thumb|upright=1.3|A graphical or bar scale. A map would also usually give its scale numerically ("1:50,000", for instance, means that one cm on the map represents 50,000cm of real space, which is 500 meters)
thumb|A bar scale with the nominal scale expressed as "1:600 000", meaning 1 cm on the map to 600,000cm6km on the ground.
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 (also called principal scale or 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 (also called point scale or 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. When scale varies noticeably, it can be accounted for as the scale factor. Tissot's indicatrix is often used to illustrate the variation of point scale across a map.
History
The foundations for quantitative map scaling goes back to ancient China with textual evidence that the idea of map scaling was understood by the second century BC. Ancient Chinese surveyors and cartographers had ample technical resources used to produce maps such as counting rods, carpenter's squares, plumb lines, compasses for drawing circles, and sighting tubes for measuring inclination. Reference frames postulating a nascent coordinate system for identifying locations were hinted by ancient Chinese astronomers that divided the sky into various sectors or lunar lodges.
The Chinese cartographer and geographer Pei Xiu of the Three Kingdoms period created a set of large-area maps that were drawn to scale. He produced a set of principles that stressed the importance of consistent scaling, directional measurements, and adjustments in land measurements in the terrain that was being mapped. 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. (This is discussed further in the examples in the following sections.)
Let P be a point at latitude <math>\varphi</math> and longitude <math>\lambda</math> on the sphere (or ellipsoid). Let Q be a neighbouring point and let <math>\alpha</math> 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 <math>\beta</math>. In general <math>\alpha\ne\beta</math>. 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
::<math>\mu(\lambda,\,\varphi,\,\alpha)=\lim_{Q\to P}\frac{P'Q'}{PQ},</math>
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 <math>(\alpha=0)</math>, the meridian scale is denoted by <math>h(\lambda,\,\varphi)</math> .
Definition: if P and Q lie on the same parallel <math>(\alpha=\pi/2)</math>, the parallel scale is denoted by <math>k(\lambda,\,\varphi)</math>.
Definition: if the point scale depends only on position, not on direction, we say that it is isotropic and conventionally denote its value in any direction by the parallel scale factor <math>k(\lambda,\varphi)</math>.
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).
Point scale for normal cylindrical projections of the sphere
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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 <math>\varphi</math> and longitude <math>\lambda</math> on the sphere. The point Q is at latitude <math>\varphi+\delta\varphi</math> and longitude <math>\lambda+\delta\lambda</math>. The lines PK and MQ are arcs of meridians of length <math>a\,\delta\varphi</math> where <math>a</math> is the radius of the sphere and <math>\varphi</math> is in radian measure. The lines PM and KQ are arcs of parallel circles of length <math>(a\cos\varphi)\delta\lambda</math> with<math>\lambda</math> 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.
thumb|350px|right|Infinitesimal elements on the sphere and a normal cylindrical projection
Normal cylindrical projections of the sphere have <math>x=a\lambda</math> and <math>y</math> 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 <math>\delta x=a\,\delta\lambda</math> and height <math>\delta y</math>. 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.)
:: parallel scale factor <math>\quad k\;=\;\dfrac{\delta x}{a\cos\varphi\,\delta\lambda\,}=\,\sec\varphi\qquad\qquad{}</math>
::meridian scale factor <math>\quad h\;=\;\dfrac{\delta y}{a\,\delta\varphi\,} = \dfrac{y'(\varphi)}{a}</math>
Note that the parallel scale factor <math>k=\sec\varphi</math>
is independent of the definition of <math>y(\varphi)</math> so it is the same for all normal cylindrical projections. It is useful to note that
::at latitude 30 degrees the parallel scale is <math>k=\sec30^{\circ}=2/\sqrt{3}=1.15</math>
::at latitude 45 degrees the parallel scale is <math>k=\sec45^{\circ}=\sqrt{2}=1.414</math>
::at latitude 60 degrees the parallel scale is <math>k=\sec60^{\circ}=2</math>
::at latitude 80 degrees the parallel scale is <math>k=\sec80^{\circ}=5.76</math>
::at latitude 85 degrees the parallel scale is <math>k=\sec85^{\circ}=11.5</math>
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.
Three examples of normal cylindrical projection
The equirectangular projection
thumb|right|350px|The equidistant projection with [[Tissot's indicatrix of deformation]]
The equirectangular projection,