Scale (map)

The scale of a map

Map

A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....

 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 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.

The scale of a map

Map

A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....

 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 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

Tissot's Indicatrix

Tissot’s indicatrix, or ellipse of distortion, is a concept developed by French mathematician Nicolas Auguste Tissot, in 1859 and 1871, to measure and illustrate distortions due to map projection...

 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 metre 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 millimetre then curvature is undetectable over a meridian distance of about 10 km and over an east-west line of about 8 km.
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

French units of measurement

In France, before the decimalised metric system of 1799, a well-defined old system existed, however with some local variants. For instance, the lieue could vary from 3.268 km in Beauce to 5.849 km in Provence...

 to one league

French units of measurement

In France, before the decimalised metric system of 1799, a well-defined old system existed, however with some local variants. For instance, the lieue could vary from 3.268 km in Beauce to 5.849 km in Provence...

 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

Theorema Egregium

Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...

 of Gauss

Gauss

Gauss may refer to:*Carl Friedrich Gauss, German mathematician and physicist**List of topics named after Carl Friedrich Gauss*GAUSS , a software package*Gauss , a crater on the moon*Gauss Gun, a type of magnetic gun....

.

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

Isotropy

Isotropy is uniformity in all directions. Precise definitions depend on the subject area. The word is made up from Greek iso and tropos . Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...

 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 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

Nicolas Auguste Tissot

Nicolas Auguste Tissot was a 19th century French cartographer, who in 1859 and 1881 published an analysis of the distortion that occurs on map projections. He devised Tissot's Indicatrix, or the distortion circle, which when plotted on a map will appear as an ellipse whose elongation depends on...

 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

Tissot's Indicatrix

Tissot’s indicatrix, or ellipse of distortion, is a concept developed by French mathematician Nicolas Auguste Tissot, in 1859 and 1871, to measure and illustrate distortions due to map projection...

. The example shown here is the Winkel tripel projection

Winkel tripel projection

The Winkel tripel projection is a modified azimuthal map projection, one of three projections proposed by Oswald Winkel in 1921...

, the standard projection for world maps made by the National Geographic Society

National Geographic Society

The National Geographic Society , headquartered in Washington, D.C. in the United States, is one of the largest non-profit scientific and educational institutions in the world. Its interests include geography, archaeology and natural science, the promotion of environmental and historical...

. The minimum distortion is on the central meridian at latitudes of 30 degrees (North and South). (Other examples ).

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

Tissot's Indicatrix

Tissot’s indicatrix, or ellipse of distortion, is a concept developed by French mathematician Nicolas Auguste Tissot, in 1859 and 1871, to measure and illustrate distortions due to map projection...

.

The equidistant projection

The equidistant projection, or the Plate Carrée

Equirectangular projection

The equirectangular projection is a very simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about 100 AD...

 (french for "flat square"), is the simplest and most ancient projection, having been in use since before the time of Ptolemy

Ptolemy

Claudius Ptolemaeus , known in English as Ptolemy , was a Roman citizen of Greek ancestry. He was a mathematician, astronomer, geographer, astrologer and a poet of a single epigram in the Greek Anthology. He lived in Egypt under the Roman Empire, and is believed to have been born in the town of...

. The sphere is mapped into a rectangle by the equations.
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

Tissot's Indicatrix

Tissot’s indicatrix, or ellipse of distortion, is a concept developed by French mathematician Nicolas Auguste Tissot, in 1859 and 1871, to measure and illustrate distortions due to map projection...

. 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

Mercator projection

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569. It became the standard map projection for nautical purposes because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as...

 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

Lambert cylindrical equal-area projection

In cartography, the Lambert cylindrical equal-area projection, Lambert cylindrical projection, or cylindrical equal-area projection is acylindrical, equal areamap projection.-History:...

 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).

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)

Universal Transverse Mercator coordinate system

The Universal Transverse Mercator coordinate system is a grid-based method of specifying locations on the surface of the Earth that is a practical application of a 2-dimensional Cartesian coordinate system...

 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

Universal Transverse Mercator coordinate system

The Universal Transverse Mercator coordinate system is a grid-based method of specifying locations on the surface of the Earth that is a practical application of a 2-dimensional Cartesian coordinate system...

 (conformal) projections the maximum scale factor can almost reach (over the mapped region); for the British OSGB

British national grid reference system

The British national grid reference system is a system of geographic grid references commonly used in Great Britain, different from using latitude and longitude....

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.