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The Mercator projection was designed in 1569 by Gerardus Mercator. This map projection is most commonly used for nautical navigation as it preserves directions when viewed in a small area. However, it distorts size and distance, especially near the poles, making landmasses like Greenland appear much larger than they actually are compared to landmasses near the equator.

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Used primarily for maritime navigation due to its ability to show directions accurately, especially places near the Equator. This is the map projection used by Google Maps!

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Known as an "interrupted" map, Goode’s Homolosine projection helps minimize distortion by snipping out parts of the oceans. It accurately portrays the size and shape of continents and is often used in thematic and educational maps to show spatial distribution. However, it interrupts the oceans, which might make it unsuitable for some uses, espeically those wanting to know the shape and size of the world's oceans.

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This map projection is ideal for thematic maps showing global distributions and characteristics, like climate patterns or population distribution since it shows the continents mostly accurately.

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The Goode homolosine projection (or interrupted Goode homolosine projection) is a pseudocylindrical, equal-area, composite map projection used for world maps. Normally it is presented with multiple interruptions. Its equal-area property makes it useful for presenting spatial distribution of phenomena.

The projection was developed in 1923 by John Paul Goode to provide an alternative to the Mercator projection for portraying global areal relationships. Goode offered variations of the interruption scheme for emphasizing the world's land and the world's oceans. Some variants include extensions that repeat regions in two different lobes of the interrupted map in order to show Greenland or eastern Russia undivided. The homolosine evolved from Goode's 1916 experiments in interrupting the Mollweide projection.

Because the Mollweide is sometimes called the "homolographic projection", Goode fused the two names "homolographic" and "sinusoidal" to create the name "homolosine". Common in the 1960s, the Goode homolosine projection is often called an "orange-peel map" because of its resemblance to the flattened rind of a hand-peeled orange. In its most common form, the map interrupts the North Atlantic, the South Atlantic, the South Pacific, the Indian Ocean, and the entire east/west meridian of the map.

Up to latitudes 40°44′11.8″N/S, the map is projected according to the sinusoidal projection's transformation. The higher latitudes are the top sections of a Mollweide projection, grafted to the sinusoidal midsection where the scale of the two projections matches. This grafting results in a kink in the meridians along the parallel of the graft. The projection's equal-area property follows from the fact that its source projections are themselves both equal-area.

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The Goode homolosine projection (or interrupted Goode homolosine projection) is a pseudocylindrical, equal-area, composite map projection used for world maps. Normally it is presented with multiple interruptions. Its equal-area property makes it useful for presenting spatial distribution of phenomena.

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The Lambert polar projection is an projection that is best for polar areas. It preserves accurate distances from the pole and directions within its hemisphere, but distorts shape and area more towards the equator. Most other map projections severely distort Antarctica and the Arctic Ocean; this one shows these polar areas most accurately.

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Frequently used for maps of polar regions and aviation charts because of its ability to accurately show direction and shapes over small areas close to the poles.

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Lambert azimuthal equal-area projection of the world. The center is 0° N 0° E. The antipode is 0° N 180° E, near Kiribati in the Pacific Ocean. That point is represented by the entire circular boundary of the map, and the ocean around that point appears along the entire boundary.

The Lambert azimuthal equal-area projection with Tissot's indicatrix of deformation.

The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the Lambert zenithal equal-area projection.

The Lambert azimuthal projection is used as a map projection in cartography. For example, the National Atlas of the US uses a Lambert azimuthal equal-area projection to display information in the online Map Maker application, and the European Environment Agency recommends its usage for European mapping for statistical analysis and display. It is also used in scientific disciplines such as geology for plotting the orientations of lines in three-dimensional space. This plotting is aided by a special kind of graph paper called a Schmidt net.

A cross sectional view of the sphere and a plane tangent to it at S. Each point on the sphere (except the antipode) is projected to the plane along a circular arc centered at the point of tangency between the sphere and plane.

To define the Lambert azimuthal projection, imagine a plane set tangent to the sphere at some point S on the sphere. Let P be any point on the sphere other than the antipode of S. Let d be the distance between S and P in three-dimensional space ( not the distance along the sphere surface). Then the projection sends P to a point P′ on the plane that is a distance d from S.

To make this more precise, there is a unique circle centered at S, passing through P, and perpendicular to the plane. It intersects the plane in two points; let P′ be the one that is closer to P. This is the projected point. See the figure. The antipode of S is excluded from the projection because the required circle is not unique. The case of S is degenerate; S is projected to itself, along a circle of radius 0.

Explicit formulas are required for carrying out the projection on a computer. Consider the projection centered at S = (0, 0, −1) on the unit sphere, which is the set of points in three-dimensional space R ³ such that . In Cartesian coordinates on the sphere and on the plane, the projection and its inverse are then described by

In spherical coordinates on the sphere (with ψ the colatitude and θ the longitude) and polar coordinates on the disk, the map and its inverse are given by

In cylindrical coordinates on the sphere and polar coordinates on the plane, the map and its inverse are given by

The projection can be centered at other points, and defined on spheres of radius other than 1, using similar formulas.

As defined in the preceding section, the Lambert azimuthal projection of the unit sphere is undefined at (0, 0, 1). It sends the rest of the sphere to the open disk of radius 2 centered at the origin (0, 0) in the plane. It sends the point (0, 0, −1) to (0, 0), the equator z = 0 to the circle of radius centered at (0, 0), and the lower hemisphere z < 0 to the open disk contained in that circle.

The projection is a diffeomorphism (a bijection that is infinitely differentiable in both directions) between the sphere (minus (0, 0, 1)) and the open disk of radius 2. It is an area-preserving (equal-area) map, which can be seen by computing the area element of the sphere when parametrized by the inverse of the projection. In Cartesian coordinates it is

This means that measuring the area of a region on the sphere is tantamount to measuring the area of the corresponding region on the disk.

On the other hand, the projection does not preserve angular relationships among curves on the sphere. No mapping between a portion of a sphere and the plane can preserve both angles and areas. (If one did, then it would be a local isometry and would preserve Gaussian curvature; but the sphere and disk have different curvatures, so this is impossible.) This fact, that flat pictures cannot perfectly represent regions of spheres, is the fundamental problem of cartography.

As a consequence, regions on the sphere may be projected to the plane with greatly distorted shapes. This distortion is particularly dramatic far away from the center of the projection (0, 0, −1). In practice the projection is often restricted to the hemisphere centered at that point; the other hemisphere can be mapped separately, using a second projection centered at the antipode.

The Lambert azimuthal projection was originally conceived as an equal-area map projection. It is now also used in disciplines such as geology to plot directional data, as follows.

A direction in three-dimensional space corresponds to a line through the origin. The set of all such lines is itself a space, called the real projective plane in mathematics. Every line through the origin intersects the unit sphere in exactly two points, one of which is on the lower hemisphere z ≤ 0. (Horizontal lines intersect the equator z = 0 in two antipodal points. It is understood that antipodal points on the equator represent a single line. See quotient topology.) Hence the directions in three-dimensional space correspond (almost perfectly) to points on the lower hemisphere. The hemisphere can then be plotted as a disk of radius using the Lambert azimuthal projection.

Thus the Lambert azimuthal projection lets us plot directions as points in a disk. Due to the equal-area property of the projection, one can integrate over regions of the real projective plane (the space of directions) by integrating over the corresponding regions on the disk. This is useful for statistical analysis of directional data, including random rigid rotation.

Not only lines but also planes through the origin can be plotted with the Lambert azimuthal projection. A plane intersects the hemisphere in a circular arc, called the trace of the plane, which projects down to a curve (typically non-circular) in the disk. One can plot this curve, or one can alternatively replace the plane with the line perpendicular to it, called the pole, and plot that line instead. When many planes are being plotted together, plotting poles instead of traces produces a less cluttered plot.

Researchers in structural geology use the Lambert azimuthal projection to plot crystallographic axes and faces, lineation and foliation in rocks, slickensides in faults, and other linear and planar features. In this context the projection is called the equal-area hemispherical projection. There is also an equal-angle hemispherical projection defined by stereographic projection.

The discussion here has emphasized an inside-out view of the lower hemisphere z ≤ 0 (as might be seen in a star chart), but some disciplines (such as cartography) prefer an outside-in view of the upper hemisphere z ≥ 0. Indeed, any hemisphere can be used to record the lines through the origin in three-dimensional space.

Animation of a Lambert projection. Each grid cell maintains its area throughout the transformation. In this animation, points on the equator remain always on the plane.

In this animated Lambert projection, the south pole is held fixed.

Let be two parameters for which and . Let be a "time" parameter (equal to the height, or vertical thickness, of the shell in the animation). If a uniform rectilinear grid is drawn in space, then any point in this grid is transformed to a point on a spherical shell of height according to the mapping

where . Each frame in the animation corresponds to a parametric plot of the deformed grid at a fixed value of the shell height (ranging from 0 to 2). Physically, is the stretch (deformed length divided by initial length) of infinitesimal line line segments. This mapping can be converted to one that keeps the south pole fixed by instead using

Regardless of the values of , the Jacobian of this mapping is everywhere equal to 1, showing that it is indeed an equal area mapping throughout the animation. This generalized mapping includes the Lambert projection as a special case when .

Application: this mapping can assist in explaining the meaning of a Lambert projection by showing it to "peel open" the sphere at a pole, morphing it to a disk without changing area enclosed by grid cells.

Borradaile, Graham J. (2003). Statistics of Earth science data. Berlin: Springer-Verlag. ISBN 3-540-43603-0.

Do Carmo; Manfredo P. (1976). Differential geometry of curves and surfaces. Englewood Cliffs, New Jersey: Prentice Hall. ISBN 0-13-212589-7.

Hobbs, Bruce E., Means, Winthrop D., and Williams, Paul F. (1976). An outline of structural geology. New York: John Wiley & Sons, Inc. ISBN 0-471-40156-0.

Ramsay, John G. (1967). Folding and fracturing of rocks. New York: McGraw-Hill.

Spivak, Michael (1999). A comprehensive introduction to differential geometry. Houston, Texas: Publish or Perish. ISBN 0-914098-70-5.

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Lambert azimuthal equal-area projection - Wikipedia

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The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772.

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The Robinson projection is a compromise map projection developed to display a pleasing appearance. It distorts size and shape to some degree to minimize overall distortion. This makes it useful for general world maps, but not for purposes that require accurate measurements.

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The Robinson is popular for world maps and atlases because it provides a balanced visual representation of the Earth, making it effective for educational purposes where a general view of the world is needed.

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From Wikipedia, the free encyclopedia

Robinson projection of the world

The Robinson projection with Tissot's indicatrix of deformation

Map of the world created by the Central Intelligence Agency, with standard parallels 38°N and 38°S

The Robinson projection is a map projection of a world map that 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 projection was devised by Arthur H. Robinson in 1963 in response to an appeal from the Rand McNally company, which has used the projection in general-purpose world maps since that time. Robinson published details of the projection's construction in 1974. The National Geographic Society (NGS) began using the Robinson projection for general-purpose world maps in 1988, replacing the Van der Grinten projection. In 1998, NGS abandoned the Robinson projection for that use in favor of the Winkel tripel projection, as the latter "reduces the distortion of land masses as they near the poles".

The Robinson projection is neither equal-area nor conformal, abandoning both for a compromise. The creator felt that this produced a better overall view than could be achieved by adhering to either. The meridians curve gently, avoiding extremes, but thereby stretch the poles into long lines instead of leaving them as points.

Hence, distortion close to the poles is severe, but quickly declines to moderate levels moving away from them. The straight parallels imply severe angular distortion at the high latitudes toward the outer edges of the map - a fault inherent in any pseudocylindrical projection. However, at the time it was developed, the projection effectively met Rand McNally's goal to produce appealing depictions of the entire world.

I decided to go about it backwards. ... I started with a kind of artistic approach. I visualized the best-looking shapes and sizes. I worked with the variables until it got to the point where, if I changed one of them, it didn't get any better. Then I figured out the mathematical formula to produce that effect. Most mapmakers start with the mathematics.

The table is indexed by latitude at 5-degree intervals; intermediate values are calculated using interpolation. Robinson did not specify any particular interpolation method, but it is reported that others used either Aitken interpolation (with polynomials of unknown degrees) or cubic splines while analyzing area deformation on the Robinson projection. The X column is the ratio of the length of the parallel to the length of the equator; the Y column can be multiplied by 0.2536 to obtain the ratio of the distance of that parallel from the equator to the length of the equator.

Coordinates of points on a map are computed as follows:

where R is the radius of the globe at the scale of the map, λ is the longitude of the point to plot, and λ ₀ is the central meridian chosen for the map (both λ and λ ₀ are expressed in radians).

Simple consequences of these formulas are:

With x computed as constant multiplier to the meridian across the entire parallel, meridians of longitude are thus equally spaced along the parallel.

With y having no dependency on longitude, parallels are straight horizontal lines.

The Central Intelligence Agency World Factbook uses the Robinson projection in its political and physical world maps.

The European Centre for Disease Prevention and Control recommends using the Robinson projection for mapping the whole world.

1. ^ John Noble Wilford (October 25, 1988). "The Impossible Quest for the Perfect Map". The New York Times.

2. ^ Snyder, John P. (1993). Flattening the Earth: 2000 Years of Map Projections. University of Chicago Press. p. 214. ISBN 0226767469.

3. ^ "National Geographic Maps - Wall Maps - World Classic (Enlarged)". National Geographic Society. This map features the Winkel Tripel projection to reduce distortion of land masses as they near the poles.

4. ^ "Selecting a Map Projection". National Geographic Society.

5. ^ Myrna Oliver (November 17, 2004). "Arthur H. Robinson, 89; Cartographer Hailed for Map's Elliptical Design". Los Angeles Times.

6. ^ New York Times News Service (November 16, 2004). "Arthur H. Robinson, 89 Geographer improved world map". Chicago Tribune.

7. ^ Ipbuker, C. (July 2005). "A Computational Approach to the Robinson Projection". Survey Review. 38 (297): 204-217. doi:10.1179/sre.2005.38.297.204. S2CID 123437786.

8. ^ "Table for Constructing the Robinson Projection". RadicalCartography.net.

9. ^ Snyder, John P.; Voxland, Philip M. (1989). An Album of Map Projections (PDF). U.S. Geological Survey Professional Paper 1453. Washington, D.C.: U.S. Government Printing Office. pp. 82-83, 222-223. doi:10.3133/pp1453.

10. ^ Richardson, Robert T. (1989). "Area deformation on the Robinson projection". The American Cartographer. 16 (4): 294-296. doi:10.1559/152304089783813936.

11. From the formulas below, this can be calculated as ^ .

12. ^ European Centre for Disease Prevention and Control. (2018). Guidelines for presentation of surveillance data: tables graphs maps. LU: Publications Office. doi:10.2900/452488.

Arthur H. Robinson (1974). "A New Map Projection: Its Development and Characteristics". In: International Yearbook of Cartography. Vol 14, 1974, pp. 145-155.

John B. Garver Jr. (1988). "New Perspective on the World". In: National Geographic, December 1988, pp. 911-913.

John P. Snyder (1993). Flattening The Earth-2000 Years of Map Projections, The University of Chicago Press. pp. 214-216.

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From Wikipedia, the free encyclopedia The Robinson projection is a map projection of a world map that 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.

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The Winkel Tripel projection, designed by German cartographer Oswald Winkle, aims to minimize three kinds (hence: tripel) of distortion: area, direction, and distance. It provides a compromise between all, distorting them to balance the projection and is commonly used by National Geographic.

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Used by various educational and reference institutions, including National Geographic, for a balanced representation that renders the world view naturally with minimal distortion, especially favored for global mapping.

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This slideshow will give you basic information about some of the most common map projections!

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A globe represents Earth with true-to-scale geographical features, providing a realistic perspective. It's the only perfectly accurate map. ALL flat maps have one or more of these types of distortion: size, shape, distance, or direction.

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A map projection is simply a way to show the 3 dimensional earth in 2 dimensions (flat). There are hundreds of different map projections, though none of them are perfect. Geographers choose the best map projection based on their needs.

Here are the five that you will commonly see in 8th grade World Geography:

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In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane.

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 the characterization of the distortions. There is no limit to the number of possible map projections. Projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds. However, "map projection" refers specifically to a cartographic projection.

Despite the name's literal meaning, projection is not limited to perspective projections, such as those resulting from casting a shadow on a screen, or the rectilinear image produced by a pinhole camera on a flat film plate. Rather, any mathematical function that transforms coordinates from the curved surface distinctly and smoothly to the plane is a projection. Few projections in practical use are perspective.[citation needed ]

Most of this article assumes that the surface to be mapped is that of a sphere. The Earth and other large celestial bodies are generally better modeled as oblate spheroids, whereas small objects such as asteroids often have irregular shapes. The surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid. Therefore, more generally, a map projection is any method of flattening a continuous curved surface onto a plane.[citation needed ]

A model globe does not distort surface relationships the way maps do, but maps can be more useful 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 be measured to find properties of the region 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 maps motivate the development of map projections.

The best known map projection is the Mercator projection. Despite its important conformal properties, it has been criticized throughout the twentieth century for enlarging area further from the equator. Equal area map projections such as the Sinusoidal projection and the Gall-Peters projection show the correct sizes of countries relative to each other, but distort angles. The National Geographic Society and most atlases favor map projections that compromise between area and angular distortion, such as the Robinson projection or the Winkel tripel projection

Many properties can be measured on the Earth's surface independent of its geography:

Map projections can be constructed to preserve some of these properties at the expense of others. Because the curved Earth's surface is not isometric to a plane, preservation of shapes inevitably leads to a variable scale and, consequently, non-proportional presentation of areas. Vice versa, 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 many purposes exist for maps, a diversity of projections have been created to suit those purposes.

Another consideration in the configuration of a projection is its compatibility with data sets to be used on the map. Data sets are geographic information; their collection depends on the chosen datum (model) of the Earth. Different datums assign slightly different coordinates to 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 other vast territories, where such differences get shrunk to imperceptibility.

Carl Friedrich Gauss'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 spheroids, ellipsoids and geoids. Since any map projection is a representation of one of those surfaces on a plane, all map projections distort.

The classical way of showing the distortion inherent in a projection is to use Tissot's indicatrix. For a given point, using 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 characterizes 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 for characterizing distortion in projections. Like Tissot's indicatrix, the Goldberg-Gott indicatrix is based on infinitesimals, and depicts flexion and skewness (bending and lopsidedness) distortions.

Rather than the original (enlarged) infinitesimal circle as in Tissot's indicatrix, some visual methods project finite shapes that span a part of the map. For example, a small circle of fixed radius (e.g., 15-degrees angular radius). Sometimes spherical triangles are used.[citation needed ] 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.

The problem of characterizing distortion globally across areas instead of at just a single point is that it 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.

The creation of a map projection involves two steps:

1. Selection of a model for the shape of the Earth or planetary body (usually choosing between a sphere or ellipsoid). Because the Earth's actual shape is irregular, information is lost in this step.

2. Transformation of geographic coordinates (longitude and latitude) to Cartesian (x,y) or polar 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.

A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a developable surface. The cylinder, cone and 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).

The developable surface may also be either or 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 ₀) and a parallel of origin (usually written φ ₀) are often used to define the origin of the map projection.

A globe is the only way to represent the Earth with constant scale throughout the entire map in all directions. A map cannot achieve that property for any area, no matter how small. It can, however, achieve constant scale along specific lines.

Some possible properties are:

The scale depends on location, but not on direction. This is equivalent to preservation of angles, the defining characteristic of a conformal map.

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 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. There are also projections (Maurer's Two-point equidistant projection, Close) where true distances from two points are preserved.

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 in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate ellipsoid. Whether spherical or ellipsoidal, the principles discussed hold without loss of generality.

Selecting a model for a shape of 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. Auxiliary latitudes are often employed in projecting the ellipsoid.

A third model is the geoid, a more complex and accurate representation of Earth's shape coincident with what mean sea level would be if there were no winds, tides, or land. Compared to the best fitting ellipsoid, a geoidal model would change the characterization of important properties such as distance, conformality and equivalence. Therefore, in geoidal projections that preserve such properties, the mapped graticule would deviate from a mapped ellipsoid's graticule. Normally the geoid is not used as an Earth model for projections, however, because Earth's shape is very regular, with the undulation of the geoid amounting to less than 100 m from the ellipsoidal model out of the 6.3 million m Earth radius. For irregular planetary bodies such as asteroids, however, sometimes models analogous to the geoid are used to project maps from.

Other regular solids are sometimes used as generalizations for smaller bodies' geoidal equivalent. For example, Io is better modeled by triaxial ellipsoid or prolated spheroid with small eccentricities. Haumea's shape is a Jacobi ellipsoid, with its major axis twice as long as its minor and with its middle axis one and half times as long as its minor. See map projection of the triaxial ellipsoid for further information.

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), conic (e.g. Albers), and 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, 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 ( 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

Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal.

The three developable surfaces (plane, cylinder, cone) provide useful models for understanding, describing, and developing map projections. However, these models are limited in two fundamental ways. For one thing, most world projections in use do not fall into any of those categories. For another thing, even most projections that do fall into those categories are not naturally attainable through physical projection. As L.P. Lee notes,

No reference has been made in the above definitions to cylinders, cones or planes. The projections are termed cylindric or conic because they can be regarded as developed on a cylinder or a cone, as the case may be, but it is as well to dispense with picturing cylinders and cones, since they have given rise to much misunderstanding. Particularly is this so with regard to the conic projections with two standard parallels: they may be regarded as developed on cones, but they are cones which bear no simple relationship to the sphere. In reality, cylinders and cones provide us with convenient descriptive terms, but little else.

Lee's objection refers to the way the terms cylindrical, conic, and planar (azimuthal) have been abstracted in the field of map projections. If maps were projected as in light shining through a globe onto a developable surface, then the spacing of parallels would follow a very limited set of possibilities. Such a cylindrical projection (for example) is one which:

1. Is rectangular;

2. Has straight vertical meridians, spaced evenly;

3. Has straight parallels symmetrically placed about the equator;

4. 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.

A normal cylindrical projection is any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude (parallels) are mapped to horizontal lines.

The mapping of meridians to vertical lines can be visualized by imagining a cylinder whose axis coincides with the Earth's axis of rotation. This cylinder is wrapped around the Earth, projected onto, and then unrolled.

By the geometry of their construction, cylindrical projections stretch distances east-west. The amount of stretch is the same at any chosen latitude on all cylindrical projections, and is given by the secant of the latitude as a multiple of the equator's scale. The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude 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; this distorts areas excessively in high latitudes (see also transverse Mercator).

North-south stretching grows with latitude faster than east-west stretching (sec² 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 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 lines -a pair of identical latitudes of opposite sign (or else the equator) at which the east-west scale matches the north-south-scale.

Normal cylindrical projections map the whole Earth as a finite rectangle, except in the first two cases, where the rectangle stretches infinitely tall while retaining constant width.

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, 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, which in its most common forms represents each meridian as two straight line segments, one from each pole to the equator.

The HEALPix projection combines an equal-area cylindrical projection in equatorial regions with the Collignon projection in polar areas.

The term "conic projection" is used to refer to any projection in which meridians are mapped to equally spaced lines radiating out from the apex and circles of latitude (parallels) are mapped to circular arcs centered on the apex.

When making a conic map, the map maker arbitrarily picks two standard parallels. Those standard parallels may be visualized as secant lines where 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.

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.

Azimuthal (projections onto a plane)

Azimuthal projections have the property that directions from a central point are preserved and therefore great circles through the central point are represented by straight lines on the map. These projections also have radial symmetry in the scales and hence in the distortions: map distances from the central point are computed by a function r( d) of the true distance d, independent of the angle; correspondingly, circles with the central point as center are mapped into circles which have as center the central point on the map.

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

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 point of perspective (along an infinite line through the tangent point and the tangent point's antipode) onto the plane:

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 ; so that even just a hemisphere is already infinite in extent.

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 . Can display up to a hemisphere on a finite circle. Photographs of Earth from far enough away, such as the Moon, 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) give this perspective. It is a generalization of near-sided perspective projection, allowing tilt.

The stereographic projection, 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/(2R cos² ). 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 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)

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.

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:

Equal-area maps preserve area measure, generally distorting shapes in order to do that. Equal-area maps are also called equivalent or authalic. These are some projections that preserve area:

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.

Great circles are displayed as straight lines:

Direction to a fixed location B (the bearing at the starting location A of the shortest route) corresponds to the direction on the map from A to B:

Compromise projections 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:

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 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 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, developed for navigational purposes, has often been used in world maps where other projections would have been more appropriate. This problem has long been recognized even outside professional circles. For example, a 1943 New York Times editorial states:

The time has come to discard [the Mercator] for something that represents the continents and directions less deceptively ... Although its usage ... has diminished ... it is still highly popular as a wall map apparently in part because, as a rectangular map, it fills a rectangular wall space with more map, and clearly because its familiarity breeds more popularity.

A controversy in the 1980s over the Peters map motivated the American Cartographic Association (now Cartography and Geographic Information Society) to produce a series of booklets (including Which Map Is Best) 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.

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

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In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane.

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Now that you have discovered information about the main map projections, use the test yourself questions at the bottom to see if you can identify which map to use in each scenario.