3 Properties of Map Projections

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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

An equidistant map projection correctly represents distances. An equidistant map projection is

possible only in a limited sense. That is, distances can be shown at the nominal map scale -the

given map scale- only from one or two points to any other point on the map or in certain

directions. If the scale on a map is correct along all meridians, the map is equidistant along the

meridians (e.g. the Plate Carree projection). If the scale on a map is correct along all parallels,

the map is equidistant along the parallels.



The Plate Carree projection as an example of an equidistant, 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

Note A map projection may possess one of the three properties, but can never have all three

properties. It can be proved that conformality and equivalence are mutually exclusive of each

other and that a projection can only be equidistant (true to scale) in certain places or directions.

There are map projections with rather special properties:

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 rumb 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 rumb lines, or lines of constant direction, are shown as straight lines.

On the Gnomonic projection, all great circle paths - the shortest routes between points on a

sphere - are shown as straight lines.



all great circles - the shortest routes between points on a sphere - are shown as straight lines

4.4 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 projections.

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 apects

of a map projection are normal , transverse and oblique . In a normal projection, the mainorientation 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 exampleJ.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 4.5 selecting a suitable map projection )

Activity The diagram below shows the developable surface of the Lambert conformal conic

projection with two standard parallels.



Answer the following questions:

1. Which developable surface is used? 2. Is it a tangential or a secant projection? 3. What is the position of the developable surface?

4. Describe some of the scale distortion characteristics.5. Are areas correctly represented?

4.5 Selecting a suitable Map Projection

Every map must begin, either consciously or unconsciously, with the choice of a map projectionand its parameters. The cartographer's task is to ensure that the right type of projection is usedfor any particular map. A well choosen map projection takes care that scale distortions remainwithin 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 sizeof the geographical area to be mapped. Ideally, the general shape of a geographical areashould match with the distortion pattern of a specific projection. For example, if an area is smalland approximately circular it is possible to create a map that minimises distortion for that area onthe basis of an Azimuthal projection. The Cylindrical projection should be the basis for a largerectangular 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 (see example figure below).

Choice of position and orientation of the projection plane for a map of Alaska

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.

4.6 Map Projections in common use

Several hundreds of map projections have been described, but only a smaller part is actuallyused. Most commonly used map projections are:



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

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

illustrated.

4.6.1 Cylindrical projections

Mercator projection The Mercator projection is a conformal cylindrical projection. Parallels and

meridians are straight lines intersecting at right angles, a requirement for conformality. Meridiansare 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 rumb-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.

The Transverse Mercator projection is based on a transverse cylinder

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)

The Transverse Mercator is the basis for the Universal Transverse Mercator projection, as well as

for the State Plane Coordinate System in some of the states of the U.S.A.

4.6.2 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 of the United States until the

1950's, early coastal charts by the U.S. Coast and Geodetic Survey, early maps in the

International Map of the World (1:1,000,000 scale) series and for topographic mapping in some

countries.

4.6.3 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 (also called Postel ) 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.

The projection principle for the Gnomonic, Stereographic and Orthographic projection



Stereographic projection The Sterographic 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 centered at the pole. Spacing gradually increases away from the pole.

Azimuthal Equidistant

Classification: Azimuthal, Neither conformal nor equal area

Aspects: Polar, oblique and equatorialEarth Shape: SphereTrue Scale at: Center

CentralClindrical

Classification: Cylindrical, Perspective neither conformal nor equal area

Aspects: EquatorialEarth Shape: SphereTrue Scale at: Equator

MercatorClassification: Cylindrical, ConformalAspects: EquatorialEarth Shape: Sphere, EllipsoidTrue Scale at: Equator

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3 Properties of Map Projections