1. Group 4

2. 1. Introduction 2. Map projections 3. Map transformations and distortions 3.1 Conformal mapping 3.2 Equal-area mapping 3.3 Equidistance mapping 3.4 Azimuthal mapping 4. Types of map projections 4.1 Azimuthal Projection 4.2 Conic Projection 4.3 Cylindrical projection 5. Conclusion 6. References

3.  Why use a map instead of a globe of Earth?  A globe is not compact and easy to handle  To show any significant part of the world in detail, an extremely large globe is needed  Measuring distances, directions and area, as well as plotting locations and routes on the curved surface of a globe is also much more difficult than on a map  Flat maps are inexpensively produced in large quantities and are easy to handle and store compared to globes.  Simpler to measure distances, plot paths and make practical use of maps, provided the proper projection is used.  The 3D globe is translated to a 2D map representation using map projection techniques.

4.  A map projection is a systematic method by which the curved surface of the earth with its lines of longitude and latitude, is represented on the flat surface of a map.  Map projection process can be considered to consists of three alterations as shown in Figure 1. The three stages are as follow:  Earth’s real shape (geoid), is represented by an ellipsoid of reference  Cartographers reduce this ellipsoid model to a globe reduced to the size (scale) chosen for the flat map (reference globe)  Globe’s surface is mathematically transformed, point by point, onto a flat surface (Robinson et al., 1995)  This process also includes the transformation of other map features such as mountains, coastlines, etc.  As a result such map features may be distorted.

5. Figure 1: The map projection process (Source: Dent, 1999)

6.  When transforming from the spherical surface to a flat surface, some distortion occurs that cannot be completely eliminated.  Impossible to transform the globe to a map without any distortion errors.  All maps contain errors because of the transformation process.  Distortions and their consequences for the appearance of the map vary with scale.  Mapping small areas (large-scale maps), distortion is not a major problem.  As map area increases to subcontinental or continental areas, distortions become a significant problem.  At such scales, alterations of area, shape, distance and direction occur.

7.  It is possible to retain the property of angular relations on a map projection.  Conformal/orthomorphic projection means that angles are preserved around points and that the shapes of small areas are preserved.  Both conformal and orthomorphic imply ‘correct form or shape’.  Latitudes and longitudes intersect at right angles, and the scale is the same in all directions about a point.  This attribute of shape does not apply to regions of any significant size.  The retaining of angles is limited to directions at points and does not necessarily apply to directions between distant points on the projection.

8.  It is also possible on a map projection to retain representation of areas so that all regions will be shown in correct relative size.  Equal-area projection means that area relationships of all parts of the globe are maintained.  Linear or distance distortion often occurs in such projections.  The intersections of latitudes and longitudes are not at right angles.  Impossible for one projection to maintain both equivalency and conformity therefore on equal-area projections the shape is often skewed.  The right-angle crossing of latitudes and longitudes is lost and this results in the distortion of shapes.

9.  Correct distance relationships require that the length of a straight line between two points on a map represents the correct great circle distance between the same points on the earth.  Equidistance projection means that the great circle distances are preserved.  The path of a great circle is the shortest distance between two points on the Earth’s surface.  There are certain limitations - the distance can be held true from one to all other points, or from a few points to others.  Impossible to hold the true distance from all points to all other points.  The distance property is never global.  Scale will be uniform along the lines whose distances are true.

10.  If correct direction is retained, a straight line drawn between two points on the maps shows the great circle route and azimuth between points.  Azimuthal projection shows true directions from one central point to all other points.  An azimuth is defined by the angle formed at the starting point of a straight line in relation to a baseline, often a line of longitude (Campbell, 2001).  The angle is usually measured in a clockwise direction, starting from north.  A more familiar definition of an azimuth angle is the compass bearing, relative to true north of a point on the horizon directly beneath an observed object.  Directions or azimuths from points other than the central point to other points are not accurate.

11.  The orientation of a projection surface may be changed as desired (Figure 2). Figure 2: Projection family (Source: Campbell, 2001)

12.  All azimuthal projections are “projected” on a plane that may be centred to the sphere anywhere.  A line perpendicular to the plane at the centre point of the projection will necessarily pass through the centre of the sphere.  It is symmetrical around the chosen centre.  The variation of the SF in all cases changes from the centre, at the same rate in every direction.  If the plane is made angular to the sphere, there is no deformation of any kind in the centre; if it is made secant the deformation will be least along the small circle.  All the great circles passing through the centre will be straight lines and show the correct azimuths, from and to the centre in relation to any point.

13.  Note that all azimuths (direction) from and to the centre are correct on an azimuthal projection.  There are various azimuthal projections possible, only five are well known: Lamberts equal area, azimuthal equidistance, Orthographic and the Gnomonic  An azimuth is defined by the angle formed at the starting point of a straight line in relation to a baseline, often a line of longitude (Campbell, 2001).

14.  One of the most popular projections used.  Accurately represents area, but it does not accurately represent angles.  Only the centre point is free of distortion, but distortion is moderate within 90° of this point.  Scale is true along the standard parallels, smaller between them, and larger outside them.  Area distortion is also relatively small between and near the standard parallels.  Useful for mid-latitude regions which are elongate in the east-west direction.

15.  Advantages:  shows areas correctly  well suited to mapping regions that do not have any large difference between their north- south and east-west extent  well suited to showing wind and ocean currents  suitable for mid-latitude areas  good for studying closely at a small area on the map  directions are correct from the centre point of the projection  Disadvantages:  not suitable for a world map, as distortions become extreme for a map of the entire Earth  Shapes become ‘bent’ toward the edges of the map.

16.  Most simple conic projection is tangent to the globe along a line of latitude, called the standard parallel.  Meridians are projected onto the conical surface, meeting at the point of the cone.  Parallel lines of latitude are projected onto the cone as rings.  Cone is ‘cut’ along any meridian to produce the final conic projection  Final projection has straight converging lines for meridians and concentric circular arcs for parallels.  The meridian opposite the cut line becomes the central meridian.  Distortion increases away from the standard parallel.  Cutting off the top of the cone produces a more accurate projection.

17. Figure 3: Conic Projection (Source: Kennedy, 2000)

18.  In this normal application there are two standard parallels.  The parallels are concentric circular arcs with equally-spaced meridians intersecting them at right angles.  Parallels are more widely spaced between the standard parallels, and more closely spaced outside them.  Although this projection is usually used with two standard parallels, it can be used with one.  To maintain equal area, scale variations along the meridians show a reciprocal pattern; the increase in east-west scale outside the standard parallels is balanced by a decrease in North-South scale.

19.  Advantages:  Well suited for large countries or other areas of greater east-west extent than north-south extent  Suitable for mid-latitude areas  Useful for areas that have nearly equal east-west and north-south dimensions  Excellent projection for studying geographical distributions (Robinson et al., 1995).  Disadvantages:  Used only for a single hemisphere;  Shapes become ‘squashed’ sideways;  Not suitable for world maps; and  With the Lambert’s projection, one pole is a point while the other pole is a curved line.

20.  Figure 4 is an example of Alber’s conic projection rendered with standard parallels 60°N and 30° N.  Figure 5 is an example of Lambert’s conical equal-area projection. The standard parallels are 90°n and the equator. Figure 4: Albers’ conical equal-area projection (Source: Furuti, 1997) Figure 5: Lambert’s conical equal-area projection (Source: Furuti, 1997)

21.  Parallels and meridians intersect at right angles (as in any conformal projection).  Areas are inaccurate in conformal projections.  Lambert's conical is also widely used for topographic maps.  It is adapted in France and recommended to the European Commission for conformal pan-European mapping at scales smaller or equal to 1:500,000.

22.  A cylindrical projection can be imagined as a cylinder that has been wrapped around a globe at the equator.  Cylindrical map projections are not so simply constructed.  Equator is usually its line of tangency.  Meridians are geometrically projected onto the cylindrical surface, parallels are mathematically projected.  This produces graticular angles of 90 degrees.  Cylinder is ‘cut’ along any meridian to produce the final cylindrical projection.  Meridians are equally spaced, while the spacing between parallel lines of latitude increases toward the poles.

23. Figure 6: Cylindrical projection (Source: Kennedy, 2000)

24.  Cylindrical conformal map projection and may be the most famous of all projections.  Designed as an aid to navigators since straight lines on the Mercator projection are rhumb lines.  These are lines of constant compass bearing or true direction.  Not ideal for a world map, but due to its rectangular grid and shape it is often published in atlases, wall maps, books and newspapers.  Extreme area distortion especially in the polar regions.  Unsuitable for general maps of large areas as it presents a highly misleading view of the world.

25.  Points are plotted using mathematical tables to determine the progressively increasing distances of the parallels from the equator.  For accurate conformal representation of the shapes of geographical areas in the higher latitudes, intervals between parallels must increase in a prescribed manner.  Accuracy of shape is maintained at the expense of an increasing distortion of size as the poles are approached.  Distances are true only along the equator, but are reasonably correct within 15° of the equator.  Poles are not shown on a Mercator map.

26.  Advantages:  shows shapes correctly  parallels and meridians are straight lines and meet at right angles  excellent to use for navigational purposes as a straight line drawn on this map shows constant bearing  distances near the equator are accurate  Disadvantages:  scale is not constant throughout the map  extreme distortions towards the poles  the poles cannot be shown on the map  not suitable for a world map

27.  Based on a spherical earth, first presented by Lambert in 1772.  An elliptical form that uses an ellipsoidal model of the Earth, was later presented by Gauss in 1822 and further analyzed by Kruger in the early 20th century  An adaptation of the Mercator projection and is also cylindrical and conformal.  The cylinder is rotated 90° (transverse) relative to the equator.  The projected surface is aligned with a ‘central’ meridian rather than the equator.  Distances are true only along the central meridian, but distances, directions, shapes and areas are reasonably accurate within 15° of the central meridian.

28.  As the map is conformal, shapes and angles within any small area are essentially true.  The graticule spacing increases away from the central meridian.  The equator is straight and the other parallels are complex curves concave toward the nearest pole.  The Gauss conform projection is used on all South African 1:10 000, 1:50 000 and 1:250 000 scale maps and images  Most South African maps are within 1° of longitude of the central meridian, thus the errors on the map will be less than 1 part in 6 000.

29. Figure 7: The Mercator projection (Source: Furuti, 1995) Figure 8: The transverse Mercator projection (Source: Furuti, 1995)

30.  A map projection is the systematic arrangement of the Earth’s latitudes and longitudes onto a plane surface.  Map projection process transforms globe of earth to a flat surface (2D map).  All projections have distortions.  Commonly used are Mercator projection, Guass conformal projection and Lambert’s azimuthal equal-area projection.

31.  Campbell, J. 2001: Map use and analysis. (4th Ed.). New York: McGraw-Hill.  Carter, J. 1997: The South African Gauss Conform map projection system. http://www.dwaf.gov.za/IWQS/gauss/gauss.html [Accessed 6 May 2008].  Clarke, K.C. 2003: Getting started with geographical information systems. (4th Ed.). Upper Saddle River: Prentice Hall.  Coetzee, R. et al. 1998: Map analysis and aerial photography. Pretoria: Vista University.  Dent, B.D. 1999: Cartography: thematic map design. (5th Ed.). Boston: WCB/McGraw-Hill.  Furuti, C.A. 1997: Map projections. www.prognos.com/furuti [Accessed 6 May 2008].  Liebenberg, E.C. 1986: Techniques for geographers. Durban: Butterworths.  Robinson, A.H. et al. 1995: Elements of cartography. (6th Ed.). New York: John Wiley & sons.