This article is about scale, nominal scale, principal scale, representative fraction, and scale factor of a map. For bar scale on a map, see Linear scale.
Large scale, medium scale, small scale.
A map is classified as small scale or large scale or sometimes medium scale. Small scale refers to world maps or maps of large regions such as 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 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 (= principal scale = representative fraction). Many maps state the nominal scale and may even display a bar scale (sometimes merely called a '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 (= point scale = particular scale).
If the region of the map is small enough to ignore Earth's curvature—a town plan, for example—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.
The terminology of scales.
Representation of scale.
Map scales may be expressed in words (a lexical scale), as a ratio, or as a fraction. 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 ratio would usually be abbreviated to 1:100M
Bar scale vs. lexical scale.
In addition to the above many maps carry one or more (graphical) bar scales. For example some modern British maps have three bar scales, one each for kilometres, miles and nautical miles.
A lexical scale on a recently published map, in a language known to the user, may be easier for a non-mathematician to visualise than a ratio: if the scale is an inch to two miles and he can see that two villages are about two inches apart on the map then it is easy to work out that they are about four miles apart on the ground.
On the other hand, 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. On the other hand ratios and fractions may be more acceptable to the numerate user since they are immediately accessible in any language. For example a scale of one inch to a furlong (1:7920) 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 but it depends 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.
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 (as tabulated above) might refer to a map at 1:500,000 as small-scale.
Mapping large areas causes noticeable distortions due to flattening the significantly curved surface of the earth. How distortion gets distributed depends on the map projection. Scale varies across the map, and the stated map scale will only be 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 100 kilometres (62 mi) and over an east-west line of about 80 km (at a latitude of 45 degrees). If surveyed to the nearest 1 millimetre (0.039 in), 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 (or ellipsoid) 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.