Scale (map)

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.
The first way is the ratio of the size of the generating globe to the size of the Earth. The generating globe is a conceptual model to which the Earth is shrunk and from which the map is projected. The ratio of the Earth's size to the generating globe's size is called the nominal scale. Many maps state the nominal scale and may even display a bar scale to represent it.
The second distinct concept of scale applies to the variation in scale across a map. It is the ratio of the mapped point's scale to the nominal scale. In this case 'scale' means the scale factor.
If the region of the map is small enough to ignore Earth's curvature, such as in a town plan, then a single value can be used as the scale without causing measurement errors. In maps covering larger areas, or the whole Earth, the map's scale may be less useful or even useless in measuring distances. The map projection becomes critical in understanding how scale varies throughout the map. When scale varies noticeably, it can be accounted for as the scale factor. Tissot's indicatrix is often used to illustrate the variation of point scale across a map.

History

The foundations for quantitative map scaling goes back to ancient China with textual evidence that the idea of map scaling was understood by the second century BC. Ancient Chinese surveyors and cartographers had ample technical resources used to produce maps such as counting rods, carpenter's square's, plumb lines, compasses for drawing circles, and sighting tubes for measuring inclination. Reference frames postulating a nascent coordinate system for identifying locations were hinted by ancient Chinese astronomers that divided the sky into various sectors or lunar lodges.
The Chinese cartographer and geographer Pei Xiu of the Three Kingdoms period created a set of large-area maps that were drawn to scale. He produced a set of principles that stressed the importance of consistent scaling, directional measurements, and adjustments in land measurements in the terrain that was being mapped.

Terminology

Representation of scale

Map scales may be expressed in words, as a ratio, or as a fraction. Examples are:

Bar scale vs. lexical scale

In addition to the above many maps carry one or more ' bar scales'''. For example, some modern British maps have three bar scales, one each for kilometres, miles and nautical miles.
A lexical scale in a language known to the user may be easier to visualise than a ratio: if the scale is an inch to two miles and the map user can see two villages that are about two inches apart on the map, then it is easy to work out that the villages are about four miles apart on the ground.
A lexical scale may cause problems if it expressed in a language that the user does not understand or in obsolete or ill-defined units. For example, a scale of one inch to a furlong will be understood by many older people in countries where Imperial units used to be taught in schools. But a scale of one pouce to one league may be about 1:144,000, depending on the cartographer's choice of the many possible definitions for a league, and only a minority of modern users will be familiar with the units used.

Large scale, medium scale, small scale

A small-scale map cover large regions, such as world maps, continents or large nations. In other words, they show large areas of land on a small space. They are called small scale because the representative fraction is relatively small.
Large-scale maps show smaller areas in more detail, such as county maps or town plans might. Such maps are called large scale because the representative fraction is relatively large. For instance a town plan, which is a large-scale map, might be on a scale of 1:10,000, whereas the world map, which is a small scale map, might be on a scale of 1:100,000,000.
The following table describes typical ranges for these scales but should not be considered authoritative because there is no standard:

Classification	Range	Examples
large scale	1:0 – 1:600,000	1:0.00001 for map of virus; 1:5,000 for walking map of town
medium scale	1:600,000 – 1:2,000,000	Map of a country
small scale	1:2,000,000 – 1:∞	1:50,000,000 for world map; 1:10²¹ for map of galaxy

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 might refer to a map at 1:500,000 as small-scale.
In the English language, the word large-scale is often used to mean "extensive". However, as explained above, cartographers use the term "large scale" to refer to less extensive maps – those that show a smaller area. Maps that show an extensive area are "small scale" maps. This can be a cause of confusion.

Scale variation

Mapping large areas causes noticeable distortions because it significantly flattens the curved surface of the earth. How distortion gets distributed depends on the map projection. Scale varies across the map, and the stated map scale is only an approximation. This is discussed in detail below.

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 and over an east-west line of about 80 km. If surveyed to the nearest, then curvature is undetectable over a meridian distance of about 10 km and over an east-west line of about 8 km. Thus a plan of New York City 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)

As proved by Gauss’s Theorema Egregium, a sphere cannot be projected onto a plane without distortion. This is commonly illustrated by the impossibility of smoothing an orange peel onto a flat surface without tearing and deforming it. The only true representation of a sphere at constant scale is another sphere such as a globe.
Given the limited practical size of globes, we must use maps for detailed mapping. Maps require projections. A projection implies distortion: A constant separation on the map does not correspond to a constant separation on the ground. While a map may display a graphical bar scale, the scale must be used with the understanding that it will be accurate on only some lines of the map.
Let P be a point at latitude and longitude on the sphere. 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 corresponding points on the projection. The angle between the direction P'Q' and the projection of the meridian is the bearing. In general. Comment: this precise distinction between azimuth and bearing 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, the meridian scale is denoted by .
Definition: if P and Q lie on the same parallel, the parallel scale is denoted by.
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.
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. 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 isotropic, 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.

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
Here we shall adopt the first of these conventions. 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 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 or 1:10000. To avoid confusion in the use of the word 'scale' this constant
scale fraction is called the representative fraction 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
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.