The scale of a 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 defined as the ratio of a distance on the map to the corresponding distance on the ground. If the region of the map is small enough for the curvature of the Earth to be neglected, then the scale may be taken as a constant ratio over the whole map. (A town plan would be an example). For maps covering larger areas, or the whole Earth, it is essential to use a map projection[1][2] from the sphere (or ellipsoid) to the plane. Such projections inevitably involve distortion and the scale can no longer be considered as constant. It is then necessary to introduce the concept of a variable point scale (or particular scale) which is defined as the ratio of the length of a small line element emanating from a point on the map to the length of the corresponding line element on the surface of the Earth. In general the point scale will vary with the position of the point and also the direction of the line element. Tissot's Indicatrix is often used to illustrate the variation of point scale. 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 (or points) of the resulting map projection. 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 scaled down by a constant ratio known as the representative fraction (RF) or principal scale. Thus we have to differentiate two uses of the word scale: the variable point scale inherent in the projection and the constant scale involved in the reduction to the printed (or screen) map.
The terminology of scales
Map scales may be expressed in words (a lexical scale), as a ratio, or as a fraction. Examples are:
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- 'one centimetre to one hundred metres' or 1:10,000 or 1/10,000
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- 'one inch to one mile' or 1:63,360 or 1/63,360
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- '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).
In addition to the above many maps carry one or more (graphical) bar scales. For example some British maps presently (2009) use three bar scales for kilometres, miles and nautical miles.
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 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. Between 1812 and 1839, many of the traditional units continued in metrified adaptations as the mesures usuelles to one league 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. Between 1812 and 1839, many of the traditional units continued in metrified adaptations as the mesures usuelles may be about 1/144,000 but it depends your choice of the many possible definitions for a league.
Scales are often qualified as small scale, typically for world maps or large regional maps, or large scale, typically for county maps or town plans. The usage of small as against large relates to the expressions as fractions. For example, a town plan may have a scale fraction of 1/10,000: this is much larger than a scale fraction 1/100,000,000 used for a world map. There is no hard and fast dividing line between small and large scales.
There are two interpretations of scale statements. If a small area has been mapped neglecting the curvature of the Earth then the above scale statements may be taken as exact. On the other hand for high accuracy and/or large area coverage the curvature must be taken into account and the above scale statements must be reinterpreted as either nominal scales or a definition of the representative fraction. These distinctions are clarified below.
Large scale maps with curvature neglected
The region over which we can take the earth as sensibly flat depends on the accuracy of the survey measurements. If measured 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 surveyed 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.[3] 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 multiplying by the inverse of the scale fraction or, equivalently, simply using dividers to transfer the separation between the points on the map to a bar scale on the map.
Point scale (or particular scale)
It is known that a sphere (or ellipsoid) cannot be projected to the plane without distortion (as illustrated by the impossibility of smoothing an orange peel onto a flat surface). More formally it follows from the Theorema Egregium of Gauss Johann Carl Friedrich Gauss (pronounced /ˈɡaʊs/; German: Gauß listen , Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy. The only true representation of a sphere at constant scale is another sphere such as the schoolroom globe. There is a limit to the practical size of such a globe and for detailed mapping we must use projections. The immediate corollary is that in any projection of the sphere to the plane the scale is variable: a constant separation on the map does not correspond to a constant separation on the ground. Graphical bar scales may be present on the map but they must be used with caution for they will be accurate on only some lines of the map. (This is discussed further in the examples in the following sections.) A good atlas will usually discuss scale variation in its preface.
Let P be a point at latitude φ and longitude λ on the sphere (or ellipsoid). Let Q be a neighbouring point and let α 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 corresponding points on the projection. The angle between the direction P'Q' and the projection of the meridian is the bearing β. In general . Comment: this precise distinction between azimuth (on the Earth's surface) and bearing (on the map) is not universally observed, many writers using the terms almost interchangeably.
Definition: the point scale 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 P and Q lie on the same meridian (α = 0), the meridian scale is denoted by .
Definition: if P and Q lie on the same parallel (α = π / 2), the parallel scale is denoted by .
Definition: if the point scale depends only on position, not on direction, we say that it is isotropic Isotropy is uniformity in all directions, it is derived from the Greek iso and tropos (direction). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary systematically, dependent on direction and conventionally denote its value in any direction by the parallel scale factor k(λ,φ).
Definition: A map projection is said to be conformal if the angle between two lines intersecting at a point P is the same as the angle between the projected lines at the projected point P'. A conformal map has an isotropic scale factor. Conversely an isotropic scale factor implies 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[1] pages 203—206.)
The representative fraction (RF) or principal scale
There are two conventions used in setting down the equations of any given projection. For example, the equirectangular cylindrical projection may be written as
- cartographers: x = aλ x = aφ
- mathematicians: x = λ x = φ
Here we shall adopt the first of these conventions (following the usage in the surveys by Snyder). Clearly the above projection equations define positions on a huge cylinder wrapped around the Earth and then unrolled. We say that these coordinates define the projection map which must be distinguished logically from the actual printed (or viewed) maps. If the definition of point scale in the previous section is in terms of the projection map then we can expect the scale factors to be close to unity. For normal tangent cylindrical projections the scale along the equator is k=1 and in general the scale changes as we move off the equator. Analysis of scale on the projection map is an investigation of the change of k away from its true value of unity.
Actual printed maps are produced from the projection map by a constant scaling denoted by a ratio such as 1:100M (for whole world maps) or 1:10000 (for such as town plans). To avoid confusion in the use of the word 'scale' this constant scale fraction is called the representative fraction (RF) of the printed map and it is to be identified with the ratio printed on the map. The actual printed map coordinates for the equirectangular cylindrical projection are
- printed map: x = (RF)aλ y = (RF)aφ
This convention allows a clear distinction of the intrinsic projection scaling and the reduction scaling.
From this point we ignore the RF and work with the projection map.
Visualisation of point scale: the Tissot indicatrix
The Winkel tripel projection with Tissot's Indicatrix of deformationConsider a small circle on the surface of the Earth centred at a point P at latitude φ and longitude λ. Since the point scale varies with position and direction the projection of the 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. Superimposing these distortion ellipses on the map projection conveys 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 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 conservation, and the study of world culture and history. The minimum distortion is on the central meridian at latitudes of 30 degrees (North and South). (Other examples[4][5]).
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 φ and longitude λ on the sphere. The point Q is at latitude φ + δφ and longitude λ + δλ. The lines PK and MQ are arcs of meridians of length aδφ where a is the radius of the sphere and φ is in radian measure. The lines PM and KQ are arcs of parallel circles of length (acosφ)δλ withλ in radian measure. In deriving a point 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.
Infinitesimal elements on the sphere and a normal cylindrical projectionNormal cylindrical projections of the sphere have x = aλ and y a function of latitude only. Therefore the infinitesimal element PMQK on the sphere projects to an infinitesimal element P'M'Q'K' which is an exact rectangle with a base δx = aδλ and height δy. By comparing the elements on sphere and projection we can immediately deduce expressions for the scale factors on parallels and meridians. (We defer the treatment of the scale in a general direction to a mathematical addendum to this page.)
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- parallel scale factor
- meridian scale factor
Note that the parallel scale factor k = secφ is independent of the definition of y(φ) so it is the same for all normal cylindrical projections. It is useful to note that
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- at latitude 30 degrees the parallel scale is
- at latitude 45 degrees the parallel scale is
- at latitude 60 degrees the parallel scale is
- at latitude 80 degrees the parallel scale is
- at latitude 85 degrees the parallel scale is
The following examples illustrate three 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
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give the scale with the diagram This is taken from the drawings at the time of the extension from Parkgate after 1886 Note that there is still just a single line through the station Go to Top
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hu, 24 Jun 2010 17:57:47 GM
It turns out that the MTA recently had to cut back on a few subway lines (I hear the V train is out), and you may have seen some of the new subway . maps. being posted around the city. Here's an interesting interactive application from the ...
