# Scale (map)

(Redirected from Representative fraction)

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. This ratio is called a scale factor. 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.

## 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 (Scale bar) 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 were recently or are currently 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 a 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 $\phi$ and longitude $\lambda$ on the sphere (or ellipsoid). Let Q be a neighboring point and let $\alpha$ 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

$k(\lambda,\,\phi,\,\alpha)=\lim_{Q\to P}\frac{P'Q'}{PQ},$

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 $k(\lambda,\phi)$— this is the case for all conformal projections. In the projection is also axi-symmetric then the scale will be independent of $\lambda$ and it is denoted by $k(\phi)$— 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

The Winkel tripel projection with Tissot's Indicatrix of deformation

Consider a small circle on the the surface of the Earth centred at a point P at latitude $\phi$ and longitude $\lambda$. 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 $\phi$ and longitude $\lambda$ on the sphere. The point Q is at latitude $\phi+\delta\phi$ and longitude $\lambda+\delta\lambda$. The lines PK and MQ are arcs of meridians which must converge at the pole: the length of PK is $a\delta\phi$ where $a$ is the radius of the sphere. The lines PM and KQ are arcs of parallel circles: the length of PM is $(a\cos\phi)\delta\lambda$ since the radius of a parallel circle at latitude $\phi$ is $(a\cos\phi)$. 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 $x=a\lambda$ and $y$ 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 $\delta x=a\delta\lambda$ and height $\delta y$. 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   $\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$
vertical scale factor     $\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\frac{y'(\phi)}{a}$

Note that the scale in the horizontal direction is independent of the definition of $y(\phi)$ 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].

$x = a\lambda \qquad\qquad y = a\phi,$

where $a$ is the radius of the sphere, $\lambda$ is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at $\lambda =0$) and $\phi$ is the latitude. Note that $\lambda$ and $\phi$ are in radians (obtained by multiplying the degree measure by a factor of $\pi/180$). The value of $\lambda$ is in the range $[{-}\pi,\pi]$ and the value of $\phi$ is in the range $[{-}\pi/2,\pi/2]$.

The results for the horizontal and vertical scale factors follow immediately from the previous section. Since $y'(\phi)=1$ we have

horizontal scale, $\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$       vertical scale $\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\,1$

The calculation of the point scale in an arbitrary direction is given below.

The equidistant projection with Tissot's Indicatrix of deformation

Thus for the equidistant projection there is no scaling in the $y$-direction. In the $x$-direction the scaling increases with latitude by the factor $\sec\phi$. This is true for all normal cylindrical projections since the parallels, of true length $2\pi a\cos\phi$ at latitude $\phi$ must be stretched by the factor $\sec\phi$ to give the width of the rectangle, namely $2\pi a$. This stretching in the $x$ 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 $\sec\phi$. 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 with Tissot's Indicatrix of deformation. (The distortion increases without limit at higher latitudes)

The Mercator projection maps the sphere to a rectangle (of infinite extent in the $y$-direction) by the equations

$x = a\lambda \qquad\qquad y = a\ln \left(\tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right)$

where a, $\lambda\,$ and $\phi \,$ are as in the previous example. Since $y'(\phi)=a\sec\phi$ the scales in the horizontal and vertical directions are:

horizontal scale,  $\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$
vertical scale     $\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\,\sec\phi$

In the mathematical addendum below we prove that the point scale in an arbitrary direction is also equal to $\sec\phi$ so the scale is isotropic, its magnitude increasing with latitude as $\sec\phi$. 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 normal cylindrical equal-area projection with Tissot's Indicatrix of deformation

Lambert's equal area projection maps the sphere to a finite rectangle by the equations

$x = a\lambda \qquad\qquad y = a\sin\phi$

where a, $\lambda$ and $\phi$ are as in the previous example. Since $y'(\phi)=\cos\phi$ the scales in the horizontal and vertical directions are

horizontal scale,  $\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$
vertical scale     $\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\,\cos\phi$

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 circles on the equator.

## The representative fraction: RF

In normal cylindrical projections, the scale along the equator is unity. This implies that the projection surface must be big; very, very big. Since $x=a\lambda$ on all normal cylindrical projections the width of the projection must be $2\pi a$, equal to the length of the equator, which is approximately 40,000 km. The extent in $y$-direction varies with the projection: on the equidistant projection it is $\pi a$ or 20,000 km, on the Lambert projection it is $2a$ 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 the analysis of variations about fractions such as 1/10,000 or 1:10M (million).

### Area Determination

• Areal scale is sometimes 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 $k=\sec\phi$. Since $\sec\phi$ 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 $\sec\phi$ is about 1.1 so Mercator is accurate to within 10% in a strip of width 50 degrees centered 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 $k=1.0004$. Since $\sec\phi$ attains this value at $\phi=1.62$ 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 isotropic scale the scale factor is a complicated function of latitude and longitude in these transverse projections.

## Modified projections

The demand that the scale satisfies $1 may be relaxed a little to $0.9996. 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

$x = 0.9996a\lambda \qquad\qquad y = 0.9996a\ln \left(\tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right).$

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,   $\quad k\;=0.9996\sec\phi.$

Thus (a) the scale on the equator is 0.9996, (b) the scale is $k=1$ at a latitude given by $\phi_1$ where $\sec\phi_1=1/0.9996=1.00004$ so that $\phi_1=1.62$ degrees and (c)  $k=1.0004$ at a latitude $\phi_2$ given by $sec\phi_2=1.0004/0.9996=1.0008$ for which $\phi_2=2.29$ 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 $\phi_1$ is where the cylindrical projection surface intersects the sphere: the radius of the parallel at $\phi_1$ 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 $0.9996 are not adopted slavishly but many projections are close. For the important UTM (conformal) projections the maximum scale factor can almost reach $k=1.001$ (over the mapped region); for the British OSGB projection (again conformal transverse Mercator) the maximum scale value is $k=1.0007$. In both these cases the scale on the central meridian is constant at $k_0=0.9996$ and the isoscales lines with $k=1$ are slightly curved lines approximately 180 km east and west of the central meridian.

For normal cylindrical projections, the geometry of the infinitesimal elements gives

$\text{(a)}\quad \tan\alpha=\frac{a\cos\phi\,\delta\lambda}{a\,\delta\phi},$
$\text{(b)}\quad \tan\beta=\frac{\delta x}{\delta y} =\frac{a\delta \lambda}{\delta y},$

from which we have the relationship between the angles $\beta \,$ and $\alpha \,$

$\text{(c)}\quad \tan\beta=\frac{a\sec\phi}{y'(\phi)} \tan\alpha.\,$

Since $y'(\phi)=a\sec\phi$ for the Mercator projection we immediately deduce the important result that $\alpha=\beta$: angles are preserved. For the equidistant and Lambert projections we have $y'(\phi)=a \,$ and $y'(\phi)=a\cos\phi \,$ respectively so the relationship between $\alpha \,$ and $\beta \,$ depends upon the latitude $\phi$. Denote the point scale at P when the infinitesimal element PQ makes an angle $\alpha \,$ with the meridian by $\mu_{\alpha}.$ It is given by the ratio of distances:

$\mu_{\alpha}=\lim_{Q\to P}\frac{P'Q'}{PQ} = \lim_{Q\to P}\frac{\sqrt{\delta x^2 +\delta y^2}} {\sqrt{ a^2\, \delta\phi^2+a^2\cos^2\!\phi\, \delta\lambda^2}}.$

Substituting $\delta x=a\delta\lambda \,$ and $\delta \phi \,$ and $\delta y \,$ from equations (a) and (b) repectively gives

$\mu_\alpha(\phi) = \sec\phi \left[\frac{\sin\alpha}{\sin\beta}\right].$

This result shows that the point scale for Mercator projection is independent of $\alpha$ since $\alpha=\beta \,$. For the other projections we must first calculate $\beta$ from $\alpha$, using equation (c), before we can find $\mu_{\alpha}$. 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 bar for maps, with both kilometers and miles units.
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.
The effective design of scale representations depends on how they are intended to be used. Depending on the type of map, its purpose, and its intended audience, scale bars serve different purposes. For example, on a street map of a small city where measuring exact distances is unnecessary, a scale bar would be simple (containing only one bar line) and give the viewer an indication of relative distance on the map. On the other hand, on a topographic map where exact distances need to be calculated, more detailed scale bars are included to aid the user in calculating precise distances. Likewise, a person well-versed in cartography will have greater use for a scale bar than the general public.[10][11][12]
Different types of map scales.

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).

## Design Considerations

A map's scale is an important part of its design. When determining a scale, it is essential to consider the map's purpose and audience. Obviously, it is important to select a scale that will fit the figure into the map, but providing ground for reference can also be important [13]. Generalization must also be taken into account when dealing with scale, as a large amount a data in a small space can be cluttered and difficult to interpret, but a small amount of data in a large space can be sparse.

## Notes

1. 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. 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.
3. Examples of Tissot's indicatrix. Some beautiful illustrations of the Tissot Indicatrix applied to a variety of projections other than normal cylindrical.
4. Further examples of Tissot's indicatrix at wiki.gis.com commons.
5. 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.
6. 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.
7. http://support.esri.com/en/knowledgebase/GISDictionary/term/scale%20bar, GIS Dictionary, Accessed September 30, 2012
8. http://www.cartography.org.uk/default.asp?contentID=747, The British Cartographic Society. Accessed September 30, 2012
9. Tyner, J. A. Principles of map design. New York, NY: The Guilford Press, 2010.Print.
10. Wikipedia contributors, Linear scale. Wikipedia, The Free Encyclopedia. Accessed 4 October 2011.
11. Reading Topographic Maps. A Free On-Line Book on How to Read Topographic Maps and Use a Compass. Accessed 4 October 2011.
12. Bar Scales. U.S. Geological Survey Open-File Report 99-430
13. 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. 108p.