Previous Page Table of Contents Next Page


SECTION 3

3.  SCALE AND PROJECTIONS

3.1  The Concept of Scale

All maps, aerial photographs and satellite images are a small representation of a portion of the earth's surface. It is their smaller than life size which is responsible for their convenience as a method for illustrating the world. If these products are to be at all useful, the relationship between the size of the graphic and the real size of the same region of the earth must be known. This fundamental concept, known as scale, is one of the most important design considerations in the field of cartography.

Establishing a scale for a map is an important design decision. Among other things, the following items are controlled by scale:

i)  the amount of data or detail which can be shown;
ii)  the size of the graphic and its suitability for production utilizing available materials and equipment;
iii)  the cost of reproduction;
iv)  the readability of any product which is an enlargement or reduction of an existing map;
v)  the regional extent of the information shown;
vi)  the degree and nature of the generalization carried out (refer to Section 7);
vii)  the suitability of an available base for a specific purpose;
viii)  the ease of use by the intended market;
ix)  the amount of time a cartographer must spend on a project.

Specifically, scale is the ratio of map distance to ground distance and its selection depends primarily on map purpose. The cartographer must also consider convenience and economy, striking a balance between the area covered, map size and the amount of detail required. Scales are often a compromise.

The use of the relative terms large-scale and small-scale can cause considerable confusion and must be carefully addressed. To understand the concept clearly, compare two maps of the same area but of significantly different scales. Select a common feature such as an airfield, a bay or an island. The map which shows the feature drawn relatively large is the large-scale map. By contrast, the map which shows the same feature as being distinctly small is by definition a small-scale map ( Figure 3.1).

Small-scale maps cover large areas with little detail while large-scale maps show great detail and only cover a small area. Most maps will be a compromise between the required detail and the area of coverage. Sometimes the needs are incompatible, as when a large area must be covered but with some parts requiring great detail. This may be solved by producing more than one map or by using portions of the map as insets at larger scales. The latter solution allows variation in small scales and greater detail in critical areas.

Figure 3.1   A comparison of a large and a small-scale map.

Figure 3.1

3.2  The Forms of Scale

Generally, once it is calculated, the scale of a map may be shown in three different standard forms. These are representative fraction, scale statement and graphic or bar scale. Other scale variations are sometimes used in addition to the standard forms.

3.2.1  Representative fraction

Representative fractions (R.F.), also known as scale ratios, relate the size of the map, or portion thereof, to its actual size on the ground. Thus an R.F. of 1:10,000 means that one unit on the map is equivalent to 10,000 units on the ground. A major advantage of this system is that it is not tied to a specific measurement system; the ratio works as well in metric as in imperial or any other convenient unit of measurement.

Comparatively low numbers after the colon are associated with largescale maps while comparatively high numbers after the colon are associated with small-scale maps. The International Cartographic Association, in an attempt to standardize the terminology, has suggested the following:

i)   R.F. larger than 1:25,000, i.e. numbers less than 25,000: large-scale maps;
ii)   1:50,000 to 1:100,000: medium-scale maps;
iii)   R.F. less than 1:200,000, i.e. numbers greater than 200,000: small-scale maps.

3.2.2   Scale statement

This is a written statement of map distance in relation to earth distance, for example, 1 inch equals 1 mile or 1 centimetre equals 1 kilometre. An R.F. could also be considered a scale statement since, for example, 1:1,000,000 could also be written as 1 centimetre equals 10 kilometres or 1 millimetre equals 1 kilometre. If this version of a scale is chosen, avoid confusion by not mixing metric and imperial units in one statement.

3.2.3  Graphic or bar scale

This device is the most common and the most useful method of depicting scale on a map or chart. It consists of one or more straight lines which are subdivided into units of ground distance or whatever the scale is designed to show (Figure 3.2). It has the considerable advantage of remaining correct even if the map is enlarged or reduced, which is not true for the other scale variants, the R.F. and scale statement.

The cartographer must remember that the scale must be designed for the user and not for the convenience of the cartographer. The subdivision units must be selected to be as even and useful as possible, whatever the R.F. For example, the common older map scale of 1 inch equals 1 mile with an R.F. of 1:63,360 should be converted by the cartographer if a metric scale is desired. By taking the original scale units of 1 inch and plotting them according to their metric scale equivalent, each subdivision unit of 1 mile would represent an inconvenient 1609.35 metres. In this case a basic subdivision unit of 1000 metres or 1 kilometre might be deemed appropriate. A calculation reveals that each unit representing 1000 metres will be 1.578 centimetres long. This is difficult to plot but is the effort the cartographer must make for turning out a useful and professional product.

Figure 3.2   Examples of graphic or bar scales.

Figure 3.2

3.2.4  Graphic scale variants

Most graphic or bar scales are designed in traditional linear measurements such as feet, miles, nautical miles, metres and kilometres. Many land sub-division systems, however, were carried out in rods, chains, furlongs and leagues. Similarly both British and American cable lengths and even fathoms have been used and may occasionally be appropriate.

Direct reading scales in units which are not purely linear are sometimes useful. Many tourist maps, for instance, incorporate walking scales calculated at an average person's pace in units of five or more minutes. Military maps have shown scales in terms of the distance a troop of marching men will cover in a given time period. Highway maps may show elapsed time travelling at a set speed. Biological maps have shown how far a migratory bird, animal or fish will travel in a particular time. These scales are sometimes more useful than the standard scale formats.

3.2.5  Grids

Grids are a system of vertical and horizontal reference lines, drawn on many maps, which enable a point to be identified by a coordinate or reference number (refer to Section 4).

Grids, however, can also be used as a scale indicator on a variety of graphics. A grid of squares having sides of known length, such as a kilometre or a mile, extended over the drawing permits an easy identification of size, area, etc. In a series of related maps such a grid provides a ready method of comparison and identification. A grid must be drawn with fine lines or it will dominate the drawing because of its geometrical, and therefore, visible nature.

3.2.6  Comparison shapes

On some mapping products which illustrate an unknown or unfamiliar geographic area a useful scaling device is the inclusion of insets of a more familiar region and the study area at the same small scale. Travel maps have long used this method, for example “London at the same scale” on a map of Tokyo.

3.2.7  Spacing of parallels

Parallels of latitude are a set of east-west lines running parallel to the equator. They are a geographic constant which can always be translated into measurements of kilometres or miles.

In latitude:

i)  10° = 1111.111 kilometres = 600 nautical miles = 691.72 statute miles;
ii)  1° = 111.11 kilometres* = 60 nautical miles = 69.172 statute miles;
iii)  1' = 1852 metres = 1 nautical mile = 1.15 statute miles.

Thus a difference of 4° in latitude must have the same length in any region of the globe (240 nautical miles), and can be used as a scale indicator or base for calculations, unless supreme accuracy is needed.

* This is a convenient average figure corresponding to 45° latitude. It varies from 110.57 kilometres at the Equator to 111.699 kilometres at the Pole.

3.2.8   Scale factor

Since the earth is essentially spherical, the only consistently accurate method of showing a large region of it is by constructing a globe which can then be given a single scale. To transfer the globe shape to a map on a flat surface requires an organized and consistent method of controlling the inevitable distortions. These mathematical and graphical techniques are known as map projections and are discussed later in this section. The employment of any map projection, however, results in the scale actually varying in different locations on the same map.

The labelled representative fraction (R.F.) on the map is referred to as the principal scale while the local scale caused by projection distortion effects is known as actual scale and will vary from place to place. The scale factor (S.F.) is a ratio of one to the other, thus:

The scale factor is 1.0 on a globe, that is the actual scale is equal to the principal scale, and is close to this figure on most large-scale maps. On small-scale maps it can easily vary from 0.5 to 2.0; this translates into a range of scales from 1:5,000,000 to 1,20,000,000 on a map whose stated scale is 1:10,000,000. On the widely used Transverse Mercator projection the S.F. of a 6° longitude zone varies only from 0.99960 to 1.00158. Similarly on the Mercator projection, much used for charting, the S.F. is limited to 1.016 in the zone between 10° North and South of the Equator.

The scale factor (S.F.) is seldom, if ever, stated on a map but it can have significant effects. The control of the scale factor is, therefore, a consideration in choosing a suitable map projection.

3.2.9  Area scale

It is sometimes necessary to construct a map so that all area proportions are correctly represented, i.e. one unit of area on the map (square centimetre, square inch, etc.) represents a particular number of the same square units on the earth. Here again, the cartographer must first select a relevant map projection which will allow this function. To prevent confusion area scales are shown graphically rather than numerically. Thus the explanation will include a square which represents a stated number of square kilometres or miles, acres, etc.

3.2.10  Variable scale

As discussed earlier, no flat map can exhibit true distance from all points in all directions simultaneously. In some map projections the distance distortion is systematic and a variable scale can be constructed to enable accurate measurements to be taken. This is particularly true on those maps containing a Mercator projection, such as some nautical and aeronautical charts. This permits the determination of scale in latitude or longitude despite the vast range in scale distortion.

3.2.11  Other scales

Ratio, nominal, ordinal, interval, value and logarithmic scales are discussed in Section 9. The scale of aerial photographs is described in Section 8.

3.3  Scale Changing

When a map or graphic is reduced or enlarged, the scale will change proportionately. If a drawing at a scale of 1 : 100, 000 is reduced to 50% of its original size, the scale will change to 1 : 200, 000. Similarly if it is enlarged to 200% of the original size, the 1 : 100, 000 graphic will now have a scale of 1 : 50, 000. All scales, and especially a scale statement or an R.F., must be carefully calculated and labelled for the reproduction scale. The amount of reduction or enlargement, if required, must be known precisely at the design stage. For this reason the cartographer must work closely with both the author and the printer when making preliminary design decisions.

Cartographic artwork may be photographically reproduced at 100%, reduced or enlarged. The scale change must be clearly identified if enlargements or reductions are required. It is best to use the designation which is built into most process cameras, the devices which are used to do the reproduction. On these instruments an identically sized reproduction is marked as 100%. To obtain a 25% reduction in size the camera must be set to 75%, and the latter is what should be specified. “Reduce to 75% of original” prevents the obvious error of setting the camera to 25%, and obtaining a drawing where every line would be 1/4 of its original size.

Similarly, to enlarge, the percentage on the camera setting should be specified. Thus, if a drawing is required where each dimension is twice that of the original, the statement “Enlarge to 200% of original” and not “100% enlargement” should be noted.

If there is still a possibility of confusion, provide the operator with a simple bar scale to place on the camera. Include two lines (lines AB and AC) of carefully measured lengths on the drawing. The instructions should read “Reduce (or enlarge) AB to AC exactly”. This prevents any confusion and enables the operator to physically check the enlargement or reduction.

3.4  Geodesy - The Form of the Earth

The fact that the earth is neither flat nor round has historically posed a problem for cartographers, particularly when producing extensive chart or map series, at small or large scales, which cover extensive geographic areas. The exact shape of the earth now becomes a major consideration. For large-scale individual maps, however, particularly those of a thematic nature, the variations are not significant.

Satellite images have ensured that the roughly spherical shape of the earth is now accepted by most people and is no longer a subject for dispute. The exact shape is, however, of distinct interest and is still actively under study. As is well known, the earth has become slightly flattened at the poles because of the effects of its rotation. The distortion is not obvious - if the earth were reduced to a globe 1 metre in diameter, the amount of polar flattening would be only about 3.5 millimetres.

Surveyors must also contend with the fact that the mass of the earth is not evenly distributed. This creates variations in the strength and direction of gravity, which controls the local horizontal and vertical surfaces with which the surveyor must work. Scientists have thus postulated in theory an irregular spheroidal shape which takes the gravity variations into account; it is called the geoid. The geoid shape is higher under the continents because of the presence of a large rock mass above sea-level, as seen in Figure 3.3.

The geoid is often described as a hypothetical surface to which the ocean would conform (i.e. sea-level) if free to adjust to the earth's gravitational attraction and the forces of centrifugal rotation. Gravity studies using satellites have now revealed that the earth's gravitational field has some distinct humps and depressions. The largest hump is near New Guinea, being some 81 metres high, while a major depression south of India dips 110 metres below reference surface.

For mapping purposes an irregular surface is highly undesirable, so the information must be transferred to a regular geometric shape which can be calculated and which closely approximates the geoid. This shape is known as the ellipsoid and is a three-dimensional reference surface (Figure 3.3). No single ellipsoid is considered to be suitable for all surveys and mapping throughout the world. For historical and political reasons a number of different figures of the earth are in current usage (Table 3.1).

Recently the International Association of Geodesy has adopted new dimensions for a reference ellipsoid, called the Geodetic Reference System 1980 (GRS80). This is the basis for a new reference mapping system, the North American Datum 1983 (NAD83).

Figure 3.3   The relationship between the regular ellipsoid surface and the irregular geoid surface under continents and over ocean basins. (After W.A. Heiskanen, 1958)

Figure 3.3
TABLE 3.1

REFERENCE ELLIPSOIDS AND THEIR DIMENSIONS
ELLIPSOIDEQUATORIAL RADIUS (Metres)FLATTENINGUSER
Everest (1830)6 377 2761/300.80India
Bessel (1841)6 377 3971/299.15Japan, Germany
Airy (1844)6 377 5631/299     Great Britain
Clarke (1860)6 378 2491/293.47France, S.Africa
Clarke (1866)6 378 3971/294.98North America
International (1924)6 378 3881/297.00International
Krasovsky (1940)6 378 2451/298.30U.S.S.R.
Astronomical Union (1965)6 378 1601/298.25
IUGG* (1979)6 377 5631/298.26North America/International

* International Union of Geodesy and Geophysics

DIMENSIONS OF THE EARTH (Based on GRS80)
KILOMETRESSTATUTE MILES (U.S.)
Equatorial diameter12 756.3 7 926.4
Polar diameter12 713.5 7 899.8
Equatorial circumference40 075.124 901.5
Radius of the sphere6 3713 949
Area of the earth          510 064 500 km²          196 936 000 mi²

3.5  Map Projections

The cartographer makes use of map projections to present the three-dimensional nature of the earth's surface in the two dimensions available on a map or chart. As discussed earlier, for the purposes of medium and small-scale graphics the basic shape of the earth can be assumed to be spherical. A small area of a large-scale map or chart can be drawn without appreciable error but for those products showing large areas, and particularly for series mapping, a projection system is vital.

Projections can be created purely graphically by projecting the earth's curved surface onto flat surfaces or developable surfaces such as cones or cylinders which can be flattened. They can also be created mathematically or by a combination of the two methods.

The ideal projection would provide correct shapes, correct areas, correct scale, correct bearings, a good overall “fit” and ease of construction. Obtaining all or even most of these properties is impossible so the cartographer must select whichever feature is the most important for a particular map, or choose a compromise projection, often one of the so called “minimum error” types.

Correct shape is a characteristic of conformal (orthomorphic) projections. It should be noted that it is only possible to keep shapes correct over small areas. Conformal projections preserve true angles and a constant scale in all directions about a given point because the parallels and meridians cross each other at right angles. This is an essential characteristic for navigational charts. Both the Mercator and the Lambert Conformal Conic are conformal projections and are widely used both for sea and for air navigational charting. As these projections preserve angles locally, they also may be used for graphics showing data based on angular measurement. These might include tidal streams, lines of gravity and magnetics, direction of surface-water movements, migrations, and bathymetry. Navigational charting using conformal projections has been undertaken for centuries, providing a ready source of data for use as base map information. This simplifies the cartographer's task.

Equal area is also known as equivalence. This property can be preserved on a map constructed from a projection such as Bonne's, but only at the expense of distorted shapes. This projection can be of great value for displaying spatial relationships and distributions. When the cartographic symbolization requires an area or quantitative symbol, such as water volume movement, an equal area projection is needed.

The attainment of full equidistance, i.e. the preservation of scale at all points on a projection, is impossible. On any projection the actual scale is continuously variable; it can vary from point to point and may also vary in different directions. It is possible, however, to maintain correct scale where a projection surface meets the sphere from which it is derived. Selection of those points in a careful manner can reduce scale errors to a minimum. Equal distance can be preserved on Zenithal projections. Equidistant projections are a useful compromise between conformal and equal area projections and they are often used for general reference graphics. The area scale changes on equidistant projections are less dramatic than those on a conformal projection and the angular errors are less than those of an equal area projection.

Map projections may be classed in several ways which are summarized in the insert included with this manual.


Previous Page Top of Page Next Page