globe onto a cylinder. ... A Winkel Tripel projection of the Earth Winkels Tripel projection was developed to show a map of the round Earth on flat paper with minimal distortion. ... The Dymaxion Map of the Earth is a projection of a global map onto the surface of a three-dimensional regular solid, which can then be unfolded to a net in many different ways and flattened to form a two-dimensional map which retains most of the relative proportional integrity... B.J.S. Cahill (Bernard Joseph Stanislaus Cahill, 1866-1944), cartographer and architect; inventor of the octahedral Butterfly Map (published 1909; patented 1913); early proponent of the San Francisco Civic Center (1899-1909); designer of the Columbarium of San Francisco. ... Waterman Butterfly Map, largest online image The Waterman Butterfly World Map Projection was created by Steve Waterman and published in 1996. ... Kavrayskiy VII projection of the Earth. ... Wagner VI projection of the Earth. ...

Other noteworthy projections

• Chamberlin trimetric
• The French cartographer Oronce Fine developed a heart-shaped projection in the sixteenth century

The Chamberlin trimetric projection is a map projection where three points are fixed on a sphere and used to triangulate the transformation onto a plane. ... Oronce Fine (in Latin, Orontius Finnaeus or Finaeus) (December 20, 1494-August 8, 1555), French mathematician and cartographer. ...

References

• Fran Evanisko, American River College, lectures for Geography 20: "Cartographic Design for GIS", Fall 2002
• Snyder, J.P., Album of Map Projections, United States Geological Survey Professional Paper 1453, United States Government Printing Office, 1989.
• Snyder, John P. (1987). Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395. United States Government Printing Office, Washington, D.C..  This paper can be downloaded from USGS pages

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.

• Paul Andersons' Gallery of Map Projections - 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"]

is the 322nd day of the year (323rd in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 322nd day of the year (323rd in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 322nd day of the year (323rd in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 322nd day of the year (323rd in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 322nd day of the year (323rd in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 322nd day of the year (323rd in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 322nd day of the year (323rd in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 322nd day of the year (323rd in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ...

See also

Physical world map (2004) with country borders and capitals—click for large, 1. ... This article does not cite its references or sources. ... Cartography or mapmaking (in Greek chartis = map and graphein = write) is the study and practice of making maps or globes. ... Cartography is the study of map making and cartographers are map makers. ... A geographic information system (GIS) is a system for capturing, storing, analyzing and managing data and associated attributes which are spatially referenced to the earth. ... Graphical projection in the visual sciences is an imaging procedure the protocols of which preclude the necessity of mathematical calculation. ... Example of orthographic drawing from a US Patent (1913), showing two views of the same object. ... Example of a dimetric axonometric drawing from a US Patent (1874). ... 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. ... An isometric drawing of a cube. ... Example of a dimetric axonometric drawing from a US Patent (1874) 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 the attendant scale and angles of presentation are determined according... This article needs to be cleaned up to conform to a higher standard of quality. ... In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k &#8722; d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ... Perspective projection is a type of drawing that graphically approximates on a planar (two-dimensional) surface (e. ... For other uses, see Plan (disambiguation). ...

External links

Wikimedia Commons has media related to:

Map projections

• G.Projector, free software by NASA GISS that can render many projections.
• An interactive JAVA applet to study deformations (area, distance and angle) of map projections
• US Geological Survey overview
• Map projections intro
• Mathworld formulae
• How Projections Work
• Table of examples and properties of all common projections, from radicalcartography.net
• PDFs of projections
• GIFs of projections
• U.S. WWII Newsmap, "Maps are Not True for All Purposes, These are three of many projections", hosted by the UNT Libraries Digital Collections
• Java applet for interactive projections
• USGS info
• Geodesy, Cartography and Map Reading from Colorado State University
• A collection of map projections and reference systems for Europe
• What is a map projection?
• The World Turned Upside Down by Katy Kramer
• PROJ.4 cartographic projections library
• Image Projections - Interactive visual comparison between different types of image projections.