The scale of a
map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
is the
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
'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
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
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
Projected is an American rock supergroup consisting of Sevendust members John Connolly and Vinnie Hornsby, Alter Bridge and Creed drummer Scott Phillips, and former Submersed and current Tremonti guitarist Eric Friedman. The band released thei ...
. 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, 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.
[This paper can be downloaded fro]
USGS pages.
It gives full details of most projections, together with introductory sections, but it does not derive any of the projections from first principles. Derivation of all the formulae for the Mercator projections may be found in ''The Mercator Projections''.[''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp. 5-8, . This is a survey of virtually all known projections from antiquity to 1993.] When scale varies noticeably, it can be accounted for as the scale factor.
Tissot's indicatrix
In cartography, a Tissot's indicatrix (Tissot indicatrix, Tissot's ellipse, Tissot ellipse, ellipse of distortion) (plural: "Tissot's indicatrices") is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 a ...
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
The earliest known written records of the history of China date from as early as 1250 BC, from the Shang dynasty (c. 1600–1046 BC), during the reign of king Wu Ding. Ancient historical texts such as the '' Book of Documents'' (early chapte ...
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
Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number.
The written fo ...
,
carpenter's square
The steel square is a tool used in carpentry. Carpenters use various tools to lay out structures that are square (that is, built at accurately measured right angles), many of which are made of steel, but the name ''steel square'' refers to a spec ...
's,
plumb lines,
compasses
A compass, more accurately known as a pair of compasses, is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, it can also be used as a tool to mark out distances, in particular, on maps. Compasses c ...
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
Pei Xiu (224–271), courtesy name Jiyan, was a Chinese cartographer, geographer, politician, and writer of the state of Cao Wei during the late Three Kingdoms period and Jin dynasty of China. He was very much trusted by Sima Zhao, and p ...
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 (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 in a language known to the user may be easier to visualise than a ratio: if the scale is an
inch
Measuring tape with inches
The inch (symbol: in or ″) is a unit of length in the British imperial and the United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelfth") ...
to two
mile
The mile, sometimes the international mile or statute mile to distinguish it from other miles, is a British imperial unit and United States customary unit of distance; both are based on the older English unit of length equal to 5,280 English ...
s 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
Lexical may refer to:
Linguistics
* Lexical corpus or lexis, a complete set of all words in a language
* Lexical item, a basic unit of lexicographical classification
* Lexicon, the vocabulary of a person, language, or branch of knowledge
* Lex ...
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
A furlong is a measure of distance in imperial units and United States customary units equal to one eighth of a mile, equivalent to 660 feet, 220 yards, 40 rods, 10 chains or approximately 201 metres. It is now mostly confined to use in hors ...
(1:7920) will be understood by many older people in countries where
Imperial unit
The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units first defined in the British Weights and Measures Act 1824 and continued to be developed thro ...
s used to be taught in schools. But a scale of one
pouce to one
league
League or The League may refer to:
Arts and entertainment
* ''Leagues'' (band), an American rock band
* ''The League'', an American sitcom broadcast on FX and FXX about fantasy football
Sports
* Sports league
* Rugby league, full contact footba ...
may be about 1:144,000, depending on the
cartographer
Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an im ...
'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
:''Contrast to
spatial scale
Spatial scale is a specific application of the term scale for describing or categorizing (e.g. into orders of magnitude) the size of a space (hence ''spatial''), or the extent of it at which a phenomenon or process occurs.
For instance, in phy ...
.''
A map is classified as small scale or large scale or sometimes medium scale. Small scale refers to
world map
A world map is a map of most or all of the surface of Earth. World maps, because of their scale, must deal with the problem of map projection, projection. Maps rendered in two dimensions by necessity distort the display of the three-dimensiona ...
s 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
Spatial scale is a specific application of the term scale for describing or categorizing (e.g. into orders of magnitude) the size of a space (hence ''spatial''), or the extent of it at which a phenomenon or process occurs.
For instance, in phy ...
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:
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.
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
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...
. Scale varies across the
map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
, 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
Survey may refer to:
Statistics and human research
* Statistical survey, a method for collecting quantitative information about items in a population
* Survey (human research), including opinion polls
Spatial measurement
* Surveying, the techniq ...
measurements. If measured only to the nearest metre, then
curvature of the earth
Spherical Earth or Earth's curvature refers to the approximation of figure of the Earth as a sphere.
The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of Greek philosophers. I ...
is undetectable over a meridian distance of about and over an east-west line of about 80 km (at a
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
of 45 degrees). If surveyed to the nearest , then curvature is undetectable over a
meridian
Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to
Science
* Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon
* ...
distance of about 10 km and over an east-west line of about 8 km.
Thus a plan of
New York City
New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the L ...
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
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
’s ''
Theorema Egregium
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
'', a sphere (or ellipsoid) cannot be projected onto a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
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
A globe is a spherical model of Earth, of some other celestial body, or of the celestial sphere. Globes serve purposes similar to maps, but unlike maps, they do not distort the surface that they portray except to scale it down. A model 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. (This is discussed further in the examples in the following sections.)
Let ''P'' be a point at latitude
and longitude
on the sphere (or
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
). 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 (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
, 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
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
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 (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 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. (See Snyder
[ 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:
: mathematicians:
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:
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
Consider a small circle on the surface of the Earth centred at a point P at latitude and longitude . Since the point scale varies with position and direction the projection of the circle on the projection will be distorted. Tissot
Tissot SA () is a Swiss watchmaker. The company was founded in Le Locle, Switzerland by Charles-Félicien Tissot and his son, Charles-Émile Tissot, in 1853. After several mergers and name changes, the group which Tissot SA belonged to was renam ...
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
In cartography, a Tissot's indicatrix (Tissot indicatrix, Tissot's ellipse, Tissot ellipse, ellipse of distortion) (plural: "Tissot's indicatrices") is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 a ...
. The example shown here is the Winkel tripel projection
The Winkel tripel projection (Winkel III), a modified azimuthal map projection of the world, is one of three projections proposed by German cartographer Oswald Winkel (7 January 1874 – 18 July 1953) in 1921. The projection is the arithmetic ...
, the standard projection for world maps made by the National Geographic Society
The National Geographic Society (NGS), headquartered in Washington, D.C., United States, is one of the largest non-profit scientific and educational organizations in the world.
Founded in 1888, its interests include geography, archaeology, and ...
. The minimum distortion is on the central meridian at latitudes of 30 degrees (North and South). (Other examplesFurther examples of Tissot's indicatrix
at Wikimedia Commons.).
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
and longitude
on the sphere. The point Q is at latitude
and longitude
. The lines PK and MQ are
arcs of meridians of length
where
is the radius of the sphere and
is in radian measure. The lines PM and KQ are arcs of parallel circles of length
with
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.
Normal cylindrical projections of the sphere have
and
equal to 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
and height
. By comparing the elements on sphere and projection we can immediately deduce expressions for the scale factors on parallels and meridians. (The treatment of scale in a general direction may be found
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
.)
:: parallel scale factor
::meridian scale factor
Note that the parallel scale factor
is independent of the definition of
so it is the same for all normal cylindrical projections. It is useful to note that
::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
In cartography, a Tissot's indicatrix (Tissot indicatrix, Tissot's ellipse, Tissot ellipse, ellipse of distortion) (plural: "Tissot's indicatrices") is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 a ...
.
Three examples of normal cylindrical projection
The equirectangular projection
The
equirectangular projection
The equirectangular projection (also called the equidistant cylindrical projection or la carte parallélogrammatique projection), and which includes the special case of the plate carrée projection (also called the geographic projection, lat/lon ...
,
[ also known as the ''Plate Carrée'' (French for "flat square") or (somewhat misleadingly) the equidistant projection, is defined by
:
where is the radius of the sphere, is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at ) and is the latitude. Note that and are in radians (obtained by multiplying the degree measure by a factor of /180). The longitude is in the range ]