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Map Projection

To produce a map of the world in a convenient way we make use of map projections. A map projection is any transformation between the curved reference surface of the earth and the flat plane of the map.

We can as well define a map projection as a set of equations which allows us to transform a set of Ellipsoidal Geographic Coordinates representing positions on the reference surface of the earth to a set of Cartesian Coordinates (x, y) representing positions on the two-dimensional surface of the map (see figure above).

Map projection equations have a number of parameters such as:

  1. Radius of the sphere (R) or equatorial (a) and polar radius (b) of the reference ellipsoid
  2. Geodetic datum
  3. Origin of the coordinate system
  4. False easting and northing
  5. Central meridian, standard parallels or centre of projection
  6. Scale factor at the central meridian or standard parallels

Information about the projection parameters is required to define a countries spatial reference system.

Scale distortions on a Map

The transformation from the curved reference surface of the earth to the flat plane of the map is never completely successful. Look at the diagram below. By flattening the curved surface of the sphere onto the map the curved surface is stretched in a non-uniform manner.

It appears that it is impossible to project the Earth on a flat piece of paper without any location distortions, therefore without any scale distortions.

The distortions increase as the distance from the central point of the projection increases. Placing the map plane so that it intersects the reference surface will reduce and mean out the scale errors.

Since no map projection maintains correct scale throughout the map, it may be important to know the extent to which the scale varies on a map.

On a world map, the distortions 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. Some maps have a scale reduction diagram, which indicates 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 also be shown on a map by a scale factor. A scale factor smaller than 1 indicates that the scale is smaller than the nominal scale, the scale given on the map. A scale factor larger than one indicates that the scale is larger than the nominal scale.

Note Scale distortions can remain within certain limits by choosing the right map projection.

Properties of Map Projections

The following properties would be present on a map projection without any scale distortions:

  • Areas are everywhere correctly represented.
  • All distances are correctly represented.
  • All directions on the map are the same as on Earth.
  • All angles are correctly represented.
  • The shape of any area is correctly represented.

It is, unfortunately, impossible to have all these properties together in one map projection.

An equivalent map projection, also known as an equal-area map projection, correctly represents areas sizes of the sphere on the map. When this type of projection is used for small-scale maps showing large regions, the distortion of angles and shapes is considerable. The Lambert cylindrical equal-area projection is an example of an equivalent map projection.

The Lambert cylindrical equal-area projection as an example of an equivalent, cylindrical projection.

A conformal map projection represents angles and shapes correctly at infinitely small locations. Shapes and angles are only slightly distorted, as the region becomes larger. At any point the scale is the same in every direction. On a conformal map projection meridians and parallels intersect at right angles (e.g. Mercator projection).

The Mercator as an example of a conformal, cylindrical projection.

On a minimum-error map projection the scale errors everywhere on the map as a whole are a minimum value (e.g. the Airy projection).

On the Mercator projection, all rhumb lines, or lines of constant direction, are shown as straight lines. A compass course or a compass bearing plotted on to a Mercator projection is a straight line, even though the shortest distance between two points on a Mercator projection - the great circle path - is not a straight line.

All rhumb lines, or lines of constant direction, are shown as straight lines.

On the Gnomonic projection, all great circle paths are:

  • the shortest routes between points on a sphere.
  • shown as straight lines.

All great circles are:

  • the shortest routes between points on a sphere
  • shown as straight lines

The classification of Map Projections

Next to their property (equivalence, equidistance, conformality), map projections can be discribed in terms of their class (azimuthal, cylindrical, conical) and aspect (normal, transverse, oblique).

The three classes of map projections are cylindrical, conical and azimuthal.The earth's surface projected on a map wrapped around the globe as a cylinder produces the cylindrical map projection. Projected on a map formed into a cone gives a conical map projection. When projected on a planar map it produces an azimuthal or zenithal map projection.

The three classes of map projections.

Projections can also be described in terms of their aspect: the direction of the projection plane's orientation (whether cylinder, plane or cone) with respect to the globe. The three possible aspects of a map projection are normal, transverse and oblique. In a normal projection, the main orientation of the projection surface is parallel to the earth's axis (as in the second figure below). A transverse projection has its main orientation perpendicular to the earth's axis. Oblique projections are all other, non-parallel and non-perpendicular, cases. The figure below provides two examples. A transverse cylindrical and an oblique conical map projection.

Both are tangent to the reference surface.

The terms polar, oblique and equatorial are also used. In a polar azimuthal projection the projection surface is tangent or secant at the pole. In a equatorial azimuthal or equatorial cylindrical projection, the projection surface is tangent or secant at the equator. In an oblique projection the projection surface is tangent or secant anywhere else.

A map projection can be tangent to the globe, meaning that it is positioned so that the projection surface just touches the globe. Alternatively, it can be secant to the globe, meaning that the projection surface intersects the globe. The figure below provides illustrations.

Three normal secant projections: cylindrical, conical and azimuthal.

A final descriptor may be the name of the inventor of the projection, such as Mercator, Lambert, Robinson, Cassini etc., but these names are not very helpful because sometimes one person invented several projections, or several people have invented the same projection. For example J.H.Lambert described half a dozen projections. Any of these might be called 'Lambert's projection', but each need additional description to be recognized.

It is now possible to describe a certain projection as, for example:

  • 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.

The question may arise here 'Why are there so many map projections? The main reason is that there is no one projection best overall (see section; Selecting a Suitable Map Projection)

The diagram below shows the developable surface of the Lambert conformal conic projection with two standard parallels.

Selecting a suitable 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.

The selection of a map projection has to be made on the basis of:

  • Shape and size of the area.
  • Position of the area.
  • Purpose of the map.

The choice of the class of a map projection should be made on the basis of the shape and size of the geographical area to be mapped. Ideally, the general shape of a geographical area should match with the distortion pattern of a specific projection. For example, if an area is small and approximately circular it is possible to create a map that minimises distortion for that area on the basis of an Azimuthal projection. The Cylindrical projection should be the basis for a large rectangular area and a Conic projection for a triangular area.

The position of the geographical area determines the aspect of a projection. Optimal is when the projection centre coincides with centre of the area, or when the projection plane is located along the main axis of the area to be mapped.

Once the class and aspect of a map projection have been selected, the choice of the property of a map projection has to be made on the basis of the purpose of the map.

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 for topographic mapping.

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 (India).

Suitable equal-area projections for 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, which is nearly conformal. In the polar 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 moderate.

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.

In conclusion, 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 property of the map projection depends on the map purpose.

Nevertheless for each country to use its own projection would make international co-operation in data exchange difficult. There are strong arguments in favour of using an international standard projection for mapping.

Map Projections in common use

There are several hundred of map projections, but only a smaller part is actually used. Most commonly used map projections are:

  • Universal Transverse Mercator (UTM).
  • Transverse Mercator (also known as Gauss-Kruger).
  • Polyconic.
  • Lambert Conformal Conic.
  • Stereographic projection.

These projections and a few other well-known map projections are briefly described and illustrated.

Cylindrical projections

The Mercator projection is a conformal cylindrical projection. Parallels and meridians are straight lines intersecting at right angles, a requirement for conformality. Meridians are equally spaced. The parallel spacing increases with distance from the Equator.

Mercator: conformal cylindrical projection

The ellipses of distortion appear as circles (indicating conformality) but increase in size away from the equator (indicating area distortion). This exaggeration of area as latitude increases makes Greenland appear to be as large as South America when, in fact, it is only a quarter of the size.

The Mercator projection is used for long distance navigation because of the straight rhumb-lines. It is more convenient to steer a rhumb-line course if the extra distance travelled is small. Often and inappropriately used as a world map in atlases and for wall charts. It presents a misleading view of the world because of excessive area distortion towards the poles.

Transverse Mercator Projection

The Transverse Mercator projection is a transverse cylindrical conformal projection based on a transverse cylinder.

Transverse Mercator projection

Versions of the Transverse Mercator Projection are used in many countries as national projection on which the topographic mapping is based. The Transverse Mercator projection is also known as the Gauss-Kruger or Gauss Conformal projection. The figure below shows the World map in Transverse Mercator projection.

The world mapped in the Transverse Mercator projection (at a small scale).

Conic Projections

Three well-known conical projections are the Lambert Conformal Conic projection, the Albers equal-area projection and the Polyconic projection.

The Lambert Conformal Conic projection in normal position is an example of a conic projection.

Polyconic projection

The Polyconic projection is neither conformal nor equal-area. The polyconic projection is projected onto cones tangent to each parallel, so the meridians are curved, not straight.

The Polyconic projection is an example of a conic projection, equidistant along the parallels.

The scale is true along the central meridian and along each parallel. The distortion increases away from the central meridian in East or West direction.

The Polyconic projection is used for early large-scale mapping and in the International Map of the World (1:1,000,000 scale) series for topographic mapping in some countries.

Azimuthal Projections

The five common azimuthal (also known as Zenithal) projections are the Stereographic projection, the Orthographic projection, the Lambert azimuthal equal-area projection, the Gnomonic projection and the azimuthal equidistant projection.

The projection is principally for; Gnomonic, Stereographic and Orthographic projection.

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.

Stereographic projection The Stereographic projection is a conformal azimuthal projection. All meridians and parallels are shown as circular arcs or straight lines. Since the projection is conformal, parallels and meridians intersect at right angles.

In the polar aspect the meridians are equally spaced straight lines, the parallels are unequally spaced circles cantered at the pole. Spacing gradually increases away from the pole.

The transverse (or equatorial) stereographic projection is an example of a conformal azimuthal projection.

The scale is constant along any circle having its centre at the projection centre, but scale increases moderately with distance from the centre. The areas increase with distance from the projection centre. The ellipses of distortion remain circles (indicating conformality).

The Stereographic projection is commonly used in the polar aspect for topographic maps of Polar Regions.

Chart Summery

CHART CHARACTERISTIC

  • Parallels of Latitude
  • Meridians of Longitude
  • Angle between Parallels and Meridians
  • Straight line crossing Meridians
  • Great Circle
  • Rhumb Line
  • Distance Scale
  • Origin of Projectors
  • Distortion of Shapes and Areas
  • Method of Production
  • Navigational Uses
  • Conformality

LAMBERT CONFORMAL

  • Arcs of Concentric Circles nearly equally spaced
  • Straight lines converging at the pole
  • 90 Degrees
  • Variable angle (Approximates Great Circle)
  • Approximated by straight line
  • Curved line
  • Nearly constant
  • Centre of sphere
  • Very little
  • Mathematical
  • Dead reckoning map reading and radio aids
  • Conformal

MERCATOR

  • Parallel straight lines unequally spaced
  • Parallel straight lines equally spaced
  • 90 Degrees
  • Constant angle (Rhumb Line)
  • Curved line (except equator and Meridians)
  • Straight line
  • Accurate close to point of projection
  • Centre of sphere
  • Increases away from Equator
  • Mathematical
  • Dead reckoning and Celestial
  • Conformal

POLAR STEREOGRAPHIC

  • Concentric circles unequally spaced
  • Straight lines radiating from the pole
  • 90 Degrees
  • Variable angle (Approximates Great Circle)
  • Approximated by straight line
  • Curved Line
  • Nearly constant except on small scale charts
  • Opposite pole
  • Increases away from pole
  • Graphic or mathematical
  • Polar navigation all types
  • Conformal