MAP PROJECTIONS

MAP PROJECTIONS

What is meant by Map Projection? A map projection is a system in which locations on the curved surface of the earth are displayed on a flat sheet or surface according to some set of rules.

The Map Projection Process:

(a) The earth’s real shape (Geoid) is thought to represent an ellipsoid through the adoption of the magnitudes of the major and minor semi-axes that best fits the real shape. The datum line is then determined.

(b)For practical measurement purposes, a shere having the same surface area as the ellipsoid is chosen as a standard and its radius calculated

(d)The spherical surface of the earth is then transformed mathematically on to a flat surface

Distorted Properties of the Earth's Surface

1. Angles, (Directions) 2. Relative distances, 3. Areas, and 4. Shapes, become distorted a portion of the earth’s is transformed from the curved surface to a plane. All these properties cannot be kept undistorted in a single projection. Usually in a projection, the distortion in one property will be kept to a minimum while other properties become exaggerated. For example, if we are interested global topographical relationships, we may choose a projection that distorts scale and direction but maintains shape.

Relevance of Map Projections to Map making

A projection provides the locational framework for all thematic maps. Maps are a common source of input data for a GIS and often input maps will be in different projections, requiring transformation of one or all maps to make coordinates compatible. We often need to know distances between places, Areas of countries, states, and parcels of land, Directions of Electronic signals, winds and headings for navigation. All these require flat maps rather that spherical globes. Thus, mathematical functions of projections are needed in mapping. Often maps are used for projects of global or regional scales so a consideration of the effect of the earth's curvature is important. Monitor screens are analogous to a flat sheet of paper thus, need to provide transformations from the curved surface to the plane for displaying data

FIGURES OF THE EARTH

A figure of the earth is a geometrical model used to generate projections; a compromise between the desire for mathematical simplicity and the need for accurate approximation of the earth's shape types in common use. Common figures that have been used to represent the earth include: a) Plane, b) Sphere c) Illipsoid

Plane Assumes the earth is flat (use no projection) used for maps only intended to depict general relationships or for maps of small areas at scales larger than 1:10,000. Planar representation has little effect on accuracy. Planar projections are usually assumed when working with air photos

Sphere Assumes the earth is perfectly spherical. Does not truly represent the earth's shape

Spheroid or ellipsoid of rotation This is the figure created by rotating an ellipse about its minor axis. The spheroid models the fact that the earth's diameter at the equator is greater than the distance between poles, by about 0.3% at global scales, the difference between the sphere and spheroid are small, about equal to the topographic variation on the earth's surface with a line width of 0.5 mm the earth would have to be drawn with a radius of 15 cm before the two models would deviate the difference is unlikely to affect mapping of the globe at scales smaller than 1:10,000,000. The spheroid is still an approximation to the actual shape of the earth. The earth is actually slightly pear shaped, slightly larger in the southern hemisphere, and has other smaller bulges therefore, different spheroids are used in different regions, each chosen to fit the observed datum of each region accurate conversion between latitude and longitude and projected coordinates requires knowledge of the specific figures of the earth that have been used. The actual shape of the earth can now be determined quite accurately by observing satellite orbits. Satellite systems, such as GPS, can determine latitude and longitude at any point on the earth's surface to accuracies of fractions of a second. Thus, it is now possible to observe otherwise invisible errors introduced by the use of an approximate figure for map projections

The Projection Challenge: The Earth is a spheroid, and the best way to represent it is with a globe. This scale model retains all of the desired properties necessary to produce the perfect map: area, distance, direction, and shape are all accurately represented. However, when this spheroid is projected on to a flat map, all these properties cannot be retained simultaneously. In fact, each projection is a compromise, showing some properties accurately, while at the same time, allowing others to be distorted. The extent to which these properties are preserved, provides another method of classifying projections.

Despite the problems related to distortion, all projections do retain one important feature, that of positional accuracy. By transforming the graticule (a gridded reference network of latitude and longitude lines, encompassing the globe) to a map, the spatial relationship between points on both surfaces is maintained.

DEVELOPABLE SURFACES

The most common methods of projection can be conceptually described by imagining the developable surface, which is a surface that can be made flat by cutting it along certain lines and unfolding or unrolling it. The points or lines where a developable surface touches the globe in projecting from the globe are called standard parallels, or points and lines of zero distortion. At these points and lines, the scale is constant and equal to that of the globe, no linear distortion is present if the developable surface touches the globe, the projection is called tangent if the surface cuts into the globe, it is called secant where the surface and the globe intersect, there is no distortion where the surface is outside the globe, objects appear bigger than in reality - scales are greater than 1 where the surface is inside the globe, objects appear smaller than in reality and scales are less than 1

Commonly used developable surfaces are:

1. Plane or Azimuthal: A flat sheet is placed in contact with a globe, and points are projected from the globe to the sheet mathematically, the projection is easily expressed as mappings from latitude and longitude to polar coordinates with the origin located at the point of contact with the paper. Produces planar projections.

2. Cone: The transformation is made to the surface of a cone tangent at a small circle (tangent case) or intersecting at two small circles (secant case) on a globe mathematically, This projection is also expressed as mappings from latitude and longitude to polar coordinates, but with the origin located at the apex of the cone. Produces Conical projections.

3. Cylinder: Developed by transforming the spherical surface to a tangent or secant cylinder. Mathematically, a cylinder wrapped around the equator is expressed with x equal to longitude, and the y coordinates some function of latitude. Produces Cylindrical projections.

Scale Factor and Transformations

In projections, the scale of the reference Globe is called the Principal Scale (PS). To derive the principal Scale we divide the earth's radius by the radius of the globe. On the reference globe therefore, the actual scale anywhere will be equal to the principal scale. The Scale Factor (SF) is the actual scale divided by the Principal Scale. The scale factor will therefore be 1.0 everywhere on the globe.

When part of the globe's surface is transformed onto a flat map, the actual scale at various places on the flat map will vary, the scale will be larger or smaller than the principal scale. This is because the Plane and the Globe are not of the same shape so that one cannot be transformed to the other without stretching, shrinking or tearing. Consequently, it is impossible to devise a transformation from a reference globe to a plane without distortion of some kind except at Standard Parallels or Standard lines. However by skillfully varying the scale factor, we can achieve the following:

a) retain some angular relationships or,

b) retain relative sizes of of figures.

Transformations:

When angular relations on a map are retained, the projection is called Conformal or Orthomorphic "correct shape". It is also possible on a map to retain representation of areas so that all regions will be shown in correct relative size. Such a projection is said to be Equal Area. When a uniform scale is retained in most parts of a map distance properties are maintained. Equidistant Projections maintain scale in all directions from one or two points. Scale properties are also maintained along standard parallels. Projections which maintain directions (azimuths) in all directions from one or two points and show Great circle arcs with correct azimuths are called Azimuthal projections.

COMMONLY USED MAP PROJECTIONS

1. Conformal (Orthomorphic)

A projection is conformal if the angles in the original features are preserved. Over small areas the shapes of objects will be preserved. However, thr preservation of shape does not hold with large regions (i.e. Greenland in Mercator projection). A line drawn with constant orientation (e.g. with respect to north) will be straight on a conformal projection. Such lines are termed a rhumb line or loxodrome parallels.

Meridians cross each other at right angles and meridians intersect parallels at right angles

Scale is the same in all directions about a point

Generally, areas near margins have a larger scale than areas near the center.

Maps with conformal projections are used for analyzing, guiding or recording motion or angular relationships. For example navigation, Topographic Maps, Meteorological charts

Good for mapping phenomena with circular radiational patterns such as hurricanes, wind directions, radiowave broadcasts, seismic wave patterns etc

There are four conformal projections in common use. These are:

a) The Mercator,

b) Transverse Mercator,

c) Lambert's Conformal Conic with two standard Parrallels and

d) Stereographic Azimuthal.

2. Equal area Projections

The representation of areas is preserved so that all regions on the projection will be represented in correct relative size. Equal area maps cannot be conformal, so most earth angles are deformed, shapes are strongly distorted as well as distances. The intersection of Meridians and parallels are not at right angles. This is an excellent projection for studying distributions on maps of the United States and Canada that have wide East-West extensions. For example population density, resources, vegetation.

Commonly used Equal Area Projections include:

a) Albert's Equal Area

b) Lambert's Equal Area

c) Cylindrical Equal area with two standard parallels

3. Azimuthal Projections

Azimuthal projections show true directions from one central point to other points. Directions from points other than the central point to other points are not accurate. Azimuthal projections may be centered anywhere with respect to the reference globe. A line perpendicular to the plane of projection will necessarily pass through the center of the globe. Consequently, all distortions are symmetrical around the chosen center. All great circles passing through the center of the projection will be straight lines and will show correct azimuth from and to the center in relation to any point. The advent of air planes, development of satellites and radio electronics and the mapping of terrestrial bodies have increased the prominence of azimuthal projections.

Commonly used azimuthal projections include:

a) Stereographic projection

b) Gnomic projection

c) Lambert's azimuthal equal-area projection

d) Orthographic projection

4. Equidistant Projections

Equidistant projections maintain relative distances from one or two points only. The distance property does not apply everywhere. Great circle distances are preserved so that in a conic projection all distances from the center are represented at the same scale. Scale is uniform along the lines whose distances are true An example of an equidistant projection is the Conic Equidistant projection and Equidistant cylindrical which takes the equator as a standard parallel. Widely used for navigational charts.

5. Other Map Projections

a) Robinsons Projections:

It is a projection that minimizes the appearance of angular and area distortions. It is neither conformal or equal-area but a compromise between the two. It is the projection used for Rand McNally and Company's World Maps, Roads etc.

b) Non-geometric projections

Molleweide

Developed in 1805, Molleweides projection has become widely used for mapping world distributions. The equator is a standard line. Parallels are straight lines parallel to the equator Meridians are curved and the one’s at right angles from the central meridian form a complete circle Maximun shape distortion occurs at the corners where the intersections of the meridians and parallels are most oblique

Used mainly for the mapping of World’s thematic distributions

CHOOSING MAP PROJECTIONS

a) Type of feature to be mapped (area based eg forests or line based)

b) Relative size of interested region

c) The shape of the interested area (east-west extension or north-south extension)

d) Projections property (conformality, equivalence, azimuthal etc)

e) Attributes of the Projection (parallel parallels, local area distortion, rectangular coordinates)

f) Amount and arrangement of distortion eg Atlases or topographic series one may choose a projection that shows the same pattern of distortion over large areas as for small areas.

g) Projection center: can the projection be centered easily for the design problem

A GUIDE FOR SELECTING PROJECTIONS

a) Continents

Bonne, Lambert Azimuthal, Albers Equal Area and Molleweide

b) Countries in Low Latitudes

Equal area, Molleweide, Hammer, Sinusoidal, Cylindrical

c) Large and Small Countries at Mid-Latitudes

Conical Projections, Lambert’s Azimuthal Equal Area, Albers Equal Area.

SUMMARY OF MAP PROJECTIONS

Projection	Type	Properties	Regional Use	General Use
Mercator	Cylindrical	conformal true direction*	World*, equatorial, east-west extent, large and medium scale	navigation large scale map series, U.S.G.S.**
Transverse Mercator	Cylindrical	Conformal	continents/oceans, equatorial/mid-latitude, north-south extent, large and medium scale	topographic large scale map series, N.T.S. and U.S.G.S.
Lambert conformal conic	Conic	conformal true direction*	continents/oceans, equatorial/mid-latitude, east-west extent, large and medium scale	mapping countries of Canada and U.S.A., National Atlas of Canada 5th ed., I.M.W. (International Map of the World)
Azimuthal equidistant	Planar	equidistant* true direction*	World, hemisphere, equatorial/mid-latitude, continents/oceans, regions/seas, polar, large scale	navigation, topographic large scale map series, U.S.G.S.
Lambert azimuthal equal-area	Planar	equal area true direction	hemisphere, continents/oceans, equatorial/mid-latitude, polar	navigation, thematic, Geomatics Canada North America reference map, U.S.G.S. maps
Polyconic	Conic	equidistant*	region/seas, north-south extent, medium and large scale	topographic map series, U.S.G.S.
Stereographic	Planar	conformal true direction*	hemisphere, polar, continents/oceans, regions/seas, equatorial/mid-latitude medium and large scale	navigation, topographic U.S.G.S. maps
van der Grinten I	Individual or unique	Compromise	World, equatorial, east-west extent	Geomatics Canada World Map, U.S.G.S. maps
Robinson	Pseudo-cylindrical	Compromise	World	thematic, reference maps National Geographic
Miller cylindrical	Cylindrical	Compromise	World	thematic, reference maps U.S.G.S. maps
Eckert IV	Pseudo-cylindrical	Equal area	World	thematic, reference maps
Sinusoidal	Pseudo-cylindrical	Equal area	World, continents/oceans equatorial, north-south extent	thematic, reference maps U.S.G.S. maps
		* Limitations apply		** United States Geological Survey - the supplier of base and thematic maps covering the U.S.A.

REVIEW QUESTIONS

1. Define the term, map projection, and give a brief description of the process.
2. What are developable shapes? Name three. Name one shape that is not developable and explain why.
3. Explain the following terms: spheroid, tangency, secancy, standard parallel and central meridian.
4. Briefly describe the polyconic projection.
5. Define the terms, great circle and rhumb line and explain their importance to navigation. Name two projections used in the construction of maps used for navigation.
6. Name the most renowned projection and describe its type, properties and use.
7. With regard to map projection, explain the significance of map scale.
8. When is a map projection equal-area, equidistant, azimuthal or conformal? Give examples of projections, each having at least one of these properties.
9. What are the common properties of all azimuthal (zenithal), projections?
10. Explain the term "interrupted" projection. What advantages does it have over other types of World projections?
11. Describe the problem of "edge matching" in regard to cartographic production and map reading.
12. Briefly describe some of the important factors involved in choosing the best map projection.
13. Select a common projection not already mentioned in this unit, and describe it in terms of the above table.

REFERENCES

Dana, Peter H. 1995. Map Projections. URL The Geographer's Craft Project. Dept. of Geography, University of Texas at Austin. Austin.

ESRI (Environmental Systems Research Institute, Inc.). 1994. Map Projections, Georeferencing spatial data. Redlands, California: ESRI.

Gersmehl, Philip J. 1991. The Language of Maps. Pathways in Geography Series, title no. 1. Indiana, Pennsylvania: Indiana University of Pennsylvania.

Greenhood, David. 1964. Mapping. Chicago: The University of Chicago Press.

Pearson II, Frederick. 1990. Map Projection: Theory and Applications. Boca Raton, Florida: CRC Press, Inc.

Raisz, Erwin. 1962. Principles of Cartography. New York: McGraw-Hill Book Company.

Robinson, Arthur H., and Sale, Randall D. 1969. Elements of Cartography. Third edition. New York: John Wiley & Sons.

Snyder, John P., and Voxland, Philip M. 1989. An Album of Map Projections. U.S. Geological Survey Professional Paper 1453. Denver: United States Government Printing Office.

Maling, D.H., 1973. Coordinate Systems and Map Projections, George Phillip and Son Limited, London.

Robinson, A.H., R.D. Sale, J.L. Morrison and P.C. Muehrcke, 1984, Elements of Cartography, 5th edition, John Wiley and Sons, New York. See pages 56-105.

Snyder, J.P., 1987. Map Projections - A Working Manual, US Geological Survey Professional Paper 1395, US Government Printing Office, Washington.

Strahler, A.N. and A.H. Strahler, 1987. Modern Physical Geography, 3rd edition, Wiley, New York. See pages 3-8 for a description of latitude and longitude and various appendices for information on coordinate systems.