The scale of a

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.

Further examples of Tissot's indicatrix

at Wikimedia Commons. ).

map
A map is a symbol
A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

is the ratio
In mathematics, a ratio indicates 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 ...

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 harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

'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 indicates 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 ...

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 music, rock supergroup (music), supergroup project consisting of Sevendust members John Connolly (musician), John Connolly and Vinnie Hornsby, Alter Bridge and Creed (band), Creed drummer Scott Phillips (musician), Sco ...

. 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
A linear scale, also called a bar scale, scale bar, graphic scale, or graphical scale, is a means of visually showing the Scale (map), scale of a map, nautical chart, engineering drawing, or architectural drawing. A scale bar is common element of ...

(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 froUSGS 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
Cartography (; from Greek χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and using maps. Combining science
Science (from the La ...

is often used to illustrate the variation of point scale across a map.
History

The foundations for quantitative map scaling goes back toancient China
The earliest known written records of the history of China date from as early as 1250 BC, from the Shang dynasty
The Shang dynasty (), also historically known as the Yin dynasty (), was a Chinese dynasty
Dynasties in Chinese h ...

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
East Asia is the eastern region of Asia, which is defined in both Geography, geographical and culture, et ...

, 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 specif ...

's, , 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
A courtesy name (), also known as a style name, is a name bestowed upon one at adulthood in addition to one's given name. This practice is a tradition in the East Asian cultural sphere, including Chi ...

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.
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)'' . 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 aninch
Measuring tape with inches
The inch (symbol: in or ″) is a unit
Unit may refer to:
Arts and entertainment
* UNIT
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television s ...

to two mile
The mile, sometimes the international mile or statute mile to distinguish it from other miles, is a British imperial unit and US customary unit
United States customary units (U.S. customary units) are a system of measurements commonly u ...

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 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
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be ...

(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
A system of measurement is a collection of units of measurement
A unit of measuremen ...

s used to be taught in schools. But a scale of one pouce
image:Poids et mesures.png, 200px, Woodcut dated 1800 illustrating the new decimal units which became the legal norm across all France on 4 November 1800
Before the French Revolution, which started in 1789, units of measurement in France, French u ...

to one league may be about 1:144,000, depending on the cartographer
Cartography (; from Greek χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and using maps. Combining science
Science (from the Latin word ''scienti ...

'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 tospatial scale
Spatial scale is a specific application of the term Scale (disambiguation), 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.
...

.''
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
A map is a symbol
A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherw ...

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 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 themap projection
In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the Surface (mathematics), surface of the globe ...

. Scale varies across the map
A map is a symbol
A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

, 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 thesurvey
Survey may refer to:
Statistics and human research
* Statistical survey
Survey methodology is "the study of survey methods".
As a field of applied statistics concentrating on Survey (human research), human-research surveys, survey methodology s ...

measurements. If measured only to the nearest metre, then curvature of the earth
The earliest documented mention of the spherical Earth concept dates from around the 5th century BC, when it appears in the writings of Greek philosophers. In the 3rd century BC, Hellenistic astronomy established the roughly spheric ...

is undetectable over a meridian distance of about and over an east-west line of about 80 km (at a latitude
In geography
Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and planets. The first person to use the ...

of 45 degrees). If surveyed to the nearest , then curvature is undetectable over a meridian
Meridian, or a meridian line may refer to
Science
* Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon
**Central meridian (planets)
* Meridian (geography), an imaginary arc on the E ...

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 to distinguish it from New York State
New York is a state
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of ...

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
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when add ...

of the scale fraction or, equivalently, simply using dividers to transfer the separation between the points on the map to a bar scale
A linear scale, also called a bar scale, scale bar, graphic scale, or graphical scale, is a means of visually showing the Scale (map), scale of a map, nautical chart, engineering drawing, or architectural drawing. A scale bar is common element of ...

on the map.
Altitude reduction

The variation in altitude, from the ground level down to the sphere's or ellipsoid's surface, also changes the scale of distance measurements.Point scale (or particular scale)

As proved byGauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of m ...

’s ''Theorema Egregium
without distortion. The Mercator projection, shown here, preserves angles but fails to preserve area.
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry
Differential geometry is a Mathe ...

'', a sphere (or ellipsoid) cannot be projected onto a plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early flying machines include all forms of aircraft studied ...

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 physical model, model of Earth, of some other astronomical object, 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 ...

.
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 $\backslash varphi$ and longitude $\backslash lambda$ on the sphere (or ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a Surface (mathemat ...

). Let Q be a neighbouring point and let $\backslash alpha$ 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 $\backslash beta$. In general $\backslash alpha\backslash ne\backslash beta$. 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
::$\backslash mu(\backslash lambda,\backslash ,\backslash varphi,\backslash ,\backslash alpha)=\backslash lim\_\backslash frac,$
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 $(\backslash alpha=0)$, the meridian scale is denoted by $h(\backslash lambda,\backslash ,\backslash varphi)$ .
Definition: if P and Q lie on the same parallel $(\backslash alpha=\backslash pi/2)$, the parallel scale is denoted by $k(\backslash lambda,\backslash ,\backslash varphi)$.
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 from the Greek ''isos'' (ἴσος, "equal") and ''tropos'' (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by ...

and conventionally denote its value in any direction by the parallel scale factor $k(\backslash lambda,\backslash varphi)$.
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 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 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: $x=a\backslash lambda$ $y=a\backslash varphi$ : mathematicians: $x=\backslash lambda$ $y=\backslash varphi$ 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: $x=(RF)a\backslash lambda$ $y=(RF)a\backslash varphi$ 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 $\backslash varphi$ and longitude $\backslash lambda$. 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 Switzerland, Swiss luxury 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 be ...

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
Cartography (; from Greek χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and using maps. Combining science
Science (from the La ...

. 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 (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 examplesat 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 $\backslash varphi$ and longitude $\backslash lambda$ on the sphere. The point Q is at latitude $\backslash varphi+\backslash delta\backslash varphi$ and longitude $\backslash lambda+\backslash delta\backslash lambda$. The lines PK and MQ are arcs of meridians of length $a\backslash ,\backslash delta\backslash varphi$ where $a$ is the radius of the sphere and $\backslash varphi$ is in radian measure. The lines PM and KQ are arcs of parallel circles of length $(a\backslash cos\backslash varphi)\backslash delta\backslash lambda$ with$\backslash lambda$ 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 $x=a\backslash lambda$ and $y$ 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 $\backslash delta\; x=a\backslash ,\backslash delta\backslash lambda$ and height $\backslash delta\; y$. 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 foundbelow
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926–1988), American blues drummer
*Fritz von Below (1853 ...

.)
:: parallel scale factor $\backslash quad\; k\backslash ;=\backslash ;\backslash dfrac=\backslash ,\backslash sec\backslash varphi\backslash qquad\backslash qquad$
::meridian scale factor $\backslash quad\; h\backslash ;=\backslash ;\backslash dfrac\; =\; \backslash dfrac$
Note that the parallel scale factor $k=\backslash sec\backslash varphi$
is independent of the definition of $y(\backslash varphi)$ so it is the same for all normal cylindrical projections. It is useful to note that
::at latitude 30 degrees the parallel scale is $k=\backslash sec30^=2/\backslash sqrt=1.15$
::at latitude 45 degrees the parallel scale is $k=\backslash sec45^=\backslash sqrt=1.414$
::at latitude 60 degrees the parallel scale is $k=\backslash sec60^=2$
::at latitude 80 degrees the parallel scale is $k=\backslash sec80^=5.76$
::at latitude 85 degrees the parallel scale is $k=\backslash sec85^=11.5$
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
Cartography (; from Greek χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and using maps. Combining science
Science (from the La ...

.
Three examples of normal cylindrical projection

The equirectangular projection

The equirectangular projection, also known as the ''Plate Carrée'' (French for "flat square") or (somewhat misleadingly) the equidistant projection, is defined by :$x\; =\; a\backslash lambda,$ $y\; =\; a\backslash varphi,$ where $a$ is the radius of the sphere, $\backslash lambda$ is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at $\backslash lambda\; =0$) and $\backslash varphi$ is the latitude. Note that $\backslash lambda$ and $\backslash varphi$ are in radians (obtained by multiplying the degree measure by a factor of $\backslash pi$/180). The longitude $\backslash lambda$ is in the range $;\; href="/html/ALL/s/\backslash pi,\backslash pi.html"\; ;"title="\backslash pi,\backslash pi">\backslash pi,\backslash pi$addendum
An addendum or appendix, in general, is an addition required to be made to a document by its author subsequent to its printing or publication. It comes from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, I ...

.
The figure illustrates the Tissot indicatrix for this projection. On the equator h=k=1 and the circular elements are undistorted on projection. At higher latitudes the circles are distorted into an ellipse given by stretching in the parallel direction only: there is no distortion in the meridian direction. The ratio of the major axis to the minor axis is $\backslash sec\backslash varphi$. Clearly the area of the ellipse increases by the same factor.
It is instructive to consider the use of bar scales that might appear on a printed version of this projection. The scale is true (k=1) on the equator so that multiplying its length on a printed map by the inverse of the RF (or principal scale) gives the actual circumference of the Earth. The bar scale on the map is also drawn at the true scale so that transferring a separation between two points on the equator to the bar scale will give the correct distance between those points. The same is true on the meridians. On a parallel other than the equator the scale is $\backslash sec\backslash varphi$ so when we transfer a separation from a parallel to the bar scale we must divide the bar scale distance by this factor to obtain the distance between the points when measured along the parallel (which is not the true distance along a great circle
A great circle, also known as an orthodrome, of a sphere
A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a circle in two-dimensional sp ...

). On a line at a bearing of say 45 degrees ($\backslash beta=45^$) the scale is continuously varying with latitude and transferring a separation along the line to the bar scale does not give a distance related to the true distance in any simple way. (But see addendum
An addendum or appendix, in general, is an addition required to be made to a document by its author subsequent to its printing or publication. It comes from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, I ...

). Even if a distance along this line of constant planar angle could be worked out, its relevance is questionable since such a line on the projection corresponds to a complicated curve on the sphere. For these reasons bar scales on small-scale maps must be used with extreme caution.
Mercator projection

TheMercator projection
The Mercator projection () is a cylindrical map projection presented by Flemish
Flemish (''Vlaams'') is a Low Franconian dialect cluster of the Dutch language. It is sometimes referred to as Flemish Dutch (), Belgian Dutch ( ), or Souther ...

maps the sphere to a rectangle (of infinite extent in the $y$-direction) by the equations
:$x\; =\; a\backslash lambda\backslash ,$
:$y\; =\; a\backslash ln\; \backslash left;\; href="/html/ALL/s/tan\_\backslash left(\backslash frac\_+\_\backslash frac\_\backslash right)\_\backslash right.html"\; ;"title="tan\; \backslash left(\backslash frac\; +\; \backslash frac\; \backslash right)\; \backslash right">tan\; \backslash left(\backslash frac\; +\; \backslash frac\; \backslash right)\; \backslash right$addendum
An addendum or appendix, in general, is an addition required to be made to a document by its author subsequent to its printing or publication. It comes from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, I ...

it is shown that the point scale in an arbitrary direction is also equal to $\backslash sec\backslash varphi$ so the scale is isotropic (same in all directions), its magnitude increasing with latitude as $\backslash sec\backslash varphi$. In the Tissot diagram each infinitesimal circular element preserves its shape but is enlarged more and more as the latitude increases.
Lambert's equal area projection

Lambert's equal area projection maps the sphere to a finite rectangle by the equations :$x\; =\; a\backslash lambda\; \backslash qquad\backslash qquad\; y\; =\; a\backslash sin\backslash varphi$ where a, $\backslash lambda$ and $\backslash varphi$ are as in the previous example. Since $y\text{'}(\backslash varphi)=\backslash cos\backslash varphi$ the scale factors are : parallel scale $\backslash quad\; k\backslash ;=\backslash ;\backslash dfrac=\backslash ,\backslash sec\backslash varphi\backslash qquad\backslash qquad$ :meridian scale $\backslash quad\; h\backslash ;=\backslash ;\backslash dfrac\; =\; \backslash ,\backslash cos\backslash varphi$ The calculation of the point scale in an arbitrary direction is givenbelow
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926–1988), American blues drummer
*Fritz von Below (1853 ...

.
The vertical and horizontal scales now compensate each other (hk=1) and in the Tissot diagram each infinitesimal circular element is distorted into an ellipse of the ''same'' area as the undistorted circles on the equator.
Graphs of scale factors

The graph shows the variation of the scale factors for the above three examples. The top plot shows the isotropic Mercator scale function: the scale on the parallel is the same as the scale on the meridian. The other plots show the meridian scale factor for the Equirectangular projection (h=1) and for the Lambert equal area projection. These last two projections have a parallel scale identical to that of the Mercator plot. For the Lambert note that the parallel scale (as Mercator A) increases with latitude and the meridian scale (C) decreases with latitude in such a way that hk=1, guaranteeing area conservation.Scale variation on the Mercator projection

The Mercator point scale is unity on the equator because it is such that the auxiliary cylinder used in its construction is tangential to the Earth at the equator. For this reason the usual projection should be called a tangent projection. The scale varies with latitude as $k=\backslash sec\backslash varphi$. Since $\backslash sec\backslash varphi$ tends to infinity as we approach the poles the Mercator map is grossly distorted at high latitudes and for this reason the projection is totally inappropriate for world maps (unless we are discussing navigation and rhumb lines). However, at a latitude of about 25 degrees the value of $\backslash sec\backslash varphi$ is about 1.1 so Mercator ''is'' accurate to within 10% in a strip of width 50 degrees centred on the equator. Narrower strips are better: a strip of width 16 degrees (centred on the equator) is accurate to within 1% or 1 part in 100. A standard criterion for good large-scale maps is that the accuracy should be within 4 parts in 10,000, or 0.04%, corresponding to $k=1.0004$. Since $\backslash sec\backslash varphi$ attains this value at $\backslash varphi=1.62$ degrees (see figure below, red line). Therefore, the tangent Mercator projection is highly accurate within a strip of width 3.24 degrees centred on the equator. This corresponds to north-south distance of about . Within this strip Mercator is ''very'' good, highly accurate and shape preserving because it is conformal (angle preserving). These observations prompted the development of the transverse Mercator projections in which a meridian is treated 'like an equator' of the projection so that we obtain an accurate map within a narrow distance of that meridian. Such maps are good for countries aligned nearly north-south (likeGreat Britain
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) and a set of 60 such maps is used for the Universal Transverse Mercator (UTM). Note that in both these projections (which are based on various ellipsoids) the transformation equations for x and y and the expression for the scale factor are complicated functions of both latitude and longitude.
Secant, or modified, projections

The basic idea of a secant projection is that the sphere is projected to a cylinder which intersects the sphere at two parallels, say $\backslash varphi\_1$ north and south. Clearly the scale is now true at these latitudes whereas parallels beneath these latitudes are contracted by the projection and their (parallel) scale factor must be less than one. The result is that deviation of the scale from unity is reduced over a wider range of latitudes. As an example, one possible secant Mercator projection is defined by :$x\; =\; 0.9996a\backslash lambda\; \backslash qquad\backslash qquad\; y\; =\; 0.9996a\backslash ln\; \backslash left(\backslash tan\; \backslash left(\backslash frac\; +\; \backslash frac\; \backslash right)\; \backslash right).$ The numeric multipliers do not alter the shape of the projection but it does mean that the scale factors are modified: ::: secant Mercator scale, $\backslash quad\; k\backslash ;=0.9996\backslash sec\backslash varphi.$ Thus * the scale on the equator is 0.9996, * the scale is ''k'' = 1 at a latitude given by $\backslash varphi\_1$ where $\backslash sec\backslash varphi\_1=1/0.9996=1.00004$ so that $\backslash varphi\_1=1.62$ degrees, *k=1.0004 at a latitude $\backslash varphi\_2$ given by $\backslash sec\backslash varphi\_2=1.0004/0.9996=1.0008$ for which $\backslash varphi\_2=2.29$ degrees. Therefore, the projection has $11.0004\; math>,\; that\; is\; an\; accuracy\; of\; 0.04\%,\; over\; a\; wider\; strip\; of\; 4.58\; degrees\; (compared\; with\; 3.24\; degrees\; for\; the\; tangent\; form).\; This\; is\; illustrated\; by\; the\; lower\; (green)\; curve\; in\; the\; figure\; of\; the\; previous\; section.\; Such\; narrow\; zones\; of\; high\; accuracy\; are\; used\; in\; the\; UTM\; and\; the\; British\; OSGB\; projection,\; both\; of\; which\; are\; secant,\; transverse\; Mercator\; on\; the\; ellipsoid\; with\; the\; scale\; on\; the\; central\; meridian\; constant\; at$ k\_0=0.9996$.\; The\; isoscale\; lines\; with$ k=1$are\; slightly\; curved\; lines\; approximately\; 180km\; east\; and\; west\; of\; the\; central\; meridian.\; The\; maximum\; value\; of\; the\; scale\; factor\; is\; 1.001\; for\; UTM\; and\; 1.0007\; for\; OSGB.\; The\; lines\; of\; unit\; scale\; at\; latitude$ \backslash varphi\_1$(north\; and\; south),\; where\; the\; cylindrical\; projection\; surface\; intersects\; the\; sphere,\; are\; the\; standard\; parallels\; of\; the\; secant\; projection.\; Whilst\; a\; narrow\; band\; with$ ,\; k-1,\; 0.0004$is\; important\; for\; high\; accuracy\; mapping\; at\; a\; large\; scale,\; for\; world\; maps\; much\; wider\; spaced\; standard\; parallels\; are\; used\; to\; control\; the\; scale\; variation.\; Examples\; are\; *Behrmann\; with\; standard\; parallels\; at\; 30N,\; 30S.\; *Gall\; equal\; area\; with\; standard\; parallels\; at\; 45N,\; 45S.$ The scale plots for the latter are shown below compared with the Lambert equal area scale factors. In the latter the equator is a single standard parallel and the parallel scale increases from k=1 to compensate the decrease in the meridian scale. For the Gall the parallel scale is reduced at the equator (to k=0.707) whilst the meridian scale is increased (to k=1.414). This gives rise to the gross distortion of shape in the Gall-Peters projection. (On the globe Africa is about as long as it is broad). Note that the meridian and parallel scales are both unity on the standard parallels.Mathematical addendum

For normal cylindrical projections the geometry of the infinitesimal elements gives ::$\backslash text\backslash quad\; \backslash tan\backslash alpha=\backslash frac,$ ::$\backslash text\backslash quad\; \backslash tan\backslash beta=\backslash frac\; =\backslash frac.$ The relationship between the angles $\backslash beta$ and $\backslash alpha$ is ::$\backslash text\backslash quad\; \backslash tan\backslash beta=\backslash frac\; \backslash tan\backslash alpha.\backslash ,$ For the Mercator projection $y\text{'}(\backslash varphi)=a\backslash sec\backslash varphi$ giving $\backslash alpha=\backslash beta$: angles are preserved. (Hardly surprising since this is the relation used to derive Mercator). For the equidistant and Lambert projections we have $y\text{'}(\backslash varphi)=a$ and $y\text{'}(\backslash varphi)=a\backslash cos\backslash varphi$ respectively so the relationship between $\backslash alpha$ and $\backslash beta$ depends upon the latitude $\backslash varphi$. Denote the point scale at P when the infinitesimal element PQ makes an angle $\backslash alpha\; \backslash ,$ with the meridian by $\backslash mu\_.$ It is given by the ratio of distances: ::$\backslash mu\_\backslash alpha\; =\; \backslash lim\_\backslash frac\; =\; \backslash lim\_\backslash frac\; .$ Setting $\backslash delta\; x=a\backslash ,\backslash delta\backslash lambda$ and substituting $\backslash delta\backslash varphi$ and $\backslash delta\; y$ from equations (a) and (b) respectively gives ::$\backslash mu\_\backslash alpha(\backslash varphi)\; =\; \backslash sec\backslash varphi\; \backslash left;\; href="/html/ALL/s/frac\backslash right.html"\; ;"title="frac\backslash right">frac\backslash right$ For the projections other than Mercator we must first calculate $\backslash beta$ from $\backslash alpha$ and $\backslash varphi$ using equation (c), before we can find $\backslash mu\_$. For example, the equirectangular projection has $y\text{'}=a$ so that ::$\backslash tan\backslash beta=\backslash sec\backslash varphi\; \backslash tan\backslash alpha.\backslash ,$ If we consider a line of constant slope $\backslash beta$ on the projection both the corresponding value of $\backslash alpha$ and the scale factor along the line are complicated functions of $\backslash varphi$. There is no simple way of transferring a general finite separation to a bar scale and obtaining meaningful results.Ratio symbol

While thecolon
Colon commonly refers to:
* Colon (punctuation) (:), a punctuation mark
* Major part of large intestine, the final section of the digestive system
Colon may also refer to:
Places
* Colon, Michigan, US
* Colon, Nebraska, US
* Kowloon, Hong Kong, s ...

is often used to express ratios, Unicode
Unicode, formally the Unicode Standard, is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's wri ...

can express a symbol specific to ratios, being slightly raised: .
See also

*Scale (analytical tool)
In the study of complex systems
A complex system is a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system, surrounded and influenced by its env ...

* Scale (ratio)
The scale ratio of a Physical model, model represents the Proportionality (mathematics), proportional ratio of a linear dimension of the model to the same feature of the original. Examples include a 3-dimensional scale model of a building or the sc ...

* Scaling (geometry)
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's metho ...

* Spatial scale
Spatial scale is a specific application of the term Scale (disambiguation), 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.
...

References

{{DEFAULTSORT:Scale (Map) Cartography Chinese inventions Measurement