4.0 Introduction 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 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 x,y), representing the position of the same point on the map plane:
The corresponding x,y) of a point on the map plane to a set of geographic coordinates (f,l) on the curved reference surface:(
Following are two examples of mapping equations for the sphere (equations for the ellipsoid are generally more complex). - The first example are the mapping equations used for the
projection:*Mercator*
- The second example is the mapping equations used for the north polar
:**stereographic projection**
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 '
4.2 Classification of map projections Map projections can be described in terms of their: (cylindrical, conical or azimuthal),**class**- point of
(tangent or secant),*secancy* (normal, transverse or oblique), and**aspect**- distortion
(equivalent, equidistant or conformal).**property**
Based on these discussions, a particular map projection can be classified. An example would be the classification ‘ - Polar stereographic azimuthal projection with secant projection plane;
- Lambert conformal conic projection with two standard parallels;
- Lambert cylindrical equal-area projection with equidistant equator;
- Transverse Mercator projection with secant projection plane.
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 distortions can be measured and shown on a map
by , 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. Tissot's IndicatrixThe 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: Demonstration of different map projections (Instituto de matematica, Brasil)
Scale distortions on a map can also be shown by means of a The 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: - the
of the area,**shape** - the
(and orientation) of the area, and*location* - the
of the map.*purpose*
In summary, the
Some map projections have rather special properties. The (or rhumb lines). 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).loxodromes
The - 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 great circles) from the Gnomonic map onto the Mercator map (figure above).orthodrome
In the 15th, 16th and 17th centuries, during the time
of great transoceanic voyaging, there was a need for conformal navigation
charts. 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 (the countries of Argentina, Colombia,
Australia, Ghana, S-Africa, Egypt use it) and the Transverse Mercator (in use for France, Spain, Morocco, Algeria). Also in use are the Lambert Conformal Conic (the Netherlands) and even non-conformal projections such as Cassini
or the Polyconic.stereographicSuitable equal-area projections for thematic and distribution maps
include those developed by 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. Albers equal-area conic 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 The projection which best fits a given country is always
the
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 and Transverse Mercator projection are the best known examples. Two other well-known normal cylindrical projections are theUniveral Transverse Mercator (UTM) (or equidistant cylindricalPlate Carrée) projection
and Lambert'sprojection. Normal cylindrical projections are typically used to map the world in its entirety (in particular areas near the equator are shown well).cylindrical equal-area
4.5.2 Conic projections Four well-known normal conical projections are the projection
and the Polyconic projection. They give useful maps of mid-latitudes for countries which have no great extent in latitude.Albers equal-area
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 projection, the perspective point
(like a source of light rays), is the centre of the Earth. For the gnomonic this point is the opposite pole to the point of tangency, and for the stereographic the perspective point is an infinite point in space on the
opposite side of the Earth. Two well known orthographic azimuthal projections
are the non-perspectiveprojection (also called Postel projection) and the Lambert azimuthal equal-area projection.azimuthal equidistant 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:
More examples of map projections are given through the following links: Demonstration of different map projections (Flex projector, ETH Zurich) Demonstration of different map projections (Instituto de matematica, Brasil) Demonstration of different map projections (H. Bottomley) Map projections grouped by use (Radical cartography) Picture gallery of map projections (TU Vienna) Picture gallery of map projections (Nevron) Understanding map projections (ESRI) Map projections for Europe (CRSeu)
Geometric Aspects of Mapping. Non-published
educational notes, ITC, Enschede, 1999.
Georeferencing and Coordinate
Transformations. Non-published educational notes. ITC, Enschede, 1996. |