# Scale (cartography)

The scale or map scale is the reduction ratio of maps , plans , relief models , terrain profiles and globes . It is defined as the ratio of a length on the map ( map route ) to its equivalent in nature ( natural route ).

The specific representation in numerical or graphical form is referred to as a scale specification . The scale is usually given as the proportion 1: scale number, so this reduction ratio is the reciprocal of the scale number (and vice versa).

## Definition of the standard

### basis

The prerequisite for the scale of a cartographic document is a system of measurements and a measurement . The reference area is usually the earth's ellipsoid , that is, the natural route is not the shortest connection between two points on the earth's surface, but always the route reduced to sea level (see geodetic line or orthodrome ).

The scale is usually given in the form 1: scale number. The values ​​are calculated as follows:

${\ displaystyle {\ text {scale}} = {\ frac {\ text {map route}} {\ text {natural route}}}}$
${\ displaystyle {\ text {scale number}} = {\ frac {\ text {natural route}} {\ text {map route}}}}$

For a correct result, of course, both lengths must be in the same unit of measurement. For example, if the map route is 1 cm long on a scale of 1: 50,000 (read: one in fifty thousand), then the natural route is 50,000 cm, i.e. 0.5 km long.

Since the scale usually refers to the linear reduction ratio, unless otherwise stated, the scale number must be squared when comparing areas. The illustration of a square kilometer on a scale of 1: 50,000 therefore only takes up a quarter of the paper surface on a scale of 1: 100,000.

Examples (common standards are highlighted in bold .)
scale Map route Natural route Typical application
00.001: 1,000 1 cm 010 m Building or land register plan
00.001: 5,000 050 m Basemap
00.01: 25,000 250 m hiking map
00.01: 50,000 500 m Cycling map (overland) - tactical map
00.1: 100,000 001 km Car Map - Tactical Map
00.1: 200,000 002 km Russian operations card
00.1: 250,000 002.5 km Operation card
00.1: 500,000 005 km General Staff Map - Tactical Pilot Chart
01: 1,000,000 010 km Weltkartewerk - Operational Navigation Chart
01: 2,500,000 025 km World map 1: 2,500,000 (Karta Mira)
1: 80,000,000 800 km World map (whole world)

Trekking and hiking maps are also created on a scale that depicts the contiguous trekking area like a national park. For example, the Oulanka National Park with the bear circuit on a scale of 1: 40,000.

### Large and small scales

 Europe map small map scale City map large map scale

Depending on the richness of content and the level of detail of the maps, a distinction is made between large , medium and small scales . The adjectives “big” and “small” refer to the size of an object on the map and not to the scale number. These terms are often confused if the difference between scale and number is not taken into account. For a large scale map, the scale number is therefore small and vice versa. For example, a 1: 25,000 map is larger-scale (the content is shown larger or in greater detail) than a 1: 100,000 map. (You can also interpret the scale as a fraction for this purpose: 1 / 25,000 is greater than 1 / 100,000.)

What is called a large or small scale is relative and depends largely on the subject or state. For example, for engineering geology a map of 1: 200,000 is already considered small-scale, for a geographer, on the other hand, only an overview map from around 1: 2,000,000. In a large state like Russia, 1: 200,000 can still be considered a large scale, while in a small state like Switzerland this is already considered a small scale.

All scales can be grouped into scale ranges, which are characterized by the same or very similar degrees of fineness of the drawing and the content. Example: Large scales such as 1: 20,000, 1: 25,000 and 1: 30,000 allow a comparable level of detail. Map authorities therefore issue their map series in series of scales whose subsequent scales are in a simple relationship to one another and are therefore easy to compare. In many countries the range extends from the scale range around 1: 5000 to 1: 1,000,000 as the smallest scale.

Map bases are kept in their original scale (including the initial scale or recording scale ). This can be greater than or equal to the working scale (also processing scale ). This in turn can be greater than or equal to the final scale in which the cartographic document finally appears.

The minimum scale that is necessary for the perceptible representation of an object is called the threshold scale .

## Types of scale

In this chapter scale are types rather than their concrete representation (scale on a document disclosures treated).

### Length scale

Normally, scales relate to lines, i.e. they are length scales. However, the spherical surface of the earth can not be mapped absolutely true to length on flat maps , because a two-dimensionally curved surface cannot be developed strictly on the map plane (see map network design ). True-to-length maps with a constant length scale over the entire map area are not possible. A certain length scale can only be achieved in certain areas or directions . In the large-scale field, this fact is of little concern for practical use. Especially with small-scale maps and in atlases, the specified length scale only applies to the map center and possibly to certain network lines that are true to length (for example the equator or a meridian ). On world maps you will therefore often find a specification such as the equatorial scale or the scale in the map center .

### Height scale

The height scale (also vertical scale) is a special case of the length scale, as it also relates to a distance. In the case of relief models or terrain profiles , the height can be displayed on a different scale than horizontal lengths. The ratio of the two measures is called the exaggeration factor, the effect is called the exaggeration .

There are five or ten fold superelevations in profiles . In didactics, most modern authors and relief makers reject exaggerations, at least on a large scale. In the case of medium or small-scale relief models in which the characteristics of the mountains would only make up a few millimeters or centimeters, an exaggeration for the purpose of clarification can definitely be useful. However, even there, more than twice the elevation is usually perceived as unnatural.

### Area scale

In contrast to maps that are absolutely true to length, true-to-area images of the earth's surface are possible on a flat map. These have an area scale . Nevertheless, the area scale is seldom used, since it is more difficult to imagine the size ratios of areas than length ratios. A well-known exception is the Peters Atlas .

### Globe scale

Since the globe is the only cartographic document that reproduces the earth in a scaled down without distortion, the term globe scale is also known for the scale of a globe .

### Variable scale

In the case of city maps, it can make sense to depict the city center, which is interesting for tourists, on a larger scale than the outlying areas with few sights. Here, for example with a Hyperboloidprojektion a sliding scale are generated. As a result, wide, outside narrowing meshes are formed in the city center for the search grid. In extreme cases, and especially with strongly curved grid lines, you get the feeling of looking through a magnifying glass.

Variable scales also result in thematic map anamorphic images, in which the size in the representation is not selected proportionally to the actual geometric size, but depending on any attribute.

## Scale specifications

In this chapter scale are disclosures treated, so the specific labeling of the different scale types on a cartographic document. A synonym for this is scale form .

### As a numerical measure

Combination of numerical and graphical scale information common today ( USGS map )

The scale is now expressed as a proportion or, less often, as a fraction . The numerical specification that we are familiar with today only came into use at the beginning of the 19th century and has established itself worldwide since the beginning of the 20th century. In addition to the map title, numerical scales are the most important feature of a map and are therefore usually placed prominently, be it directly after the map title, be it in large font with the legend or clearly visible on the edge of the map.

Round scales are mostly used in cartography because they are easier to calculate with. Sometimes, for example, for reasons of space, maps are published on the map sheet or with non-metric systems of measurement in non-circular scales (examples: city map of Zurich 1: 12,600; topographic map of Great Britain 1: 63,360 corresponding to 6  inches by 1  mile ).

In the case of elevations in relief models and profiles, the scale can be specified in two ways:

• Specification of the length and height scale: "Length scale 1: 5000, height scale 1: 2500"
• Specification of the length scale and the superelevation factor: "Scale 1: 5000, twice superelevated"

Area scales are given, for example, in the form "1 square centimeter corresponds to 6000 square kilometers". Often times, globes simply indicate the diameter.

### As a graphical benchmark

In addition to the numerical scale, a graphical scale, a so-called scale bar , is often indicated on the edge of the map. Old maps can have up to twenty scale bars for different measurement systems, for example those for German, geographical and nautical miles , for hours or Leugen .

In the case of conformal world maps, in which the scale increases according to law, for example from the equator to the poles, scale diagrams can also be found. These are nothing more than pyramid-shaped scale bars, which, however, unlike the scale bars on old maps, are all numbered in the same unit.

As a variant of the graphic scales, atlases occasionally also contain squares with area information or a comparative map of a known country on the edge of the map (for example, the outline of Switzerland on an Asia map of the same scale).

### The benchmark for digital data

The points stored in a digital geographic information system (GIS) as "columns of numbers" are initially free of scale.

If this vector data comes directly from a survey of the earth's surface, for example, it can theoretically be used independent of scale. However, if they have been vectorized from a paper map, i.e. were already generalized beforehand, they should only be visualized on a relatively narrow scale. For example, it doesn't make sense to read out coordinates to the nearest meter (even if that's technically possible).

In the case of raster data derived from vector data or scanned from paper maps (e.g. Top50 of the German Land Surveying Offices on CD-ROM), the standards usually given in the product documentation are only to be understood as an indication of the reference map used as the data basis, since the maps are zoomed on the screen can change the scale. The output scale or screen scale can accordingly be different from the scale of the database or the digitization scale . As a rule, a numerical or graphical scale that changes with zooming is displayed on the screen.

## history

### Development of the theoretical basis

The idea of ​​a scale and the scale of maps was probably known to nautical chart makers as early as the 13th century. But since most map makers had no mathematical training until the 17th century, their maps were neither provided with an exact projection nor with a scale according to today's understanding. In addition, the technical prerequisites for sufficient position determination and measurements were missing.

Longer distances in particular have always been estimated and given in day trips . Measuring devices attached to the bike also provided reasonably useful data for the route in question. With these methods, distances covered could be determined, but not the “beeline”. The result was very inaccurate maps. Only in the 17th century, after the establishment of observatories in Europe, precise position determinations began, which since the 18th century have been linked and condensed by triangulations . This was the first time that the prerequisite for accurate and thus true-to-scale maps was given.

However, the countless pre-metric systems of measurement made surveying extremely difficult. With the development of the metric system in France around 1800, round and thus easily comparable scales became possible for the first time. Therefore, it was not until the 19th century that the equation map route: natural route  = 1: scale number became established . After 1880 the insight emerged that the scale and the map projection are jointly responsible for the most faithful representation of a section of the spherical earth's surface on a flat map and should not be chosen independently of each other.

Simultaneously with the advent of scale indications in maps, literary preoccupation with the scale began. In 1893 , Lewis Carroll was the first to mention the 1: 1 scale map as a theoretical thought game.

### Development of the scale information

Scale bar without figures (portolan map of the Black Sea by Diego Homem approx. 1559)

Scale information is still missing on medieval, Christian mappae mundi , as they were not intended as true-to-scale maps according to today's conception.

The first portolan map , the Pisan map from the last quarter of the 13th century, already has a scale indication. The concept of the scale bar, which looks like a ladder and from which the French term “échelle” is derived, soon developed on postage cards. However, the distances have not yet been quantified, so that it is not always clear which system of measurement has been used. Portolan maps are very accurate, but the scales of the various maritime basins (e.g. the Mediterranean and Black Sea) are sometimes different.

The first world map with a graphic scale is the world map by Andreas Walsperger from 1448. Walsperger supplemented his map by also recording instructions for the use of the scale bar. The first city map of the Renaissance with a scale bar is the Albertine Plan of Vienna from 1421/1422. The first prints of Ptolemy's Geography (from 1477) did not yet contain any scale information. It was not until the Ptolemy edition of 1513 that the concept of the graphic scale was implemented in printed form and made known to a wider audience. Since then, several scale bars have usually been indicated on the maps. So z. B. the Tabula Hungarie (printed in 1528) by Lazarus Secretarius and Georg Tannstetter a scale bar, a total of 80 German miles long. Each mile (about 7.5 km) was broken down into four parts on a second scale bar. A legend explained the application to the reader.

However, such scale bars sometimes had little to do with the scale of the map, since measurements were not yet common and less talented copyists misunderstood the scale bar as a purely graphic element and therefore simply changed its size.

When traveling by stagecoach emerged in the 18th century, scale bars became particularly important. Up to twenty scale bars in the most varied of measurement systems were specified. In contrast to this, world maps and history maps , which were not used for travel or trade, were often published without any scale indications even in the 18th century.

From the beginning of the 19th century, with a slight delay to the development of the metric system of measurement, numerical scales came into use. At first they were written as a fraction. An early example is an overview map of the Austrian Empire from 1822, which shows a "rejuvenation of the scale of 1 / 864,000 of nature". Heinrich Berghaus earned himself by specifying a numerical scale with his magazine Hertha . At the 7th International Congress of Geographers in 1899, the fractional spelling was recommended internationally.

In the 20th century, the now generally accepted numerical scale specification in the form of a proportion established itself worldwide. As a rule, numerical and graphical scale information is combined on official maps today ; with tourist maps one often only finds scale bars.

### Determination of the scale for old maps

The most important library regulations stipulate that the yardstick for the formal indexing of cartographic documents must be recorded. This is not a problem with modern maps, in which the scale is given in numerical form. It is different, however, if only graphical scales (scale bars) in outdated lengths or no scale information at all are available. However, it only makes sense to try to determine the scale if the old map is based on a survey. This is not the case for many maps made before the 18th century.

There are currently at least four different ways in which an ancient map can be scaled:

1. The non-metric length dimensions of the scale bars are converted into metric dimensions.
2. Known routes on the old map are compared to the same routes on modern maps.
3. The scale is determined from the distance between the circles of latitude .
4. The old map is compared with a modern map with the help of analysis software and the scale is calculated from this and graphical error representations (e.g. distortion grids ) are generated.

Often the paper distortion also plays a role, that is, the nominal scale is not identical to the actual scale . Since the paper distortion (difference in scale) is not exactly known, scales that have been determined subsequently should always be rounded sensibly or the presumed target scale given. Examples:

• Scale determined by comparing several routes on a French map 1: 86.617, presumable and therefore rounded scale 1: 86.400 (standard scale of the French system of measurement before 1800)
• Scale 1: 4,208,740 determined with the help of analysis software, meaningful rounded scale specification 1: 4,200,000

This is often not done because the scale calculation is difficult for non-specialized library staff.

## Linguistic aspects

The German language knows two adjective forms, to scale next to, less often, to scale . To be more precise, one says large-scale and small-scale (in addition to the rather unusual large-scale and small-scale ). For more information, see the definition of the scale .