User:Peter Mercator/Draft for Scale (map) version II

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. The scale factor may be taken as approximately constant for maps of small regions (town plans, OSGB maps covering Great Britain, USGS state maps) but for larger regions (continents), or the whole Earth, the variation of scale of over the map may be appreciable. Scale variation is always present but it is usually negligible for maps of small regions. The variation in scale arises because maps are constructed by projections from the curved surface of a sphere (or an ellipsoid) to the plane,  a process which introduces unavoidable distortion.

It is useful to consider the scale of a map as the product of a principal scale and a particular scale: the first is invariant and it is a measure of the ratio of the actual sizes of the map (sheet) and the mapped region; the second is a variable. These terms are defined below.

from scale (ratio)
Map scales require careful discussion. A town plan may be constructed as an exact scale drawing, but for larger areas a map projection is necessary and no projection can represent the Earth's surface at a uniform scale. In general the scale of a projection depends on position and direction. The variation of scale may be considerable in small scale maps which may cover the globe. In large scale maps of small areas the variation of scale may be insignificant for most purposes but it is always present. The scale of a map projection must be interpreted as a nominal scale. (The usage large and small in relation to map scales relates to their expressions as fractions. The fraction 1/10,000 used for a local map is much larger than 1/100,000,000 used for a global map. There is no fixed dividing line between small and large scales.)

from Linear scale
On large scale maps and charts, those covering a small area, and engineering and architectural drawings, the linear scale can be very simple, a line marked at intervals to show the distance on the earth or object which the distance on the scale represents. A person using the map can use a pair of dividers (or, less precisely, two fingers) to measure a distance by comparing it to the linear scale. The length of the line on the linear scale is equal to the distance represented on the earth multiplied by the map or chart's scale. In most projections, scale varies with latitude, so on small scale maps, covering large areas and a wide range of latitudes, the linear scale must show the scale for the range of latitudes covered by the map. One of these is shown below. Since most nautical charts are constructed using the Mercator projection whose scale varies substantially with latitude, linear scales are not used on charts with scales smaller than approximately 1/80,000.[1][2] Mariners generally use the nautical mile, which, because a nautical mile is approximately equal to a minute of latitude, can be measured against the latitude scale at the sides of the chart.

from Map
Many, but not all, maps are drawn to a scale, expressed as a ratio such as 1:10,000, meaning that 1 of any unit of measurement on the map corresponds exactly, or approximately, to 10,000 of that same unit on the ground. The scale statement may be taken as exact when the region mapped is small enough for the curvature of the Earth to be neglected, for example in a town planner's city map. Over larger regions where the curvature cannot be ignored we must use map projections from the curved surface of the Earth (sphere or ellipsoid) to the plane. The impossibility of flattening the sphere to the plane implies that no map projection can have constant scale: on most projections the best we can achieve is accurate scale on one or two lines (not necessarily straight) on the projection. Thus for map projections we must introduce the concept of point scale, which is a function of position, and strive to keep its variation within narrow bounds. Although the scale statement is nominal it is usually accurate enough for all but the most precise of measurements. Large scale maps, say 1:10,000, cover relatively small regions in great detail and small scale maps, say 1:10,000,000, cover large regions such as nations, continents and the whole globe. The large/small terminology arose from the practice of writing scales as numerical fractions: 1/10,000 is larger than 1/10,000,000. There is no exact dividing line between large and small but 1/100,000 might well be considered as a medium scale. Examples of large scale maps are the 1:25,000 maps produced for hikers; on the other hand maps intended for motorists at 1:250,000 or 1:1,000,000 are small scale. It is important to recognize that even the most accurate maps sacrifice a certain amount of accuracy in scale to deliver a greater visual usefulness to its user. For example, the width of roads and small streams are exaggerated when they are too narrow to be shown on the map at true scale; that is, on a printed map they would be narrower than could be perceived by the naked eye. The same applies to computer maps where the smallest unit is the pixel. A narrow stream say must be shown to have the width of a pixel even if at the map scale it would be a small fraction of the pixel width.

Principal and particular scales
The basic data of the cartographer is the latitude and longitude of all surface features plotted on a spherical or an ellipsoidal approximation of the geoid which, in its turn, approximates the actual surface of the Earth. The construction of a paper (or screen) map may be conceptually divided into two stages. In the first stage the data is plotted on a small generating globe (or ellipsoid) at a size comparable to that of the required map. This involves reduction by a constant fraction which is called the principal scale associated with the map. Alternative names are the nominal scale and the representative fraction (normally abbreviated to RF). There is no distortion associated with this stage.

The second stage is a mathematical projection from the generating globe (or ellipsoid) to the plane, producing the finished map. This step necessarily involves distortion and in general the ratio of the distance between two nearby points on the map and the corresponding points on the generating globe is not constant. The scale factor defined in this way is the particular scale (or point-scale) although in mathematical treatments of map projections this terminology is often reduced to scale factor or simply scale. The particular scale is a function of position which may be calculated from the mathematical formulae defining the projection: examples of such calculations are given below. In most cases the particular scale is equal to unity at some points or on some lines of the final map. Tissot's Indicatrix may be used to illustrate the variation of particular scale over the map. The deviation of the particular scale from unity is a measure of the distortion introduced by the projection. As an example, for the Mercator projection the particular scale on the equator is unity but it becomes larger at high latitudes where the distortion is clearly evident.

The actual scale to be used in relating map distance to true ground distance is the product of the RF and the particular scale.

Representation factor (RF) or principal scale
The RF, or principal scale, should be indicated on any map or plan. It may be expressed in words (a lexical scale), as a ratio, as a fraction or as a graphical bar scale. Examples are:
 * 'one centimetre to one hundred metres'   or    1:10,000   or    1/10,000 or    '1 to 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 ratio would usually be abbreviated to 1:100M)

Lexical scales
A lexical scale (in a known language and in known units) is easy to comprehend: if the scale is one inch to two miles then two locations which are four inches apart on the map are separated by about eight miles apart on the ground (assuming minimal scale variation). The interpretation of lexical scales on old maps is often problematic: a scale of one pouce to one league may be about 1:144,000 but it depends on the cartographer's choice of the many possible definitions for a league.

Bar scales
The RF may also be indicated by one or more (graphical) bar scales. This allows simple transference (by dividers or ruler) from map intervals to the scale bar if the scale variation is minimal. Bar scales on world maps may be worse than useless. (See below)

For example some British maps presently (2009) use three bar scales for kilometres, miles and nautical miles. The following illustration illustrates two scales on an NOAA chart with a principal scale of 1:10,000.

Large scale, medium scale, small scale
A map is classified as small scale or large scale or sometimes medium scale. Small scale refers to maps of large regions or world maps: they show large areas of land on a small space. Large scale maps such as county maps or town plans show smaller areas in more detail. The terminology reflects the relative sizes of the representative fractions: an RF of 1/100,000,000 for a world map is much smaller than an RF of 1/10,000 for a town plan.

The following table describes typical ranges for these scales but there are no hard and fast dividing lines between them. The terms are sometimes used in the absolute sense of the table, but other times in a relative sense. For example, a map reader whose work refers solely to large-scale maps (as tabulated above) might refer to a map at 1:500,000 as small-scale.

Large-scale maps with curvature neglected
The region over which the earth can be regarded as flat depends on the accuracy of the survey measurements. If measured only to the nearest metre, then curvature of the earth 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 surveyed to the nearest 1 mm, then curvature is undetectable over a meridian distance of about 10 km and over an east-west line of about 8 km. Thus a city plan of New York accurate to one metre or a building site plan accurate to one millimetre would both satisfy the above conditions for the neglect of curvature. They can be treated by plane surveying and mapped by scale drawings in which any two points at the same distance on the drawing are at the same distance on the ground. True ground distances are calculated by measuring the distance on the map and then dividing by the RF or, equivalently, simply using dividers to transfer the separation between the points on the map to a bar scale on the map. For example if the RF is 1/10,000 then any interval of 1cm on the map or plan corresponds to 100 metres on the ground.

Point scale (or particular scale)
If a region is surveyed with an accuracy great enough to detect the curvature of the Earth it is necessary to use a map projection from the generating globe to the plane. The resulting distortion is commonly illustrated by the impossibility of smoothing an orange peel onto a flat surface without tearing and deforming it. (More formally it follows from Gauss’s Theorema Egregium). The distortion induces a varying particular scale on the map

Let P be a point at latitude $$\phi$$ and longitude $$\lambda$$ on the generating globe (or ellipsoid). Let Q be a neighbouring point. Let P' and Q' be corresponding points on the map projection, then the particular scale is simply the ratio P'Q'/PQ. As an example suppose that the RF is 1:500M then the Earth, of radius 6378km is reduced to a generating globe of radius 1.27cm and circumference of 8.01cm. In this case neighbouring points would a millimetre or two apart. If the map is projected to a cylinder tangential at the equator then the width of the resulting map will also be 8.01cm and the particular scale between neighbouring equatorial points on map and globe will be equal to unity. At any point away from the equator the particular scale will differ from unity.

In general the particular scale may also depend upon $$\alpha$$, the azimuth angle between the element PQ and the meridian at P and a more formal definition of particular scale is:

Definition: the particular or point scale, $$\mu$$, at a point P is the ratio of the distances P'Q' and PQ in the limit that Q approaches P. We write this as
 * $$\mu(\lambda,\,\phi,\,\alpha)=\lim_{Q\to P}\frac{P'Q'}{PQ},$$

where the notation indicates that the point scale is a function of the position of P and also the direction of the element PQ. Examples in later sections show the calculation of the particular scale.

Definition: if P and Q lie on the same meridian $$(\alpha=0)$$, the meridian scale is denoted by $$h(\lambda,\,\phi)$$.

Definition: if P and Q lie on the same parallel $$(\alpha=\pi/2)$$, the parallel scale is denoted by $$k(\lambda,\,\phi)$$.

Definition: if the point scale depends only on position, not on direction, we say that it is isotropic and conventionally denote its value in any direction by the parallel scale factor $$k(\lambda,\phi)$$.

Definition: A map projection is said to be conformal if the angle between a pair of lines intersecting at a point P is the same as the angle between the projected lines at the projected point P', for all pairs of lines intersecting at point P. A conformal map has an isotropic scale factor. Conversely isotropic scale factors across the map imply a conformal projection.

Isotropy of scale implies that small elements are stretched equally in all directions, that is the shape of a small element is preserved. This is the property of orthomorphism (from Greek 'right shape'). The qualification 'small' means that at some given accuracy of measurement no change can be detected in the scale factor over the element. Since conformal projections have an isotropic scale factor they have also been called orthomorphic projections. For example the Mercator projection is conformal since it is constructed to preserve angles and its scale factor is isotopic, a function of latitude only: Mercator does preserve shape in small regions.

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.)

British maps
The Ordnance Survey of Great Britain (OSGB) produces a number of highly accurate maps based on survey data defined on the OSGB36 datum which, in turn, is based on the Airy 1830 reference ellipsoid with major and minor axes equal to 6377563.396m and 6356256.910m respectively.

The OSGB publishes maps at several different principal scales: this section describes the 1:50,000 series which have replaced the original 1:63,360 (one inch to one mile) series. The conceptual map construction described above posits a reference ellipsoid with axes equal to 127.551m and 127.125m. The data is then projected by using the Redfearn series for the Transverse Mercator projection with a central meridian at 2°W. The resulting map for the portion of the projection covering Great Britain would be 14m wide and 26m high but this is dissected into 204 separate (non-empty) sheets, each measuring 80cm square. The actual map construction is of course direct from the data set to finished map sheets stored as digital images.

Each separate map sheet cover is presently (2013) endorsed with two statements: "1:50,000 scale" and "2 cm to 1 km — 1¼ inches to 1 mile". These must be interpreted as the RF or principal scale or nominal scale for the map. The particular scale on the central meridian is assigned the value of 0.9996012717 : the values at all other points are calculated with the Redfearn series for the Transverse Mercator projection. On a typical sheet (e.g. Edinburgh) the variation of particular scale over the width of the sheet is from 0.999686 on the west to 0.999624 on the east but there is no variation in the north-south direction at this level of accuracy. The corresponding variation in true scale, the product of the RF and the particular scale, varies from 1:50,016 to 1:50,019. Other sheets have slightly different values but over the whole of Britain the particular scale factor is within the range 0.9996 to 1.0007. The particular scale attains the value of unity on two slightly curved lines which are close to the grid lines marked as 180$$\,$$km east and west of the central meridian grid line; this happens only on some map sheets. The small variations described here are negligible for most map users and the RF, or nominal scale, is adequate for most purposes.

Universal Transverse Mercator (UTM)
The UTM is the basis for a world wide mapping at high accuracy. It is now based on WGS84 ellipsoid but it was previously based predominantly on the 1924 International ellipsoid.

Which accurate maps are available?

Need a specific sheet example, preferably not at 1:50000. Possibly one which includes the central meridian for some zone or other.

Need precise coordinates at corners of sheet to evaluate scale.

Small scale maps with scale variation
Small scale maps, such as world maps, are often shown with with a scale as a ratio or with a bar scale. In general this may be interpreted as an RF or particular scale but it is not useful for the interpretation of arbitrary distances on the map

Images


Robinson (secant) 1:35,000,000



Miller cylindrical 1:85,000,000 on 0° + bar scale (equator presumed)



BAr scale with (0°) appended



Finnish map of world RF 1:100,000,000 bar scales for latitudes 0 20 40 60 70



world with bar scale and lexical scale which are not tied to the equator