Scale (map)
The scale of a map is usually defined as the ratio of a single unit of distance on the map to the corresponding distance on the ground. Although this is true for accurate large scale maps, covering a restricted area, it is not true for map projections^{[1]} of the curved surface of the Earth to the plane. For such projections we must use the concept of a point scale which may vary with position and direction. In the study of point scale it is convenient to define the projection formulae in such a way that the scale is unity, or nearly so, on some lines of the resulting map. Clearly such a map projection must be comparable to the size of the Earth and, in order to represent it on a small sheet of paper, it must be reduced. The ratio defined on the printed map projection is then re-interpreted as the representative fraction, or in short, the RF. Point scale and RF are defined below with examples drawn from simple projections. Tissot's Indicatrix is used to illustrate the variation of point scale in the examples.
Contents
- 1 Maps as scale drawings: constant scale throughout
- 2 Point scale
- 3 Point scale for normal cylindrical projections of the sphere
- 4 Three examples of normal cylindrical projection
- 5 The representative fraction: RF
- 6 Scale variation on the Mercator projection
- 7 Modified projections
- 8 Mathematical addendum
- 9 Depictions of Scale on Maps
- 10 Notes
Maps as scale drawings: constant scale throughout
The region over which we can take the earth as sensibly flat depends on the accuracy of the survey measurements. If we measure only to the nearest meter then curvature is undetectable over a meridian distance of about 100 km and over an east-west line of about 80 km (at a latitude of 45 degrees). If we can survey to the nearest millimeter then curvature is undetectable over a meridian distance of about 10 km and over an east-west line of about 8 km.^{[2]} Mapping a small planar region to a paper map is simply a matter of scale-drawing. The linear scale is constant and it can be expressed in four ways: in words (a lexical scale), as a ratio, as a fraction and as a graphical (bar) scale. Examples are:
- 'one centimetre to one hundred metres' or 1:10,000 or 1/10,000
- 'one inch to one mile' or 1:63,360 or 1/63,360
- 'one centimetre to one thousand kilometres' or 1:100,000,000 or 1/100,000,000
The first two are examples of large scale maps whilst the third is an example of a small scale map. This usage relates to the expressions as fractions. The fraction 1/10,000 is much larger than 1/100,000,000. There is no hard and fixed dividing line between small and large scales.
A (graphical) bar scale may be a suitable physical ruler which can be used to measure distance on the map, or maps, if the ruler has several scales. The maps themselves usually have one or more scales drawn on them. (For example there may be both Imperial and metric scales).
Lexical scales are to be deprecated whereas ratios and fractions are much more acceptable since they are immediately accessible in any language. Old maps may cause difficulties if they possess only a lexical scale in rare, old or even archaic units. For example a scale of one inch to a furlong is not too difficult to interpret in countries where Imperial units are, or were recently, in use. (It is 1/7920). A scale of one pouce to one league may be about 1/144,000 but it depends on the choice of definition for league.
True ground distances are calculated by measuring the distance on the map (in any measure) and then multiplying by the inverse of the scale fraction. True ground areas are calculated by measuring the area on the map (in any measure) and then multiplying by the inverse of the scale fraction squared.
Point scale
It is known that a sphere (or ellipsoid) cannot be projected to the plane at a constant scale. This is illustrated by the impossibility of smoothing an orange peel onto a flat surface. More formally it follows from the Theorema Egregium of Gauss.
Suppose P is a point at latitude and longitude on the sphere (or ellipsoid). Let Q be a neighbouring point and let be the angle between the element PQ and the meridian at P: this angle is the azimuth angle of the element PQ. Let P' and Q' be points on the projection corresponding to P and Q.
Definition: the point scale k at P is the ratio of the two distances P'Q' and PQ in the limit that Q approaches P. We write this as
where the notation indicates that the point scale is a function of the position of P and also the direction of the element PQ.
Definition: if the point scale depends only on position and not on direction we say that it is isotropic and denote the scale by — this is the case for all conformal projections. In the projection is also axi-symmetric then the scale will be independent of and it is denoted by — this is the case for the (normal) Mercator projection.
Definition: on a conformal projection with an isotropic scale, points which have the same scale value may be joined to form the isoscale lines. These are not plotted on maps for end users but they feature in many of the standard texts.(See Snyder^{[1]} pages 203—206.)
Visualisation of point scale: the Tissot indicatrix
Consider a small circle on the the surface of the Earth centred at a point P at latitude and longitude . Since the point scale varies with position and magnitude the corresponding small circle on the projection will be distorted. Tissot proved that, as long as the distortion is not too great, the circle will become an ellipse on the projection. In general the dimension, shape and orientation of the ellipse will change over the projection and by superimposing these distortion ellipses on the map projection we can convey the way in which the point scale is changing over the map. The distortion ellipse is known as Tissot's Indicatrix. The example shown here is the Winkel tripel projection, the standard projection for world maps made by the National Geographic Society. The minimum distortion is on the central meridian at latitudes of 30 degrees (North and South). (Other examples ^{[3]}^{[4]}).
Point scale for normal cylindrical projections of the sphere
The key to a quantitative understanding of scale is to consider an infinitesimal element on the sphere. The figure shows a point P at latitude and longitude on the sphere. The point Q is at latitude and longitude . The lines PK and MQ are arcs of meridians which must converge at the pole: the length of PK is where is the radius of the sphere. The lines PM and KQ are arcs of parallel circles: the length of PM is since the radius of a parallel circle at latitude is . In deriving a property of the projection at P it suffices to take an infinitesimal element PMQK of the surface. In the limit of Q approaching P such an element tends to an infinitesimally small planar rectangle.
Consider the infinitesimal element on the sphere and the corresponding infinitesimal in the plane of the projection. For all the normal cylindrical projections of the sphere we know that and is some function of latitude only. Therefore on the projection the meridians are vertical (without approximation) and the parallels horizontal so that the element P'M'Q'K' is also a rectangle with a base and height . We defer the treatment of the scale in a general direction to a mathematical addendum to this page. Here we simply note that
- horizontal scale factor
- vertical scale factor
Note that the scale in the horizontal direction is independent of the definition of so it is the same for all normal cylindrical projections. The following examples illustrate three such normal cylindrical projections and in each case the variation of scale with position and direction is illustrated by the use of Tissot's Indicatrix.
Three examples of normal cylindrical projection
The equidistant projection
The equidistant projection, or the Plate Carrée (french for "flat square"), is the simplest and most ancient projection, having been in use since before the time of Ptolemy. The sphere is mapped into a rectangle by the equations^{[1]}^{[2]}.
where is the radius of the sphere, is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at ) and is the latitude. Note that and are in radians (obtained by multiplying the degree measure by a factor of ). The value of is in the range and the value of is in the range .
The results for the horizontal and vertical scale factors follow immediately from the previous section. Since we have
- horizontal scale, vertical scale
The calculation of the point scale in an arbitrary direction is given below.
Thus for the equidistant projection there is no scaling in the -direction. In the -direction the scaling increases with latitude by the factor . This is true for all normal cylindrical projections since the parallels, of true length at latitude must be stretched by the factor to give the width of the rectangle, namely . This stretching in the direction alone is illustrated by using Tissot's Indicatrix. The semi-minor axis of each ellipse is equal to that of the undistorted circles on the equator where the scale is unity. The mafor axis of each is stretched by the factor of . The area of any ellipse is greater than that of the circles showing that the projection does not preserve area.
Mercator projection
The Mercator projection maps the sphere to a rectangle (of infinite extent in the -direction) by the equations
where a, and are as in the previous example. Since the scales in the horizontal and vertical directions are:
- horizontal scale,
- vertical scale
In the mathematical addendum below we prove that the point scale in an arbitrary direction is also equal to so the scale is isotropic, its magnitude increasing with latitude as . In the Tissot diagram each infinitesimal circular element preserves its shape but is enlarged more and more as the latitude increases.
Lambert's equal area projection
Lambert's equal area projection maps the sphere to a finite rectangle by the equations
where a, and are as in the previous example. Since the scales in the horizontal and vertical directions are
- horizontal scale,
- vertical scale
The calculation of the point scale in an arbitrary direction is given below.
The vertical and horizontal scales now compensate each other and in the Tissot diagram each infinitesimal circular element is distorted into an ellipse of the same area as the undistorted cirles on the equator.
The representative fraction: RF
In the normal cylindrical projections the scale along the equator is unity. This implies that the projection surface must be big; very, very big. Since on all normal cylindrical projections the width of the projection must be , equal to the length of the equator, which is approximately 40,000 km. The extent in -direction varies with the projection: on the equidistant projection it is or 20,000 km, on the Lambert projection it is or about 12,760 km and on the Mercator projection it is infinite. The (fictitious) projection maps at this size may well be called super maps. Actual printed maps are then produced from the super map by a constant scaling denoted by a ratio or a fraction or in words. This fraction is called the representative fraction or simply "RF" in short.
The advantage of this distinction between varying scale on the super map and a constant RF is that for the former we can analyse scale variation around a value of unity. This is preferable to analysis of variations about fractions such as 1/10,000 or 1:10M (million).
Area Determination
- Areal scale is some times used to describe the relationship between the area of a feature plotted on a map and the area of the same feature on the earth’s surface. Areal scale can be simply calculated as the square of the map’s linear scale. ^{[5]}
- If the feature has a relatively regular geometric shape, the size of features can be estimated directly with a graphic areal scale.^{[6]}
Scale variation on the Mercator projection
The Mercator point scale is unity on the equator but varies with latitude as . Since tends to infinity as we approach the poles the Mercator map is grossly distorted at high lines. For this reason the projection is totally inappropriate for world maps (unless we are discussing navigation and rhumb lines). However, at a latitude of about 25 degrees the value of is about 1.1 so Mercator is accurate to within 10% in a strip of width 50 degrees centred on the equator. Narrower strips are better: a strip of width 16 degrees is accurate to within 1% or 1 part in 100.
A standard criterion for good large scale maps is that the accuracy should be within 4 parts in 10,000, or 0.04%, corresponding to . Since attains this value at degrees and the Mercator projection is highly accurate within a strip of width 3.24 degrees centred on the equator. This corresponds to north-south distance of about 360 km or 200 miles.
Therefore Mercator is very good in a narrow strip near the equator and is well suited to accurate mapping of territory there. This is the motivation for the universal application of the Transverse Mercator projection in which the formulae of the normal Mercator projection are applied treating some meridian as an equator: the result is a highly accurate map in a narrow strip near that meridian. By repeating this for many meridians we can cover the whole globe with highly accurate maps. For example the Universal Transverse Mercator (UTM) system employs 60 separate transverse projections to cover the ellipsoidal globe (at least between latitudes 80S and 85N). Although the transverse Mercator projection is conformal with an istropic scale the scale factor is a complicated function of latitude and longitude in these transverse projections.
Modified projections
The demand that the scale satisfies may be relaxed a little to . In this case we still have a scale variation that is within 0.04% of true scale so the mapping is still highly accurate. As an example suppose that the Mercator projection given above is modified to
This alteration does not alter the shape of the projection but it does mean that the scale factors given above are reduced. In fact the scale factor remains isotropic but we now have
- modified Mercator scale,
Thus (a) the scale on the equator is 0.9996, (b) the scale is at a latitude given by where so that degrees and (c) at a latitude given by for which degrees. This modification of the transformation equations means that the projection is just as accurate in a wider strip (4.58 degrees).
Clearly the line of unit scale at latitude is where the cylindrical projection surface intersects the sphere: the radius of the parallel at must equal the radius of the projection cylinder. This is an example of a secant (in the sense of cutting) projection as opposed to a tangent (touching) projection. The same ideas may be used to extend the zone of accuracy of all cylindrical and conical projections.
The limits are not adopted slavishly but many projections are close. For the important UTM (conformal) projections the maximum scale factor can almost reach (over the mapped region); for the British OSGB projection (again conformal transverse Mercator) the maximum scale value is . In both these cases the scale on the central meridian is constant at and the isoscales lines with are slightly curved lines approximately 180 km east and west of the central meridian.
Mathematical addendum
For normal cylindrical projections the geometry of the infinitesimal elements gives
from which we have the relationship between the angles and
Since for the Mercator projection we immediately deduce the important result that : angles are preserved. For the equidistant and Lambert projections we have and respectively so the relationship between and depends upon the latitude . Denote the point scale at P when the infinitesimal element PQ makes an angle with the meridian by It is given by the ratio of distances:
Substituting and and from equations (a) and (b) repectively gives
This result shows that the point scale for Mercator projection is independent of since . For the other projections we must first calculate from , using equation (c), before we can find . The results of such calculations can be used to calculate the shape of the Tissot indicatrices shown above.
Depictions of Scale on Maps
It is usually important to include some indication of the scale of a map, for two reasons: 1) map readers may need to make measurements of distance or area, and 2) it effectively conveys a general sense of the size of the region being portrayed and the features therein. There are three primary ways of depicting the scale of a map:
- Scale Bars are a map element used to graphically represent the scale of a map. A scale bar is typically a line marked like a ruler in units proportional to the map's scale, thus showing how a given distance on the map is equivalent to a distance in the real world. ^{[7]} There are many different forms of scale bar which vary from a simple box of a set length, to more detailed bars with smaller divisions, allowing for more precise measuring. Map readers can often quickly calculate distances by using a ruler. ^{[8]} A graphic scale bar has the advantage that it changes size appropriately when the map is enlarged or reduced, or when the physical size of the map depiction cannot be predicted, such as on a computer monitor or mobile device screen.
- A verbal scale or word scale uses a word statement to explain the scale; for example, "1 inch equals 1 mile" means that 1 inch on the map will equal 1 mile in reality. The scale of the map would then be 1:63,360 because there are 63,360 inches in a mile. When using a verbal scale, numbers are often rounded to get a more desirable result.^{[9]}
- A representative fraction explains the mathematical relationship between any one unit on the map and the number of ground units that one unit on the map represents; for example, "1:64,000." The RF has the advantage of being a unitless measure, so it is possible to apply the fraction to any unit — centimeters, inches, feet, miles, etc. So 1:64,000 could mean 1 cm is 64,000 cm or 1 inch is 64,000 inches.
In terms of visual hierarchy scale bars are often placed at the bottom of the map document and are easy to find but should not readily stand out on the map, it is an aid to the reader not the focus of the map. It serves as one of two purposes: for dimensionality or measurement. It should not be ornate. It should be placed for balance and clarity. ^{[9]}
A good scale bar should have rounded units that are easy for the reader to understand, such as 1 mile as opposed to 1.36 miles. Usually multiples of 5 and 10 are best. When making a map, it is important to remember that scale does not always stay consistent, especially on small scale maps, so a single scale bar would not be accurate for the whole map. Although scale bars are important they should not call too much attention to themselves as they are there to assist the reader in understanding the map. In some maps, it may not be necessary or appropriate to depict the scale, especially world maps (in which scale is rarely constant across the map), thematic maps (in which scale may not matter much), or cartograms and other schematic maps (which are not drawn to scale).
Notes
- ↑ ^{1.0} ^{1.1} ^{1.2} Snyder, John P. (1987). Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395. United States Government Printing Office, Washington, D.C.This paper can be downloaded from USGS pages. It gives full details of most projections, together with interesting introductory sections, but it does not derive any of the projections from first principles. Derivation of all the formulae for the Mercator projections may be found in The Mercator Projections — see below.
- ↑ ^{2.0} ^{2.1} The Mercator Projections A pedagogical derivation of the formulae describing many of the variants of the Mercator projections. In particular normal and transverse projections on the sphere and ellipsoid along with their modified versions. It concludes with the Redfearn formulae used for the Transverse Mercator on the ellipsoid. The mathematics of this page is contained within Chapter 2.
- ↑ Examples of Tissot's indicatrix. Some beautiful illustrations of the Tissot Indicatrix applied to a variety of projections other than normal cylindrical.
- ↑ Further examples of Tissot's indicatrix at wiki.gis.com commons.
- ↑ Arthur H. Robinson, Joel L. Morrison, Phillip C. Muehrcke, A. Jon Kimerling, and Stephen C. Guptill, (1995). Elements of Cartography, Sixth Edition, John Wiley & Sons, Inc. 674p.
- ↑ Arthur H. Robinson, Joel L. Morrison, Phillip C. Muehrcke, A. Jon Kimerling, and Stephen C. Guptill, (1995). Elements of Cartography, Sixth Edition, John Wiley & Sons, Inc. 674p.
- ↑ http://support.esri.com/en/knowledgebase/GISDictionary/term/scale%20bar, GIS Dictionary, Accessed September 30, 2012
- ↑ http://www.cartography.org.uk/default.asp?contentID=747, The British Cartographic Society. Accessed September 30, 2012
- ↑ ^{9.0} ^{9.1} Tyner, J. A. Principles of map design. New York, NY: The Guilford Press, 2010.Print.
- ↑ Wikipedia contributors, Linear scale. Wikipedia, The Free Encyclopedia. Accessed 4 October 2011.
- ↑ Reading Topographic Maps. A Free On-Line Book on How to Read Topographic Maps and Use a Compass. Accessed 4 October 2011.
- ↑ Bar Scales. U.S. Geological Survey Open-File Report 99-430
Brewer, Y. A. (2005). Designing better maps, a guide for gis users. Esri Press.Retrieved from http://books.google.com/books/feeds/volumes?q=1589480899