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cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
of a
globe A globe is a spherical model of Earth, of some other celestial body, or of the celestial sphere. Globes serve purposes similar to maps, but unlike maps, they do not distort the surface that they portray except to scale it down. A model glo ...
on a plane. In a map projection,
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
, often expressed as
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
and
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. 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 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 Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
,
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
, and
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s. 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 a shadow on a screen, or the rectilinear image produced by a
pinhole camera A pinhole camera is a simple camera without a lens but with a tiny aperture (the so-called '' pinhole'')—effectively a light-proof box with a small hole in one side. Light from a scene passes through the aperture and projects an inverted image ...
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 Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
and other large
celestial bodies An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often us ...
are generally better modeled as
oblate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ci ...
s, whereas small objects such as
asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...
s often have irregular shapes. The surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid. Therefore, more generally, a map projection is any method of flattening a continuous curved surface onto a plane. The most well-known map projection is the
Mercator projection The Mercator projection () is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and s ...
. 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 map projections such as the
Sinusoidal projection The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson–Flamsteed or the Mercator equal-area projection. Jean Cossin of Dieppe was one of the first mapmakers to use the sinusoidal, appearing i ...
and the
Gall–Peters projection The Gall–Peters projection is a rectangular, equal-area map projection. Like all equal-area projections, it distorts most shapes. It is a cylindrical equal-area projection with latitudes 45° north and south as the regions on the map that hav ...
show the correct sizes of countries relative to each other, but distort angles. The
National Geographic Society The National Geographic Society (NGS), headquartered in Washington, D.C., United States, is one of the largest non-profit scientific and educational organizations in the world. Founded in 1888, its interests include geography, archaeology, ...
and most atlases favor map projections that compromise between area and angular distortion, such as the
Robinson projection The Robinson projection is a map projection of a world map which shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of readily showing the whole globe as a flat image. The Robinson ...
and the
Winkel tripel projection The Winkel tripel projection (Winkel III), a modified azimuthal map projection of the world, is one of three projections proposed by German cartographer Oswald Winkel (7 January 1874 – 18 July 1953) in 1921. The projection is the arithmetic ...
.


Metric properties of maps

Many properties can be measured on the Earth's surface independently of its geography: *
Area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
*
Shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...
* Direction * Bearing *
Distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
Map projections can be constructed to preserve some of these properties at 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 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 In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted. ...
(model) of the Earth. Different datums assign slightly different coordinates to 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.


Distortion

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
's '' Theorema Egregium'' proved that a sphere's surface cannot be represented on a plane without distortion. The same applies to other reference surfaces used as models for the Earth, such as oblate
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...
s, ellipsoids, and
geoid The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
s. 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 In cartography, a Tissot's indicatrix (Tissot indicatrix, Tissot's ellipse, Tissot ellipse, ellipse of distortion) (plural: "Tissot's indicatrices") is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 ...
. For a given point, using 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.


Other distortion metrics

Many other ways have been described of showing 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 A circle of a sphere is a circle that lies on a sphere. Such a circle can be formed as the intersection of a sphere and a plane, or of two spheres. Circles of a sphere are the spherical geometry analogs of generalised circles in Euclidean sp ...
of fixed radius (e.g., 15 degrees
angular radius The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is ...
). Sometimes
spherical triangle Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...
s are used. In the first half of the 20th century, projecting a human head onto different projections was common to show how distortion varies across one projection as compared to another. In dynamic media, shapes of familiar coastlines and boundaries can be dragged across an interactive map to show how the projection distorts sizes and shapes according to position on the map. Another way to visualize local distortion is through grayscale or color gradations whose shade represents the magnitude of the angular deformation or areal inflation. Sometimes both are shown simultaneously by blending two colors to create a
bivariate map A bivariate map or multivariate map is a type of thematic map that displays two or more variables on a single map by combining different sets of symbols.Nelson, J. (2020)Multivariate Mapping ''The Geographic Information Science & Technology Bo ...
. To measure distortion globally across areas instead of at just a single point necessarily involves choosing priorities to reach a compromise. Some schemes use distance distortion as a proxy for the combination of angular deformation and areal inflation; such methods arbitrarily choose what paths to measure and how to weight them in order to yield a single result. Many have been described.


Design and construction

The creation of a map projection involves two steps: # Selection of a model for the shape of the Earth or planetary body (usually choosing between a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
or
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as th ...
). Because the Earth's actual shape is irregular, information is lost in this step. # Transformation of geographic coordinates (
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
and
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
) to Cartesian (''x'',''y'') or
polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates *Polar climate, the cli ...
(''r'', ''θ'') plane coordinates. In large-scale maps, Cartesian coordinates normally have a simple relation to eastings and northings defined as a grid superimposed on the projection. In small-scale maps, eastings and northings are not meaningful, and grids are not superimposed. Some of the simplest map projections are literal projections, as obtained by placing a light source at some definite point relative to the globe and projecting its features onto a specified surface. Although most projections are not defined in this way, picturing the light source-globe model can be helpful in understanding the basic concept of a map projection.


Choosing a projection surface

A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a ''
developable surface In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). ...
''. The
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an ...
,
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
and the plane are all developable surfaces. The sphere and ellipsoid do not have developable surfaces, so any projection of them onto a plane will have to distort the image. (To compare, one cannot flatten an orange peel without tearing and warping it.) One way of describing a projection is first to project from the Earth's surface to a developable surface such as a cylinder or cone, and then to unroll the surface into a plane. While the first step inevitably distorts some properties of the globe, the developable surface can then be unfolded without further distortion.


Aspect of the projection

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 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).


Notable lines

The developable surface may also be either ''
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
'' 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 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'') and a parallel of origin (usually written ''φ'') are often used to define the origin of the map projection.


Scale

A
globe A globe is a spherical model of Earth, of some other celestial body, or of the celestial sphere. Globes serve purposes similar to maps, but unlike maps, they do not distort the surface that they portray except to scale it down. A model glo ...
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: * The scale depends on location, but not on direction. This is equivalent to preservation of angles, the defining characteristic of a
conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
. * Scale is constant along any parallel in the direction of the parallel. This applies for any cylindrical or pseudocylindrical projection in normal aspect. * Combination of the above: the scale depends on latitude only, not on longitude or direction. This applies for the
Mercator projection The Mercator projection () is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and s ...
in normal aspect. * Scale is constant along all straight lines radiating from a particular geographic location. This is the defining characteristic of an equidistant projection such as the
azimuthal equidistant projection The azimuthal equidistant projection is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimut ...
. There are also projections (Maurer's two-point equidistant projection, Close) where true distances from ''two'' points are preserved.


Choosing a model for the shape of the body

Projection construction is also affected by how the shape of the Earth or planetary body is approximated. In the following section on projection categories, the earth is taken as a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as th ...
. Whether spherical or ellipsoidal, the principles discussed hold without loss of generality. 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 map In modern mapping, a topographic map or topographic sheet is a type of map characterized by large- scale detail and quantitative representation of relief features, usually using contour lines (connecting points of equal elevation), but histori ...
s and for other large- and medium-scale maps that need to accurately depict the land surface. Auxiliary latitudes are often employed in projecting the ellipsoid. A third model is the
geoid The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
, a more complex and accurate representation of Earth's shape coincident with what
mean sea level There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ...
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 In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
and equivalence. Therefore, in geoidal projections that preserve such properties, the mapped
graticule Graticule may refer to: * An oscilloscope graticule scale * The reticle pattern in an optical instrument * Graticule (cartography), a grid of lines on a map See also * Grid (disambiguation) Grid, The Grid, or GRID may refer to: Common us ...
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 The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extende ...
amounting to less than 100 m from the ellipsoidal model out of the 6.3 million m
Earth radius Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, den ...
. For irregular planetary bodies such as
asteroids An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...
, however, sometimes models analogous to 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 , discoverer = , discovered = , earliest_precovery_date = March 22, 1955 , mpc_name = (136108) Haumea , pronounced = , adjectives = Haumean , note = yes , alt_names = , named_after = Haumea , mp_category = , orbit_ref = , epoc ...
'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 the triaxial ellipsoid In geodesy, a map projection of the triaxial ellipsoid maps Earth or some other astronomical body modeled as a triaxial ellipsoid to the plane. Such a model is called the reference ellipsoid. In most cases, reference ellipsoids are spheroids, and s ...
for further information.


Classification

A fundamental projection classification is based on the 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 __NOTOC__ Mercator (Latin for "merchant") may refer to: People * Marius Mercator (c. 390–451), a Catholic ecclesiastical writer * Arnold Mercator, a 16th-century cartographer * Gerardus Mercator, a 16th-century cartographer ** Mercator 1569 ...
), conic (e.g., Albers), and plane (e.g.,
stereographic Stereoscopy (also called stereoscopics, or stereo imaging) is a technique for creating or enhancing the illusion of depth in an image by means of stereopsis for binocular vision. The word ''stereoscopy'' derives . Any stereoscopic image is ...
). 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, pseudocylindrical, pseudoazimuthal, retroazimuthal, and polyconic. Another way to classify projections is according to properties of the model they preserve. Some of the more common categories are: * Preserving direction (''azimuthal or zenithal''), a trait possible only from one or two points to every other point * Preserving shape locally ('' conformal'' or ''orthomorphic'') * Preserving area (''equal-area'' or ''equiareal'' or ''equivalent'' or ''authalic'') * Preserving distance (''equidistant''), a trait possible only between one or two points and every other point * Preserving shortest route, a trait preserved only by the
gnomonic projection A gnomonic map projection is a map projection which displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achi ...
Because the sphere is not a
developable surface In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). ...
, it is impossible to construct a map projection that is both equal-area and conformal.


Projections by surface

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, 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 a cylindrical projection (for example) is one which: # Is rectangular; # Has straight vertical meridians, spaced evenly; # Has straight parallels symmetrically placed about the equator; # Has parallels constrained to where they fall when light shines through the globe onto the cylinder, with the light source someplace along the line formed by the intersection of the prime meridian with the equator, and the center of the sphere. (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 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 a course of constant bearing is always plotted as a straight line.


Cylindrical

A normal cylindrical projection is any projection in which meridians are mapped to equally spaced vertical lines and
circles of latitude A circle of latitude or line of latitude on Earth is an abstract east– west small circle connecting all locations around Earth (ignoring elevation) at a given latitude coordinate line. Circles of latitude are often called parallels bec ...
(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 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 In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
as a multiple of the equator's scale. The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude is given by φ): * North-south stretching equals east-west stretching ( sec ''φ''): The east-west scale matches the north-south scale: conformal cylindrical or
Mercator __NOTOC__ Mercator (Latin for "merchant") may refer to: People * Marius Mercator (c. 390–451), a Catholic ecclesiastical writer * Arnold Mercator, a 16th-century cartographer * Gerardus Mercator, a 16th-century cartographer ** Mercator 1569 ...
; this distorts areas excessively in high latitudes (see also transverse Mercator). * North-south stretching grows with latitude faster than east-west stretching (sec ''φ''): The cylindric perspective (or 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 (sec ''φ''). * North-south distances neither stretched nor compressed (1):
equirectangular projection The equirectangular projection (also called the equidistant cylindrical projection or la carte parallélogrammatique projection), and which includes the special case of the plate carrée projection (also called the geographic projection, lat/lon ...
or "plate carrée". * North-south compression equals the cosine of the latitude (the reciprocal of east-west stretching): equal-area cylindrical. This projection has many named specializations differing only in the scaling constant, such as the Gall–Peters or Gall orthographic (undistorted at the 45° parallels), Behrmann (undistorted at the 30° parallels), and Lambert cylindrical equal-area (undistorted at the equator). Since this projection scales north-south distances by the reciprocal of east-west stretching, it preserves area at the expense of shapes. 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 line Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or recipr ...
s—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 the whole Earth as a finite rectangle, except in the first two cases, where the rectangle stretches infinitely tall while retaining constant width.


Pseudocylindrical

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 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: *
Sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often i ...
, which was the first pseudocylindrical projection developed. On the map, as in reality, the length of each parallel is proportional to the cosine of the latitude. The area of any region is true. *
Collignon projection The Collignon projection is an equal-area pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881. For the smallest choices of the parameters chosen for this projecti ...
, which in its most common forms represents each meridian as two straight line segments, one from each pole to the equator.


Hybrid

The
HEALPix HEALPix (sometimes written as Healpix), an acronym for Hierarchical Equal Area isoLatitude Pixelisation of a 2-sphere, is an algorithm for pixelisation of the 2-sphere and the associated class of map projections. The pixelisation algorithm was de ...
projection combines an equal-area cylindrical projection in equatorial regions with the
Collignon projection The Collignon projection is an equal-area pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881. For the smallest choices of the parameters chosen for this projecti ...
in polar areas.


Conic

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 A circle of latitude or line of latitude on Earth is an abstract east– west small circle connecting all locations around Earth (ignoring elevation) at a given latitude coordinate line. Circles of latitude are often called parallels bec ...
(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 line Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or recipr ...
s where the cone intersects the globe—or, if the map maker chooses the same parallel twice, as the tangent line where the cone is tangent to the globe. The resulting conic map has low distortion in scale, shape, and area near those standard parallels. Distances along the parallels to the north of both standard parallels or to the south of both standard parallels are stretched; distances along parallels between the standard parallels are compressed. When a single standard parallel is used, distances along all other parallels are stretched. Conic projections that are commonly used are: * Equidistant conic, which keeps parallels evenly spaced along the meridians to preserve a constant distance scale along each meridian, typically the same or similar scale as along the standard parallels. * Albers conic, which adjusts the north-south distance between non-standard parallels to compensate for the east-west stretching or compression, giving an equal-area map. * Lambert conformal conic, which adjusts the north-south distance between non-standard parallels to equal the east-west stretching, giving a conformal map.


Pseudoconic

* Bonne, an equal-area projection on which most meridians and parallels appear as curved lines. It has a configurable standard parallel along which there is no distortion. * Werner cordiform, upon which distances are correct from one pole, as well as along all parallels. * American polyconic and other projections in the
polyconic projection class Polyconic can refer either to a class of map projections or to a specific projection known less ambiguously as the American polyconic projection. Polyconic as a class refers to those projections whose parallels are all non-concentric circular arc ...
.


Azimuthal (projections onto a plane)

Azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematical ...
al projections have the property that directions from a central point are preserved and therefore
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
s through the central point are represented by straight lines on the map. These projections also have radial symmetry in the scales and hence in the distortions: map distances from the central point are computed by a function ''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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to the Earth, with the central point as tangent point. The radial scale is ''r′''(''d'') and the transverse scale ''r''(''d'')/(''R'' sin ) where ''R'' is the radius of the Earth. Some azimuthal projections are true
perspective projection Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation ...
s; that is, they can be constructed mechanically, projecting the surface of the Earth by extending lines from a point of perspective (along an infinite line through the tangent point and the tangent point's antipode) onto the plane: * The
gnomonic projection A gnomonic map projection is a map projection which displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achi ...
displays
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
s as straight lines. Can be constructed by using a point of perspective at the center of the Earth. ''r''(''d'') = ''c'' tan ; so that even just a hemisphere is already infinite in extent. * The
orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogona ...
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 . Can display up to a hemisphere on a finite circle. Photographs of Earth from far enough away, such as the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
, approximate this perspective. * Near-sided perspective projection, which simulates the view from space at a finite distance and therefore shows less than a full hemisphere, such as used in '' The Blue Marble 2012''). * 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 The International Space Station (ISS) is the largest Modular design, modular space station currently in low Earth orbit. It is a multinational collaborative project involving five participating space agencies: NASA (United States), Roscosmos ( ...
) give this perspective. It is a generalization of near-sided perspective projection, allowing tilt. * The
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
, which is conformal, can be constructed by using the tangent point's antipode as the point of perspective. ''r''(''d'') = ''c'' tan ; the scale is ''c''/(2''R'' cos ). Can display nearly the entire sphere's surface on a finite circle. The sphere's full surface requires an infinite map. Other azimuthal projections are not true perspective projections: * Azimuthal equidistant: ''r''(''d'') = ''cd''; it is used by
amateur radio Amateur radio, also known as ham radio, is the use of the radio frequency spectrum for purposes of non-commercial exchange of messages, wireless experimentation, self-training, private recreation, radiosport, contesting, and emergency commu ...
operators to know the direction to point their antennas toward a point and see the distance to it. Distance from the tangent point on the map is proportional to surface distance on the Earth (; for the case where the tangent point is the North Pole, see the
flag of the United Nations The flag of the United Nations consists of the emblem depicting the white azimuthal equidistant projection of the world map, centred on the North Pole, with two white olive branches placed on to its right and left, located on the sky blue ba ...
) * 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  * 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. ''r''(''d'') = ''c'' ln ); locations closer than at a distance equal to the constant ''d''0 are not shown.
Reprinted in:


Polyhedral

Polyhedral map projections use a
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...
to subdivide the globe into faces, and then projects each face to the globe. The most well-known polyhedral map projection is Buckminster Fuller's
Dymaxion map The Dymaxion map or Fuller map is a projection of a world map onto the surface of an icosahedron, which can be unfolded and flattened to two dimensions. The flat map is heavily interrupted in order to preserve shapes and sizes. The projection ...
.


Projections by preservation of a metric property


Conformal

Conformal, or orthomorphic, map projections preserve angles locally, implying that they map infinitesimal circles of constant size anywhere on the Earth to infinitesimal circles of varying sizes on the map. In contrast, mappings that are not conformal distort most such small circles into ellipses of distortion. An important consequence of conformality is that relative angles at each point of the map are correct, and the local scale (although varying throughout the map) in every direction around any one point is constant. These are some conformal projections: *
Mercator __NOTOC__ Mercator (Latin for "merchant") may refer to: People * Marius Mercator (c. 390–451), a Catholic ecclesiastical writer * Arnold Mercator, a 16th-century cartographer * Gerardus Mercator, a 16th-century cartographer ** Mercator 1569 ...
:
Rhumb line In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north. Introduction The effect of following a rhumb l ...
s are represented by straight segments * Transverse Mercator *
Stereographic Stereoscopy (also called stereoscopics, or stereo imaging) is a technique for creating or enhancing the illusion of depth in an image by means of stereopsis for binocular vision. The word ''stereoscopy'' derives . Any stereoscopic image is ...
: Any
circle of a sphere A circle of a sphere is a circle that lies on a sphere. Such a circle can be formed as the intersection of a sphere and a plane, or of two spheres. Circles of a sphere are the spherical geometry analogs of generalised circles in Euclidean space ...
, great and small, maps to a circle or straight line. * Roussilhe * Lambert conformal conic * Peirce quincuncial projection *
Adams hemisphere-in-a-square projection The Adams hemisphere-in-a-square is a conformal map projection for a hemisphere. It is a transverse version of the Peirce quincuncial projection, and is named after American cartographer Oscar Sherman Adams, who published it in 1925.. When it i ...
*
Guyou hemisphere-in-a-square projection The Guyou hemisphere-in-a-square projection is a conformal map projection for the hemisphere. It is an oblique aspect of the Peirce quincuncial projection. History The projection was developed by of France in 1887. Formal description The pro ...


Equal-area

Equal-area maps preserve area measure, generally distorting shapes in order to do so. Equal-area maps are also called ''equivalent'' or ''authalic''. These are some projections that preserve area: * Albers conic * Bonne * Bottomley * Collignon * Cylindrical equal-area * Eckert II, IV and VI * Equal Earth * Gall orthographic (also known as Gall–Peters, or Peters, projection) * Goode's homolosine *
Hammer A hammer is a tool, most often a hand tool, consisting of a weighted "head" fixed to a long handle that is swung to deliver an impact to a small area of an object. This can be, for example, to drive nails into wood, to shape metal (as wi ...
* Hobo–Dyer * Lambert azimuthal equal-area * Lambert cylindrical equal-area * Mollweide *
Sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often i ...
* Strebe 1995 * Snyder's equal-area polyhedral projection, used for
geodesic grid A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron. Construction A geodesic grid is a global Earth reference that uses triangular tiles based on the subdivision of a polyhedron (usually the icosahedron, a ...
s. * Tobler hyperelliptical * Werner


Equidistant

If the length of the line segment connecting two projected points on the plane is proportional to the geodesic (shortest surface) distance between the two unprojected points on the globe, then we say that distance has been preserved between those two points. An equidistant projection preserves distances from one or two special points to all other points. The special point or points may get stretched into a line or curve segment when projected. In that case, the point on the line or curve segment closest to the point being measured to must be used to measure the distance. * Plate carrée: Distances from the two poles are preserved, in equatorial aspect. * Azimuthal equidistant: Distances from the center and edge are preserved. * Equidistant conic: Distances from the two poles are preserved, in equatorial aspect. * Werner cordiform Distances from the
North Pole The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Ma ...
are preserved, in equatorial aspect. * Two-point equidistant: Two "control points" are arbitrarily chosen by the map maker; distances from each control point are preserved.


Gnomonic

Great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
s are displayed as straight lines: *
Gnomonic projection A gnomonic map projection is a map projection which displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achi ...


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


Compromise projections

Compromise projections give up the idea of perfectly preserving metric properties, seeking instead to strike a balance between distortions, or to simply make things look right. Most of these types of projections distort shape in the polar regions more than at the equator. These are some compromise projections: *
Robinson Robinson may refer to: People and names * Robinson (name) Fictional characters * Robinson Crusoe, the main character, and title of a novel by Daniel Defoe, published in 1719 Geography * Robinson projection, a map projection used since the 1960 ...
* van der Grinten * Miller cylindrical * Winkel Tripel * Buckminster Fuller's Dymaxion * B. J. S. Cahill's Butterfly Map * Kavrayskiy VII projection * Wagner VI projection * Chamberlin trimetric *
Oronce Finé Oronce Finé (or Fine; Latin: ''Orontius Finnaeus'' or ''Finaeus''; it, Oronzio Fineo; 20 December 1494 – 8 August 1555) was a French mathematician, cartographer, editor and book illustrator. Life Born in Briançon, the son and grandson of p ...
's cordiform * AuthaGraph projection


Which projection is best?

The mathematics of projection do not permit any particular map projection to be best for everything. Something will always be distorted. Thus, many projections exist to serve the many uses of maps and their vast range of scales. Modern national mapping systems typically employ a transverse Mercator or close variant for large-scale maps in order to preserve
conformality In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
and low variation in scale over small areas. For smaller-scale maps, such as those spanning continents or the entire world, many projections are in common use according to their fitness for the purpose, such as Winkel tripel,
Robinson Robinson may refer to: People and names * Robinson (name) Fictional characters * Robinson Crusoe, the main character, and title of a novel by Daniel Defoe, published in 1719 Geography * Robinson projection, a map projection used since the 1960 ...
and Mollweide. Reference maps of the world often appear on compromise projections. Due to distortions inherent in any map of the world, the choice of projection becomes largely one of aesthetics. Thematic maps normally require an equal area projection so that phenomena per unit area are shown in correct proportion. However, representing area ratios correctly necessarily distorts shapes more than many maps that are not equal-area. The
Mercator projection The Mercator projection () is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and s ...
, developed for navigational purposes, has often been used in world maps where other projections would have been more appropriate.Bauer, H.A. (1942). "Globes, Maps, and Skyways (Air Education Series)". New York. p. 28Raisz, Erwin Josephus. (1938). ''General Cartography''. New York: McGraw–Hill. 2d ed., 1948. p. 87.Robinson, Arthur Howard. (1960). ''Elements of Cartography'', second edition. New York: John Wiley and Sons. p. 82. This problem has long been recognized even outside professional circles. For example, a 1943 ''
New York Times ''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid ...
'' editorial states: A controversy in the 1980s over the Peters map motivated the American Cartographic Association (now the
Cartography and Geographic Information Society The Cartography and Geographic Information Society (CaGIS) is a learned society in the fields of cartography and geographic information science. It includes the U.S. National Committee for the International Cartographic Association (ICA). It starte ...
) to produce a series of booklets (including ''Which Map Is Best''American Cartographic Association's Committee on Map Projections, 1986. ''Which Map is Best'' p. 12. Falls Church: American Congress on Surveying and Mapping.) designed to educate the public about map projections and distortion in maps. In 1989 and 1990, after some internal debate, seven North American geographic organizations adopted a resolution recommending against using any rectangular projection (including Mercator and Gall–Peters) for reference maps of the world.


See also

*
Geodetic datum A geodetic datum or geodetic system (also: geodetic reference datum, geodetic reference system, or geodetic reference frame) is a global datum reference or reference frame for precisely representing the position of locations on Earth or other pla ...
*
Geographic information system A geographic information system (GIS) is a type of database containing geographic data (that is, descriptions of phenomena for which location is relevant), combined with software tools for managing, analyzing, and visualizing those data. In a ...
(GIS) *
Geoinformatics Geoinformatics is the science and the technology which develops and uses information science infrastructure to address the problems of geography, cartography, geosciences and related branches of science and engineering, such as Land Surveying. ...
*
Grid reference A projected coordinate system, also known as a projected coordinate reference system, a planar coordinate system, or grid reference system, is a type of spatial reference system that represents locations on the Earth using cartesian coordin ...
*
List of map projections This is a summary of map projections that have articles of their own on Wikipedia or that are otherwise notable. Because there is no limit to the number of possible map projections, there can be no comprehensive list. Table of projections * ...
*
Plan (drawing) Plans are a set of drawings or two-dimensional diagrams used to describe a place or object, or to communicate building or fabrication instructions. Usually plans are drawn or printed on paper, but they can take the form of a digital file. Pla ...
* Rubbersheeting *
South-up map orientation South-up map orientation is the orientation of a map with south up, at the top of the map, amounting to a 180-degree rotation of the map from the standard convention of north-up. Maps in this orientation are sometimes called upside down maps o ...
* UV mapping *
World map A world map is a map of most or all of the surface of Earth. World maps, because of their scale, must deal with the problem of projection. Maps rendered in two dimensions by necessity distort the display of the three-dimensional surface of ...
* Spherical image projection


References


Citations


Sources

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


External links

*  , U.S. Geological Survey Professional Paper 1453, by John P. Snyder (USGS) and Philip M. Voxland (U. Minnesota), 1989.
A Cornucopia of Map Projections
a visualization of distortion on a vast array of map projections in a single image.
G.Projector
free software can render many projections (
NASA The National Aeronautics and Space Administration (NASA ) is an independent agency of the US federal government responsible for the civil space program, aeronautics research, and space research. NASA was established in 1958, succeedin ...
GISS The Goddard Institute for Space Studies (GISS) is a laboratory in the Earth Sciences Division of NASA's Goddard Space Flight Center affiliated with the Columbia University Earth Institute. The institute is located at Columbia University in ...
).
Color images of map projections and distortion
(Mapthematics.com).

(KartoWeb.itc.nl).

Henry Bottomley (SE16.info).

(MathWorld).
MapRef: The Internet Collection of MapProjections and Reference Systems in Europe

PROJ.4 – Cartographic Projections Library

Projection Reference
Table of examples and properties of all common projections (RadicalCartography.net). *  , Melita Kennedy (
Esri Esri (; Environmental Systems Research Institute) is an American multinational geographic information system (GIS) software company. It is best known for its ArcGIS products. With a 43% market share, Esri is the world's leading supplier of GIS ...
).
World Map Projections
Stephen Wolfram Stephen Wolfram (; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer science, mathematics, and theoretical physics. In 2012, he was named a fellow of the American Ma ...
based on work by Yu-Sung Chang (
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
).
Compare Map Projections

"the true size" page show size of countries without distortion from Mercator projection
{{DEFAULTSORT:Map Projection Cartography Infographics Descriptive geometry Geodesy