Maps are one of the world’s oldest types of document. For quite some time it was thought that our planet was flat, and during those days, a map simply was a miniature representation of a part of the world. Now that we know that the Earth’s surface is curved in a specific way, we know that a map is in fact a flattened representation of some part of the planet. The field of map projections concerns itself with the ways of translating the curved surface of the Earth into a flat map.
4.1 What is a map projection?
A map projection is a mathematically described technique of how to represent the Earth’s curved surface on a flat map. To represent parts of the surface of the Earth on a flat paper map or on a computer screen, the curved horizontal reference surface must be mapped onto the 2D mapping plane. The reference surface for large-scale mapping is usually an oblate ellipsoid, and for small-scale mapping, a sphere. Mapping onto a 2D mapping plane means transforming each point on the reference surface with geographic coordinates (f,l) to a set of Cartesian coordinates (x,y) representing positions on the map plane (figure below).
The actual mapping cannot usually be visualized as a true geometric projection, directly onto the mapping plane as illustrated in the figure above. This is mostly achieved through mapping equations. A forward mapping equation transforms the geographic coordinates (f,l) of a point on the curved reference surface to a set of planar Cartesian coordinates (x,y), representing the position of the same point on the map plane:
The corresponding inverse mapping equation transforms mathematically the planar Cartesian coordinates (x,y) of a point on the map plane to a set of geographic coordinates (f,l) on the curved reference surface:
(f, l) = f (x, y)
Following are two examples of mapping equations for the sphere (equations for the ellipsoid are generally more complex).
Map projection equations have a significant role in projection change (section 5 on coordinate transformations). Interested readers can find an extensive list of mapping equations with numerical examples in 'Map Projections - A Working manual' by J. P. Snyder. A number of equations are also given at World of Mathematics and in the OGP Guidance note 7: Coordinate Conversions and Transformations including Formulas.
4.2 Classification of map projections
Map projections can be described in terms of their:
Based on these discussions, a particular map projection can be classified. An example would be the classification ‘conformal conic projection with two standard parallels’ having the meaning that the projection is a conformal map projection, that the intermediate surface is a cone, and that the cone intersects the ellipsoid (or sphere) along two parallels; i.e. the cone is secant and the cone’s symmetry axis is parallel to the rotation axis. This would amount to the projection of the figure above (conical projection with a secant projection plane). Other examples are:
4.3 Scale distortions on a map
A map projection without distortions would correctly represent shapes, angles, areas, distances and directions, everywhere on the map. Unfortunately, any map projection is associated with scale distortions. There is simply no way to flatten out a piece of ellipsoidal or spherical surface without stretching some parts of the surface more than others (figure below). The amount and which kind of distortions a map will have depends largely - next to size of the area being mapped - on the type of the map projection that has been selected.
Since there is no map projection that maintains correct scale all over the map, it may be important to know the extent to which the scale varies on a map. On a world map, the scale variations are evident where landmasses are wrongly sized or out of shape and the meridians and parallels do not intersect at right angles or are not spaced uniformly. These maps may have a scale reduction diagram to indicate the map scale at different locations, helping the map-reader to become aware of the distortions. On maps at larger scales, maps of countries or even city maps, the distortions are not evident to the eye. However, the map user should be aware of the distortions if he or she computes distances, areas or angles on the basis of measurements taken from these maps.
Scale distortions can be measured and shown on a map by ellipses of distortion. The ellipse of distortion, also known as Tissot's Indicatrix, shows the shape of an infinite small circle with a fixed scale on the Earth as it appears when plotted on the map. Every circle is plotted as circle or an ellipse or, in extreme cases, as a straight line. The size and shape of the ellipse shows how much the scale is changed and in what direction.
The indicatrices on the map in the figure below have a varying degrees of flattening, but the areas of the indicatrices everywhere on the map are the same, which means that areas are represented correctly on the map. The distortion property of the map projection is therefore equal-area (or equivalent).
When the indicatrices are circles everywhere on the map, the angles and consequently shapes (of small areas) are shown correctly on the map. The distortion property of the map projection is therefore conformal (e.g. the Mercator projection). A Java tool for the demonstration of map projections with an option to show Tissot's indicatrices is given through the following external link:
Scale distortions on a map can also be shown by means of a scale factor (ratio of the scale at a given point to the true scale). Scale distortions exist at locations where the scale factor is smaller or larger than 1. E.g. a scale factor at a given point on the map is equal to 0.99960 signifies that 1000 metres on the reference surface of the Earth will actually measure 999.6 metres on the map. This is a contraction of 40 centimetre per kilometre.
The nominal map scale (given map scale) divided by the scale factor will give the actual scale. E.g. a scale factor of 0.99960 at a given point on a map with a nominal scale of 1:10,000 (one to ten thousand) will give a scale of 1:10,004 (10,000 divided by 0.99960) at the given point. This is a smaller scale than the nominal map scale. A scale factor of 2 at a given point on a map with a nominal scale of 1:10M (one to ten million) will give a scale of 1:5M (10 million divided by 2) at the given point. This is a larger scale than the nominal map scale.
Scale distortions for both, tangent and secant map surfaces, are illustrated in the figures below. Distortions increase as the distance from the central point (tangent plane) or closed line(s) of intersection increases.
On a secant map projection - the application of a scale factor of less than 1.0000 to the central point or the central meridian has the effect of making the projection secant - the overall distortions are less than on one that uses a tangent map surface. Most countries have derived there map coordinate system from a projection with a secant map surface for this reason.
4.4 Choosing a map projection
Every map must begin, either consciously or unconsciously, with the choice of a map projection and its parameters. The cartographer's task is to ensure that the right type of projection is used for any particular map. A well chosen map projection takes care that scale distortions remain within certain limits and that map properties match to the purpose of the map.
Generally, normal cylindrical projections are typically used to map the world in its entirety (in particular areas near the equator are shown well). Conical projections are often used to map the different continents (the mid-latitudes regions are shown well), while the polar azimuthal projections may be used to map the polar areas. Transverse and oblique aspects of many projections can be used for most parts of the world, though they are usually more difficult to construct.
In theory, the selection of a map projection for a particular area can be made on the basis of:
In summary, the ideal map projection for any country would either be an azimuthal, cylindrical, or conic projection, depending on the shape of the area, with a secant projection plane located along the main axis of the country or the area of interest. The selected distortion property depends largely on the purpose of the map.
Some map projections have rather special properties. The Mercator projection was originally designed to display accurate compass bearings for sea travel. Any straight line drawn on this projection represents an actual compass bearing. These true direction lines are rhumb lines (or loxodromes). Thus, the route of constant direction between two locations is a always a straight line. For navigation, this is the easiest route to follow, but not necessary the shortes route (figure below).
The gnomonic projection is a useful projection for defining routes of navigation for sea and air travel, because great circles - the shortest routes between points on a sphere - are shown as straight lines. Thus, the shortest route between any two locations is always a straight line. No other projection has this special property. In combination with the Mercator map where all lines of constant direction are shown as straight lines it assist navigators and aviators to determine appropriate courses. Changes in direction for following the shortest route can be determined by plotting the shortest route (great circle or orthodrome) from the Gnomonic map onto the Mercator map (figure above).
In the 15th, 16th and 17th centuries, during the time of great transoceanic voyaging, there was a need for conformal navigation charts. Mercator's projection - conformal cylindrical - met a real need, and is still in use today when a simple, straight course is needed for navigation. Because conformal projections show angles correctly, they are suitable for sea, air, and meteorological charts. This is useful for displaying the flow of oceanic or atmospheric currents, for instance.
For topographic and large-scale maps, conformality and equidistance are important properties. The equidistant property, possible only in a limited sense, however, can be improved by using secant projection planes. The Universal Transverse Mercator (UTM) projection is a conformal cylindrical projection using a secant cylinder so it meets conformality and reasonable equidistance. Other projections currently used for topographic and large-scale maps are the Transverse Mercator (the countries of Argentina, Colombia, Australia, Ghana, S-Africa, Egypt use it) and the Lambert Conformal Conic (in use for France, Spain, Morocco, Algeria). Also in use are the stereographic (the Netherlands) and even non-conformal projections such as Cassini or the Polyconic.
Suitable equal-area projections for thematic and distribution maps include those developed by Lambert, whether azimuthal, cylindrical, or conical. These do, however, have rather noticeable shape distortions. A better projection is the Albers equal-area conic projection with two standard parallels, which is nearly conformal. In the normal aspect, they are excellent for mid-latitude distribution maps and do not contain the noticeable distortions of the Lambert projections.
An equidistant map, in which the scale is correct along a certain direction, is seldom desired. However, an equidistant map is a useful compromise between the conformal and equal-area maps. Shape and area distortions are often reasonably well preserved. An example is the equidistant cylindrical projection (also called Plate Carrée projection), where the meridians are true to scale map (i.e. no distortion in North-South direction).
The projection which best fits a given country is always the minimum-error projection of the selected class. The use of minimum-error projections is however exceptional. Their mathematical theory is difficult and the equidistant projections of the same class will provide a very similar map.
4.5 Map projections in common use
A variety of map projections have been developed, each with its own specfic qualities. Only a limited amount are frequently used. Here are some well-known projections described and illustrated. They are grouped into cylindrical, conical and azimuthal projections.
4.5.1 Cylindrical projections
Probably one of the best known cylindrical projection is Mercator's cylindrical projection. The transverse case and occasionally the oblique case of the Mercator projection are used in several countries for topographic mapping purposes. The Transverse Mercator and Univeral Transverse Mercator (UTM) projection are the best known examples. Two other well-known normal cylindrical projections are the equidistant cylindrical (or Plate Carrée) projection and Lambert's cylindrical equal-area projection. Normal cylindrical projections are typically used to map the world in its entirety (in particular areas near the equator are shown well).
4.5.2 Conic projections
Four well-known normal conical projections are the Lambert conformal conic projection, the simple conic projection, the Albers equal-area projection and the Polyconic projection. They give useful maps of mid-latitudes for countries which have no great extent in latitude.
4.5.3 Azimuthal projections
Azimuthal (or zenithal or planar) projections are made upon a plane tangent (or secant) to the reference surface. All azimuthal projections possess the property of maintaining correct azimuths, or true directions from the centre of the map. In the polar cases, the meridians all radiate out from the pole at their correct angular distance apart. A subdivision may be made into perspective and non-perspective azimuthal projections. In the perspective projections, the actual mapping can be visualized as a true geometric projection, directly onto the mapping plane; illustrations are in the figure below. For the gnomonic projection, the perspective point (like a source of light rays), is the centre of the Earth. For the stereographic this point is the opposite pole to the point of tangency, and for the orthographic the perspective point is an infinite point in space on the opposite side of the Earth. Two well known non-perspective azimuthal projections are the azimuthal equidistant projection (also called Postel projection) and the Lambert azimuthal equal-area projection.
Three perspective azimuthal projections: Gnomonic, stereographic and orthographic (source: ESRI).
4.6 Map projections overview
In summary, a short list of map projections grouped by class:
List of map projections in common use. See also summary tables on the USGS map projections page.
More examples of map projections are given through the following links:
4.7 Main references
R.A. Knippers. Coordinate systems and Map projections.. Non-published educational notes, ITC, Enschede, 1998.
R.A. Knippers. Geometric Aspects of Mapping. Non-published educational notes, ITC, Enschede, 1999.
P. Stefanovic. Georeferencing and Coordinate Transformations. Non-published educational notes. ITC, Enschede, 1996.