3312:
4524:
1528:
2178:
1199:
3867:
2849:
3906:
4507:
3641:
3854:
1077:(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.
50:
5317:
1276:
3648:
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.
134:
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.
452:
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
3841:
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
3647:
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
407:
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
2741:
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
296:
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
4514:
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
2771:
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
1254:
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
911:
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
3901:
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.
4515:
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.
2840:). 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.
4017:
175:
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.
5120:
2772:
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
3836:
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 360 km (220 mi). Within this strip Mercator is
42:
1068:
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
4695:
3548:
2663:
1734:
923:(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
4609:
167:
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.
4830:
1820:
351:
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.
444:
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
685:
3172:
2798:
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
3616:
5893:
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
3242:
3656:
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
3850:. 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.
2725:
3719:
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
5494:
5312:{\displaystyle \mu _{\alpha }=\lim _{Q\to P}{\frac {P'Q'}{PQ}}=\lim _{Q\to P}{\frac {\sqrt {\delta x^{2}+\delta y^{2}}}{\sqrt {a^{2}\,\delta \varphi ^{2}+a^{2}\cos ^{2}\varphi \,\delta \lambda ^{2}}}}.}
4356:
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
3754:
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.
3918:
1647:. 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
3376:
5673:
4282:
1955:
2012:
1088:
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
151:
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
2836:) 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
4166:
4999:
4884:
3091:
3465:
1494:
4068:
45:
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)
5358:
1622:
201:'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)
4615:
3474:
2589:
1660:
577:. 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.
1382:
1353:
2153:
2106:
119:
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
854:
768:
4539:
2059:
899:
4950:
2580:
2834:
1413:
575:
4315:
4199:
4347:
814:
5381:
5110:
4493:
3691:
1857:
1176:
1135:
5586:
4910:
3834:
3808:
3748:
3717:
3301:
3275:
2915:
2796:
2766:
1092:(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
4444:
4388:
4229:
4113:
3899:
2997:
2262:
2230:
1886:
1562:
1009:
980:
728:
232:
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.
4750:
3042:
3021:
2331:
1063:
1037:
5617:
5080:
2160:
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
1524:
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.
927:. 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
5738:
5559:
5404:
5059:
4416:
are slightly curved lines approximately 180 km east and west of the central meridian. The maximum value of the scale factor is 1.001 for UTM and 1.0007 for OSGB.
3782:
3419:
3399:
2486:
2431:
2391:
2371:
2351:
2305:
1645:
1514:
1453:
1324:
1304:
1248:
1228:
498:
478:
5718:
5539:
5019:
4740:
3628:
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
1741:
522:
5698:
5519:
5039:
4720:
549:
239:
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
135:
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.
4495:
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
3847:
371:
Mapping large areas causes noticeable distortions because it significantly flattens the curved surface of the earth. How distortion gets distributed depends on the
3100:
2411:
4414:
528:
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
2882:
2285:
1582:
1433:
949:
There are two conventions used in setting down the equations of any given projection. For example, the equirectangular cylindrical projection may be written as
2537:
2466:
3555:
595:
453:
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.)
1584:
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
3323:
3179:
4912:: angles are preserved. (Hardly surprising since this is the relation used to derive Mercator). For the equidistant and Lambert projections we have
3843:
2670:
85:'s surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.
363:
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.
3303:. In the Tissot diagram each infinitesimal circular element preserves its shape but is enlarged more and more as the latitude increases.
300:
The following table describes typical ranges for these scales but should not be considered authoritative because there is no standard:
286:, continents or large nations. In other words, they show large areas of land on a small space. They are called small scale because the
5947:
3874:
The basic idea of a secant projection is that the sphere is projected to a cylinder which intersects the sphere at two parallels, say
5922:
4012:{\displaystyle x=0.9996a\lambda \qquad \qquad y=0.9996a\ln \left(\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\right).}
3758:
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
3311:
395:
is undetectable over a meridian distance of about 100 kilometres (62 mi) and over an east-west line of about 80 km (at a
259:'s choice of the many possible definitions for a league, and only a minority of modern users will be familiar with the units used.
5414:
81:
of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the
3332:
692:
where the notation indicates that the point scale is a function of the position of P and also the direction of the element PQ.
5627:
6017:
1250:. Since the point scale varies with position and direction the projection of the circle on the projection will be distorted.
919:
elements are stretched equally in all directions, that is the shape of a small element is preserved. This is the property of
4234:
412:
of the scale fraction or, equivalently, simply using dividers to transfer the separation between the points on the map to a
96:
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
1897:
4523:
2177:
1962:
1527:
941:. These are not plotted on maps for end users but they feature in many of the standard texts. (See Snyder pages 203—206.)
4118:
4022:
The numeric multipliers do not alter the shape of the projection but it does mean that the scale factors are modified:
1198:
6066:
5740:. There is no simple way of transferring a general finite separation to a bar scale and obtaining meaningful results.
3870:
Comparison of tangent and secant cylindrical, conic and azimuthal map projections with standard parallels shown in red
3866:
1264:
252:
248:
4349:, that is an accuracy of 0.04%, over a wider strip of 4.58 degrees (compared with 3.24 degrees for the tangent form).
1458:
1286:
understanding of scale is to consider an infinitesimal element on the sphere. The figure shows a point P at latitude
359:
is often used to mean "extensive". However, as explained above, cartographers use the term "large scale" to refer to
4033:
1267:. The minimum distortion is on the central meridian at latitudes of 30 degrees (North and South). (Other examples).
937:
on a conformal projection with an isotropic scale, points which have the same scale value may be joined to form the
2190:
4690:{\displaystyle {\text{(b)}}\quad \tan \beta ={\frac {\delta x}{\delta y}}={\frac {a\,\delta \lambda }{\delta y}}.}
3543:{\displaystyle \quad k\;=\;{\dfrac {\delta x}{a\cos \varphi \,\delta \lambda \,}}=\,\sec \varphi \qquad \qquad {}}
2658:{\displaystyle \quad k\;=\;{\dfrac {\delta x}{a\cos \varphi \,\delta \lambda \,}}=\,\sec \varphi \qquad \qquad {}}
1729:{\displaystyle \quad k\;=\;{\dfrac {\delta x}{a\cos \varphi \,\delta \lambda \,}}=\,\sec \varphi \qquad \qquad {}}
408:
the ground. True ground distances are calculated by measuring the distance on the map and then multiplying by the
221:. For example, some modern British maps have three bar scales, one each for kilometres, miles and nautical miles.
5327:
4955:
4840:
3047:
1591:
195:'one centimetre to one hundred metres' or 1:10,000 or 1/10,000
3424:
2848:
5813:
1260:
908:
399:
of 45 degrees). If surveyed to the nearest 1 millimetre (0.039 in), then curvature is undetectable over a
6033:
Some illustrations of the Tissot Indicatrix applied to a variety of projections other than normal cylindrical.
6030:
1358:
1329:
4604:{\displaystyle {\text{(a)}}\quad \tan \alpha ={\frac {a\cos \varphi \,\delta \lambda }{a\,\delta \varphi }},}
2113:
2066:
5792:
823:
737:
287:
224:
A lexical scale in a language known to the user may be easier to visualise than a ratio: if the scale is an
5759:
3316:
2853:
2738:
2182:
2161:
2019:
1256:
1251:
1203:
1181:
This convention allows a clear distinction of the intrinsic projection scaling and the reduction scaling.
869:
409:
392:
164:
136:
35:
2806:
1391:
554:
4287:
4171:
587:
at P is the ratio of the two distances P'Q' and PQ in the limit that Q approaches P. We write this as
156:
4320:
2307:
is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at
779:
6071:
6061:
6042:
5363:
5085:
4456:
3664:
1830:
1140:
1099:
5564:
4915:
4889:
2545:
189:
Map scales may be expressed in words (a lexical scale), as a ratio, or as a fraction. Examples are:
5787:
5749:
4825:{\displaystyle {\text{(c)}}\quad \tan \beta ={\frac {a\sec \varphi }{y'(\varphi )}}\tan \alpha .\,}
3813:
3787:
3727:
3696:
3280:
3254:
2890:
2861:
2775:
2745:
148:
4422:
4360:
4207:
4091:
3877:
2921:
2235:
2203:
198:'one inch to one mile' or 1:63,360 or 1/63,360
5802:
2197:(French for "flat square") or (somewhat misleadingly) the equidistant projection, is defined by
1862:
1538:
985:
956:
701:
3905:
3026:
3005:
2310:
1815:{\displaystyle \quad h\;=\;{\dfrac {\delta y}{a\,\delta \varphi \,}}={\dfrac {y'(\varphi )}{a}}}
1042:
1016:
5064:
3857:
Scale variation near the equator for the tangent (red) and secant (green) Mercator projections.
5995:
5943:
5918:
5834:
5723:
5544:
5386:
5044:
4506:
3761:
3404:
3384:
3277:
so the scale is isotropic (same in all directions), its magnitude increasing with latitude as
2471:
2416:
2376:
2356:
2336:
2290:
1627:
1499:
1438:
1309:
1289:
1233:
1213:
1193:
483:
463:
436:
5703:
5524:
5004:
4725:
3167:{\displaystyle k\;=\;{\dfrac {\delta x}{a\cos \varphi \,\delta \lambda \,}}=\,\sec \varphi .}
507:
5975:
5683:
5504:
5024:
4705:
534:
441:
4446:(north and south), where the cylindrical projection surface intersects the sphere, are the
403:
distance of about 10 km and over an east-west line of about 8 km. Thus a plan of
6010:
5850:
5591:
3640:
2396:
5940:
Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
4393:
30:
This article is about the general mapping concept. For its graphical representation, see
379:, and the stated map scale is only an approximation. This is discussed in detail below.
2867:
2270:
1567:
1418:
425:
387:
The region over which the earth can be regarded as flat depends on the accuracy of the
372:
97:
4353:
This is illustrated by the lower (green) curve in the figure of the previous section.
3853:
2491:
2436:
6055:
5807:
5797:
3611:{\displaystyle \quad h\;=\;{\dfrac {\delta y}{a\,\delta \varphi \,}}=\,\cos \varphi }
1210:
Consider a small circle on the surface of the Earth centred at a point P at latitude
680:{\displaystyle \mu (\lambda ,\,\varphi ,\,\alpha )=\lim _{Q\to P}{\frac {P'Q'}{PQ}},}
404:
270:
244:
152:
53:
A bar scale with the nominal scale expressed as "1:600 000", meaning 1 cm on the map
4531:
For normal cylindrical projections the geometry of the infinitesimal elements gives
1888:
so it is the same for all normal cylindrical projections. It is useful to note that
41:
2800:
1385:
400:
256:
217:
49:
31:
5885:
Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395
866:
and conventionally denote its value in any direction by the parallel scale factor
356:
5061:. Denote the point scale at P when the infinitesimal element PQ makes an angle
3842:
that meridian. Such maps are good for countries aligned nearly north-south (like
1455:
is in radian measure. The lines PM and KQ are arcs of parallel circles of length
862:
if the point scale depends only on position, not on direction, we say that it is
5838:
3721:
3237:{\displaystyle h\;=\;{\dfrac {\delta y}{a\,\delta \varphi \,}}=\,\sec \varphi .}
4510:
Scale variation for the Lambert (green) and Gall (red) equal area projections.
5925:. This is a survey of virtually all known projections from antiquity to 1993.
524:
be the angle between the element PQ and the meridian at P: this angle is the
100:. The ratio of the Earth's size to the generating globe's size is called the
3251:
it is shown that the point scale in an arbitrary direction is also equal to
2856:
of deformation. (The distortion increases without limit at higher latitudes)
501:
413:
388:
283:
160:
113:
5890:
2393:
are in radians (obtained by multiplying the degree measure by a factor of
5979:
2720:{\displaystyle \quad h\;=\;{\dfrac {\delta y}{a\,\delta \varphi \,}}=\,1}
1255:
point scale is changing over the map. The distortion ellipse is known as
863:
396:
17:
4527:
Infinitesimal elements on the sphere and a normal cylindrical projection
1531:
Infinitesimal elements on the sphere and a normal cylindrical projection
5753:
240:
236:
172:
139:
is often used to illustrate the variation of point scale across a map.
3912:
As an example, one possible secant Mercator projection is defined by
3621:
The calculation of the point scale in an arbitrary direction is given
27:
Ratio of distance on a map to the corresponding distance on the ground
2730:
For the calculation of the point scale in an arbitrary direction see
1275:
243:(1:7920) will be understood by many older people in countries where
1184:
From this point we ignore the RF and work with the projection map.
5987:
5983:
5770:
4522:
4505:
3904:
3852:
3639:
3310:
2847:
2176:
1526:
1197:
446:
431:
89:
82:
78:
48:
40:
5720:
and the scale factor along the line are complicated functions of
5756:
can express a symbol specific to ratios, being slightly raised:
5501:
For the projections other than Mercator we must first calculate
2768:. Clearly the area of the ellipse increases by the same factor.
229:
225:
376:
74:
5887:. United States Government Printing Office, Washington, D.C.
5489:{\displaystyle \mu _{\alpha }(\varphi )=\sec \varphi \left.}
1274:
1271:
Point scale for normal cylindrical projections of the sphere
323:
1:0.00001 for map of virus; 1:5,000 for walking map of town
112:). Many maps state the nominal scale and may even display a
5915:
Flattening the Earth: Two Thousand Years of Map Projections
3724:). However, at a latitude of about 25 degrees the value of
2864:
maps the sphere to a rectangle (of infinite extent in the
391:
measurements. If measured only to the nearest metre, then
147:
The foundations for quantitative map scaling goes back to
3371:{\displaystyle x=a\lambda \qquad \qquad y=a\sin \varphi }
5668:{\displaystyle \tan \beta =\sec \varphi \tan \alpha .\,}
3326:
maps the sphere to a finite rectangle by the equations
3315:
Lambert's normal cylindrical equal-area projection with
4277:{\displaystyle \sec \varphi _{2}=1.0004/0.9996=1.0008}
1073:
which must be distinguished logically from the actual
5726:
5706:
5686:
5630:
5594:
5567:
5547:
5527:
5507:
5417:
5389:
5366:
5330:
5123:
5088:
5067:
5047:
5027:
5007:
4958:
4918:
4892:
4843:
4753:
4728:
4708:
4618:
4542:
4459:
4425:
4396:
4363:
4323:
4290:
4237:
4210:
4174:
4121:
4094:
4036:
3921:
3880:
3816:
3790:
3764:
3730:
3699:
3667:
3569:
3558:
3488:
3477:
3427:
3407:
3387:
3335:
3283:
3257:
3192:
3182:
3113:
3103:
3050:
3029:
3008:
2924:
2893:
2870:
2809:
2778:
2748:
2684:
2673:
2603:
2592:
2548:
2494:
2474:
2439:
2419:
2399:
2379:
2359:
2339:
2313:
2293:
2273:
2238:
2206:
2116:
2069:
2022:
1965:
1950:{\displaystyle k=\sec 30^{\circ }=2/{\sqrt {3}}=1.15}
1900:
1865:
1833:
1785:
1755:
1744:
1674:
1663:
1630:
1594:
1570:
1541:
1502:
1461:
1441:
1421:
1394:
1361:
1332:
1312:
1292:
1263:, the standard projection for world maps made by the
1236:
1216:
1143:
1102:
1045:
1019:
988:
959:
872:
826:
782:
740:
704:
598:
557:
537:
510:
486:
466:
440:, a sphere (or ellipsoid) cannot be projected onto a
212:
In addition to the above many maps carry one or more
116:(sometimes merely called a "scale") to represent it.
5942:. Springer (published March 17, 2008). p. 567.
4502:
Gall equal area with standard parallels at 45N, 45S.
2007:{\displaystyle k=\sec 45^{\circ }={\sqrt {2}}=1.414}
953:
cartographers:
1535:Normal cylindrical projections of the sphere have
1188:
Visualisation of point scale: the Tissot indicatrix
945:
The representative fraction (RF) or principal scale
345:1:50,000,000 for world map; 1:10 for map of galaxy
5732:
5712:
5700:on the projection both the corresponding value of
5692:
5667:
5611:
5588:. For example, the equirectangular projection has
5580:
5553:
5533:
5513:
5488:
5398:
5375:
5352:
5311:
5104:
5074:
5053:
5033:
5013:
4993:
4944:
4904:
4878:
4824:
4734:
4714:
4689:
4603:
4487:
4438:
4408:
4382:
4341:
4309:
4276:
4223:
4193:
4161:{\displaystyle \sec \varphi _{1}=1/0.9996=1.00004}
4160:
4107:
4062:
4011:
3893:
3828:
3802:
3776:
3742:
3711:
3685:
3610:
3542:
3471:parallel scale
3459:
3413:
3393:
3370:
3295:
3269:
3236:
3166:
3085:
3036:
3015:
2991:
2909:
2876:
2828:
2790:
2760:
2719:
2657:
2574:
2531:
2480:
2460:
2425:
2405:
2385:
2365:
2345:
2325:
2299:
2279:
2256:
2224:
2147:
2100:
2053:
2006:
1949:
1880:
1851:
1814:
1728:
1639:
1616:
1576:
1556:
1508:
1488:
1447:
1427:
1407:
1376:
1347:
1318:
1298:
1242:
1222:
1170:
1129:
1096:printed map:
1057:
1031:
1003:
974:
893:
848:
808:
762:
722:
679:
569:
543:
516:
492:
472:
297:scale map, might be on a scale of 1:100,000,000.
247:used to be taught in schools. But a scale of one
5933:
5931:
5187:
5138:
3632:area as the undistorted circles on the equator.
629:
2168:Three examples of normal cylindrical projection
1489:{\displaystyle (a\cos \varphi )\delta \lambda }
5406:from equations (a) and (b) respectively gives
4063:{\displaystyle \quad k\;=0.9996\sec \varphi .}
3661:projection. The scale varies with latitude as
4499:Behrmann with standard parallels at 30N, 30S.
2803:). On a line at a bearing of say 45 degrees (
2110:at latitude 85 degrees the parallel scale is
2063:at latitude 80 degrees the parallel scale is
2016:at latitude 60 degrees the parallel scale is
1959:at latitude 45 degrees the parallel scale is
1894:at latitude 30 degrees the parallel scale is
8:
5910:
5908:
5906:
5904:
3846:) and a set of 60 such maps is used for the
5353:{\displaystyle \delta x=a\,\delta \lambda }
4994:{\displaystyle y'(\varphi )=a\cos \varphi }
4879:{\displaystyle y'(\varphi )=a\sec \varphi }
3086:{\displaystyle y'(\varphi )=a\sec \varphi }
1617:{\displaystyle \delta x=a\,\delta \lambda }
4041:
3652:Scale variation on the Mercator projection
3622:
3567:
3563:
3486:
3482:
3460:{\displaystyle y'(\varphi )=\cos \varphi }
3248:
3190:
3186:
3111:
3107:
2837:
2731:
2682:
2678:
2601:
2597:
1753:
1749:
1672:
1668:
1648:
1326:on the sphere. The point Q is at latitude
1084:are produced from the projection map by a
1013:mathematicians:
5725:
5705:
5685:
5680:If we consider a line of constant slope
5664:
5629:
5593:
5572:
5566:
5546:
5526:
5506:
5453:
5422:
5416:
5388:
5365:
5343:
5329:
5297:
5289:
5277:
5267:
5254:
5246:
5240:
5228:
5212:
5202:
5190:
5153:
5141:
5128:
5122:
5093:
5087:
5071:
5066:
5046:
5026:
5006:
5001:respectively so the relationship between
4957:
4917:
4891:
4842:
4821:
4772:
4754:
4752:
4727:
4707:
4666:
4660:
4637:
4619:
4617:
4588:
4576:
4561:
4543:
4541:
4474:
4460:
4458:
4430:
4424:
4395:
4368:
4362:
4322:
4295:
4289:
4260:
4248:
4236:
4215:
4209:
4179:
4173:
4144:
4132:
4120:
4099:
4093:
4035:
3986:
3973:
3920:
3885:
3879:
3815:
3789:
3763:
3729:
3698:
3666:
3598:
3590:
3583:
3568:
3557:
3538:
3526:
3518:
3511:
3487:
3476:
3426:
3406:
3386:
3334:
3282:
3256:
3221:
3213:
3206:
3191:
3181:
3151:
3143:
3136:
3112:
3102:
3049:
3033:
3028:
3012:
3007:
2969:
2956:
2923:
2906:
2892:
2869:
2820:
2808:
2777:
2747:
2713:
2705:
2698:
2683:
2672:
2653:
2641:
2633:
2626:
2602:
2591:
2547:
2518:
2504:
2493:
2473:
2438:
2418:
2398:
2378:
2358:
2338:
2312:
2292:
2272:
2237:
2205:
2133:
2115:
2086:
2068:
2039:
2021:
1991:
1982:
1964:
1934:
1929:
1917:
1899:
1864:
1832:
1784:
1776:
1769:
1754:
1743:
1724:
1712:
1704:
1697:
1673:
1662:
1629:
1607:
1593:
1569:
1540:
1501:
1460:
1440:
1420:
1398:
1393:
1360:
1331:
1311:
1291:
1235:
1215:
1142:
1101:
1044:
1018:
987:
958:
871:
839:
825:
795:
781:
753:
739:
703:
644:
632:
618:
611:
597:
556:
536:
509:
504:). Let Q be a neighbouring point and let
485:
465:
383:Large-scale maps with curvature neglected
255:may be about 1:144,000, depending on the
5878:
5876:
5874:
5872:
5870:
3865:
1377:{\displaystyle \lambda +\delta \lambda }
1348:{\displaystyle \varphi +\delta \varphi }
302:
171:The Chinese cartographer and geographer
6043:Further examples of Tissot's indicatrix
5866:
5826:
5561:using equation (c), before we can find
5112:It is given by the ratio of distances:
4317:degrees. Therefore, the projection has
4088: = 1 at a latitude given by
3097:parallel scale
2148:{\displaystyle k=\sec 85^{\circ }=11.5}
2101:{\displaystyle k=\sec 80^{\circ }=5.76}
6005:
6004:
5993:
5965:
5963:
5961:
5959:
3421:are as in the previous example. Since
3044:are as in the previous example. Since
849:{\displaystyle k(\lambda ,\,\varphi )}
763:{\displaystyle h(\lambda ,\,\varphi )}
263:Large scale, medium scale, small scale
3306:
7:
4702:The relationship between the angles
4419:The lines of unit scale at latitude
2054:{\displaystyle k=\sec 60^{\circ }=2}
1859:is independent of the definition of
1827:Note that the parallel scale factor
894:{\displaystyle k(\lambda ,\varphi )}
776:if P and Q lie on the same parallel
698:if P and Q lie on the same meridian
4081:the scale on the equator is 0.9996,
4030:secant Mercator scale,
3848:Universal Transverse Mercator (UTM)
5889:This paper can be downloaded from
2829:{\displaystyle \beta =45^{\circ }}
1657:parallel scale factor
1408:{\displaystyle a\,\delta \varphi }
1202:The Winkel tripel projection with
570:{\displaystyle \alpha \neq \beta }
355:In the English language, the word
25:
5917:, John P. Snyder, 1993, pp. 5-8,
5752:is often used to express ratios,
4310:{\displaystyle \varphi _{2}=2.29}
4194:{\displaystyle \varphi _{1}=1.62}
1516:in radian measure. In deriving a
931:preserve shape in small regions.
420:Point scale (or particular scale)
282:map cover large regions, such as
6031:Examples of Tissot's indicatrix.
4342:{\displaystyle 1<k<1.0004}
3862:Secant, or modified, projections
2181:The equidistant projection with
1435:is the radius of the sphere and
1259:. The example shown here is the
809:{\displaystyle (\alpha =\pi /2)}
5376:{\displaystyle \delta \varphi }
5105:{\displaystyle \mu _{\alpha }.}
5041:depends upon the latitude
4759:
4624:
4548:
4488:{\displaystyle |k-1|<0.0004}
4037:
3938:
3937:
3686:{\displaystyle k=\sec \varphi }
3559:
3537:
3536:
3478:
3349:
3348:
3324:Lambert's equal area projection
3307:Lambert's equal area projection
2674:
2652:
2651:
2593:
1852:{\displaystyle k=\sec \varphi }
1745:
1723:
1722:
1664:
1171:{\displaystyle y=(RF)a\varphi }
1130:{\displaystyle x=(RF)a\lambda }
915:Isotropy of scale implies that
907:A map projection is said to be
5581:{\displaystyle \mu _{\alpha }}
5434:
5428:
5194:
5145:
4973:
4967:
4945:{\displaystyle y'(\varphi )=a}
4933:
4927:
4905:{\displaystyle \alpha =\beta }
4858:
4852:
4803:
4797:
4475:
4461:
3442:
3436:
3065:
3059:
2884:-direction) by the equations
2575:{\displaystyle y'(\varphi )=1}
2563:
2557:
2526:
2495:
2455:
2440:
2173:The equirectangular projection
1875:
1869:
1802:
1796:
1477:
1462:
1159:
1150:
1118:
1109:
888:
876:
843:
830:
803:
783:
757:
744:
717:
705:
636:
622:
602:
1:
5849:km, as per definition of the
4837:For the Mercator projection
3829:{\displaystyle \varphi =1.62}
3803:{\displaystyle \sec \varphi }
3743:{\displaystyle \sec \varphi }
3712:{\displaystyle \sec \varphi }
3296:{\displaystyle \sec \varphi }
3270:{\displaystyle \sec \varphi }
2910:{\displaystyle x=a\lambda \,}
2852:The Mercator projection with
2791:{\displaystyle \sec \varphi }
2761:{\displaystyle \sec \varphi }
2287:is the radius of the sphere,
1738:meridian scale factor
4439:{\displaystyle \varphi _{1}}
4383:{\displaystyle k_{0}=0.9996}
4224:{\displaystyle \varphi _{2}}
4108:{\displaystyle \varphi _{1}}
3894:{\displaystyle \varphi _{1}}
3552:meridian scale
2992:{\displaystyle y=a\ln \left}
2257:{\displaystyle y=a\varphi ,}
2225:{\displaystyle x=a\lambda ,}
5833:The text "1cm = 6km" is an
3176:meridian scale
2737:The figure illustrates the
2582:the previous section gives
2353:is the latitude. Note that
1881:{\displaystyle y(\varphi )}
1557:{\displaystyle x=a\lambda }
1520:property of the projection
1265:National Geographic Society
1004:{\displaystyle y=a\varphi }
975:{\displaystyle x=a\lambda }
723:{\displaystyle (\alpha =0)}
208:Bar scale vs. lexical scale
6088:
4453:Whilst a narrow band with
4450:of the secant projection.
4390:. The isoscale lines with
3037:{\displaystyle \varphi \,}
3016:{\displaystyle \lambda \,}
2326:{\displaystyle \lambda =0}
2191:equirectangular projection
1384:. The lines PK and MQ are
1191:
1058:{\displaystyle y=\varphi }
1032:{\displaystyle x=\lambda }
423:
375:. Scale varies across the
29:
5075:{\displaystyle \alpha \,}
3750:is about 1.1 so Mercator
1137:
1039:
982:
5972:The Mercator Projections
5896:The Mercator Projections
5883:Snyder, John P. (1987).
5814:On Exactitude in Science
5733:{\displaystyle \varphi }
5554:{\displaystyle \varphi }
5399:{\displaystyle \delta y}
5054:{\displaystyle \varphi }
3777:{\displaystyle k=1.0004}
3414:{\displaystyle \varphi }
3394:{\displaystyle \lambda }
2481:{\displaystyle \varphi }
2426:{\displaystyle \lambda }
2386:{\displaystyle \varphi }
2366:{\displaystyle \lambda }
2346:{\displaystyle \varphi }
2300:{\displaystyle \lambda }
1640:{\displaystyle \delta y}
1509:{\displaystyle \lambda }
1448:{\displaystyle \varphi }
1319:{\displaystyle \lambda }
1299:{\displaystyle \varphi }
1261:Winkel tripel projection
1243:{\displaystyle \lambda }
1223:{\displaystyle \varphi }
925:orthomorphic projections
493:{\displaystyle \lambda }
473:{\displaystyle \varphi }
6016:CS1 maint: postscript (
5970:Osborne, Peter (2013),
5938:Selin, Helaine (2008).
5793:Scale (analytical tool)
5713:{\displaystyle \alpha }
5534:{\displaystyle \alpha }
5014:{\displaystyle \alpha }
4735:{\displaystyle \alpha }
4204:k=1.0004 at a latitude
3636:Graphs of scale factors
3093:the scale factors are:
1090:representative fraction
517:{\displaystyle \alpha }
460:be a point at latitude
331:1:600,000 – 1:2,000,000
288:representative fraction
185:Representation of scale
110:representative fraction
5988:Latex code and figures
5734:
5714:
5694:
5693:{\displaystyle \beta }
5669:
5613:
5582:
5555:
5535:
5515:
5514:{\displaystyle \beta }
5490:
5400:
5377:
5354:
5313:
5106:
5076:
5055:
5035:
5034:{\displaystyle \beta }
5015:
4995:
4946:
4906:
4880:
4826:
4736:
4716:
4715:{\displaystyle \beta }
4691:
4605:
4528:
4511:
4489:
4440:
4410:
4384:
4343:
4311:
4278:
4225:
4195:
4162:
4109:
4064:
4013:
3909:
3895:
3871:
3858:
3830:
3810:attains this value at
3804:
3778:
3744:
3713:
3687:
3644:
3612:
3544:
3467:the scale factors are
3461:
3415:
3395:
3372:
3320:
3297:
3271:
3238:
3168:
3087:
3038:
3017:
2993:
2911:
2878:
2857:
2830:
2792:
2762:
2721:
2659:
2586:parallel scale,
2576:
2533:
2482:
2462:
2427:
2407:
2387:
2367:
2347:
2327:
2301:
2281:
2258:
2226:
2186:
2149:
2102:
2055:
2008:
1951:
1882:
1853:
1816:
1730:
1641:
1618:
1588:rectangle with a base
1578:
1558:
1532:
1510:
1490:
1449:
1429:
1409:
1378:
1349:
1320:
1300:
1279:
1244:
1224:
1207:
1172:
1131:
1059:
1033:
1005:
976:
912:conformal projection.
895:
850:
810:
764:
724:
681:
571:
545:
544:{\displaystyle \beta }
518:
494:
474:
393:curvature of the earth
66:
46:
34:. For other uses, see
6045:at Wikimedia Commons.
5735:
5715:
5695:
5670:
5614:
5583:
5556:
5536:
5516:
5491:
5401:
5378:
5355:
5314:
5107:
5082:with the meridian by
5077:
5056:
5036:
5016:
4996:
4947:
4907:
4881:
4827:
4737:
4717:
4692:
4606:
4526:
4519:Mathematical addendum
4509:
4490:
4441:
4411:
4385:
4344:
4312:
4279:
4226:
4196:
4163:
4110:
4065:
4014:
3908:
3896:
3869:
3856:
3831:
3805:
3779:
3745:
3714:
3688:
3643:
3613:
3545:
3462:
3416:
3396:
3373:
3314:
3298:
3272:
3239:
3169:
3088:
3039:
3018:
2994:
2912:
2879:
2851:
2831:
2793:
2763:
2722:
2660:
2577:
2534:
2483:
2463:
2428:
2413:/180). The longitude
2408:
2388:
2368:
2348:
2328:
2302:
2282:
2259:
2227:
2180:
2150:
2103:
2056:
2009:
1952:
1883:
1854:
1817:
1731:
1642:
1619:
1579:
1559:
1530:
1511:
1491:
1450:
1430:
1410:
1379:
1350:
1321:
1301:
1278:
1245:
1225:
1201:
1173:
1132:
1060:
1034:
1006:
977:
896:
851:
811:
765:
725:
682:
572:
546:
519:
495:
475:
290:is relatively small.
88:The first way is the
52:
44:
5980:10.5281/zenodo.35392
5724:
5704:
5684:
5628:
5612:{\displaystyle y'=a}
5592:
5565:
5545:
5525:
5505:
5415:
5387:
5364:
5328:
5121:
5086:
5065:
5045:
5025:
5005:
4956:
4916:
4890:
4841:
4751:
4726:
4706:
4616:
4540:
4457:
4423:
4394:
4361:
4321:
4288:
4235:
4208:
4172:
4119:
4092:
4034:
3919:
3878:
3814:
3788:
3762:
3728:
3697:
3665:
3556:
3475:
3425:
3405:
3385:
3333:
3281:
3255:
3247:In the mathematical
3180:
3101:
3048:
3027:
3006:
2922:
2891:
2868:
2807:
2776:
2746:
2671:
2590:
2546:
2492:
2472:
2437:
2417:
2406:{\displaystyle \pi }
2397:
2377:
2357:
2337:
2311:
2291:
2271:
2236:
2204:
2193:, also known as the
2114:
2067:
2020:
1963:
1898:
1863:
1831:
1742:
1661:
1628:
1592:
1568:
1539:
1500:
1459:
1439:
1419:
1392:
1359:
1330:
1310:
1290:
1234:
1214:
1141:
1100:
1043:
1017:
986:
957:
870:
824:
780:
738:
702:
596:
555:
535:
508:
484:
464:
5788:Geographic distance
4409:{\displaystyle k=1}
3317:Tissot's indicatrix
2862:Mercator projection
2854:Tissot's indicatrix
2844:Mercator projection
2183:Tissot's indicatrix
2162:Tissot's indicatrix
1257:Tissot's indicatrix
1204:Tissot's indicatrix
137:Tissot's indicatrix
92:of the size of the
6067:Chinese inventions
5803:Scaling (geometry)
5730:
5710:
5690:
5665:
5609:
5578:
5551:
5531:
5511:
5486:
5396:
5373:
5350:
5309:
5201:
5152:
5102:
5072:
5051:
5031:
5011:
4991:
4942:
4902:
4876:
4822:
4732:
4712:
4687:
4601:
4529:
4512:
4485:
4448:standard parallels
4436:
4406:
4380:
4339:
4307:
4274:
4221:
4191:
4158:
4105:
4060:
4009:
3910:
3891:
3872:
3859:
3826:
3800:
3774:
3740:
3709:
3683:
3645:
3608:
3593:
3540:
3521:
3457:
3411:
3391:
3368:
3321:
3293:
3267:
3234:
3216:
3164:
3146:
3083:
3034:
3013:
2989:
2907:
2874:
2858:
2826:
2788:
2758:
2717:
2708:
2655:
2636:
2572:
2529:
2478:
2458:
2423:
2403:
2383:
2363:
2343:
2323:
2297:
2277:
2254:
2222:
2187:
2145:
2098:
2051:
2004:
1947:
1878:
1849:
1812:
1810:
1779:
1726:
1707:
1637:
1614:
1574:
1554:
1533:
1506:
1486:
1445:
1425:
1405:
1374:
1345:
1316:
1296:
1280:
1240:
1220:
1208:
1168:
1127:
1055:
1029:
1001:
972:
891:
846:
806:
760:
720:
677:
643:
567:
541:
514:
500:on the sphere (or
490:
470:
157:carpenter's square
67:
47:
6006:|postscript=
6003:External link in
5835:abuse of notation
5477:
5360:and substituting
5304:
5303:
5234:
5186:
5181:
5137:
4807:
4757:
4682:
4655:
4622:
4596:
4546:
3994:
3981:
3592:
3520:
3215:
3145:
2977:
2964:
2877:{\displaystyle y}
2739:Tissot indicatrix
2707:
2635:
2468:and the latitude
2280:{\displaystyle a}
1996:
1939:
1809:
1778:
1706:
1577:{\displaystyle y}
1428:{\displaystyle a}
1386:arcs of meridians
1194:Tissot indicatrix
672:
628:
437:Theorema Egregium
349:
348:
342:1:2,000,000 – 1:∞
334:Map of a country
65:km on the ground.
16:(Redirected from
6079:
6046:
6040:
6034:
6028:
6022:
6021:
6014:
6008:
6007:
6001:
5999:
5991:
5982:. (Supplements:
5967:
5954:
5953:
5935:
5926:
5912:
5899:
5888:
5880:
5854:
5848:
5844:
5831:
5777:
5773:
5768:
5765:
5763:
5739:
5737:
5736:
5731:
5719:
5717:
5716:
5711:
5699:
5697:
5696:
5691:
5674:
5672:
5671:
5666:
5618:
5616:
5615:
5610:
5602:
5587:
5585:
5584:
5579:
5577:
5576:
5560:
5558:
5557:
5552:
5540:
5538:
5537:
5532:
5520:
5518:
5517:
5512:
5495:
5493:
5492:
5487:
5482:
5478:
5476:
5465:
5454:
5427:
5426:
5405:
5403:
5402:
5397:
5382:
5380:
5379:
5374:
5359:
5357:
5356:
5351:
5318:
5316:
5315:
5310:
5305:
5302:
5301:
5282:
5281:
5272:
5271:
5259:
5258:
5245:
5244:
5235:
5233:
5232:
5217:
5216:
5204:
5203:
5200:
5182:
5180:
5172:
5171:
5163:
5154:
5151:
5133:
5132:
5111:
5109:
5108:
5103:
5098:
5097:
5081:
5079:
5078:
5073:
5060:
5058:
5057:
5052:
5040:
5038:
5037:
5032:
5020:
5018:
5017:
5012:
5000:
4998:
4997:
4992:
4966:
4951:
4949:
4948:
4943:
4926:
4911:
4909:
4908:
4903:
4885:
4883:
4882:
4877:
4851:
4831:
4829:
4828:
4823:
4808:
4806:
4796:
4787:
4773:
4758:
4755:
4741:
4739:
4738:
4733:
4721:
4719:
4718:
4713:
4696:
4694:
4693:
4688:
4683:
4681:
4673:
4661:
4656:
4654:
4646:
4638:
4623:
4620:
4610:
4608:
4607:
4602:
4597:
4595:
4583:
4562:
4547:
4544:
4494:
4492:
4491:
4486:
4478:
4464:
4445:
4443:
4442:
4437:
4435:
4434:
4415:
4413:
4412:
4407:
4389:
4387:
4386:
4381:
4373:
4372:
4348:
4346:
4345:
4340:
4316:
4314:
4313:
4308:
4300:
4299:
4283:
4281:
4280:
4275:
4264:
4253:
4252:
4230:
4228:
4227:
4222:
4220:
4219:
4200:
4198:
4197:
4192:
4184:
4183:
4167:
4165:
4164:
4159:
4148:
4137:
4136:
4114:
4112:
4111:
4106:
4104:
4103:
4069:
4067:
4066:
4061:
4018:
4016:
4015:
4010:
4005:
4001:
4000:
3996:
3995:
3987:
3982:
3974:
3900:
3898:
3897:
3892:
3890:
3889:
3835:
3833:
3832:
3827:
3809:
3807:
3806:
3801:
3783:
3781:
3780:
3775:
3749:
3747:
3746:
3741:
3718:
3716:
3715:
3710:
3692:
3690:
3689:
3684:
3617:
3615:
3614:
3609:
3594:
3591:
3578:
3570:
3549:
3547:
3546:
3541:
3539:
3522:
3519:
3497:
3489:
3466:
3464:
3463:
3458:
3435:
3420:
3418:
3417:
3412:
3400:
3398:
3397:
3392:
3377:
3375:
3374:
3369:
3302:
3300:
3299:
3294:
3276:
3274:
3273:
3268:
3243:
3241:
3240:
3235:
3217:
3214:
3201:
3193:
3173:
3171:
3170:
3165:
3147:
3144:
3122:
3114:
3092:
3090:
3089:
3084:
3058:
3043:
3041:
3040:
3035:
3022:
3020:
3019:
3014:
2998:
2996:
2995:
2990:
2988:
2984:
2983:
2979:
2978:
2970:
2965:
2957:
2916:
2914:
2913:
2908:
2883:
2881:
2880:
2875:
2835:
2833:
2832:
2827:
2825:
2824:
2797:
2795:
2794:
2789:
2767:
2765:
2764:
2759:
2726:
2724:
2723:
2718:
2709:
2706:
2693:
2685:
2667:meridian scale
2664:
2662:
2661:
2656:
2654:
2637:
2634:
2612:
2604:
2581:
2579:
2578:
2573:
2556:
2538:
2536:
2535:
2532:{\displaystyle }
2530:
2522:
2508:
2488:is in the range
2487:
2485:
2484:
2479:
2467:
2465:
2464:
2461:{\displaystyle }
2459:
2433:is in the range
2432:
2430:
2429:
2424:
2412:
2410:
2409:
2404:
2392:
2390:
2389:
2384:
2372:
2370:
2369:
2364:
2352:
2350:
2349:
2344:
2332:
2330:
2329:
2324:
2306:
2304:
2303:
2298:
2286:
2284:
2283:
2278:
2263:
2261:
2260:
2255:
2231:
2229:
2228:
2223:
2154:
2152:
2151:
2146:
2138:
2137:
2107:
2105:
2104:
2099:
2091:
2090:
2060:
2058:
2057:
2052:
2044:
2043:
2013:
2011:
2010:
2005:
1997:
1992:
1987:
1986:
1956:
1954:
1953:
1948:
1940:
1935:
1933:
1922:
1921:
1887:
1885:
1884:
1879:
1858:
1856:
1855:
1850:
1821:
1819:
1818:
1813:
1811:
1805:
1795:
1786:
1780:
1777:
1764:
1756:
1735:
1733:
1732:
1727:
1725:
1708:
1705:
1683:
1675:
1646:
1644:
1643:
1638:
1624:and height
1623:
1621:
1620:
1615:
1583:
1581:
1580:
1575:
1563:
1561:
1560:
1555:
1515:
1513:
1512:
1507:
1495:
1493:
1492:
1487:
1454:
1452:
1451:
1446:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1383:
1381:
1380:
1375:
1354:
1352:
1351:
1346:
1325:
1323:
1322:
1317:
1305:
1303:
1302:
1297:
1249:
1247:
1246:
1241:
1229:
1227:
1226:
1221:
1177:
1175:
1174:
1169:
1136:
1134:
1133:
1128:
1064:
1062:
1061:
1056:
1038:
1036:
1035:
1030:
1010:
1008:
1007:
1002:
981:
979:
978:
973:
900:
898:
897:
892:
855:
853:
852:
847:
815:
813:
812:
807:
799:
769:
767:
766:
761:
729:
727:
726:
721:
686:
684:
683:
678:
673:
671:
663:
662:
654:
645:
642:
576:
574:
573:
568:
550:
548:
547:
542:
523:
521:
520:
515:
499:
497:
496:
491:
479:
477:
476:
471:
303:
129:particular scale
94:generating globe
64:
60:
21:
6087:
6086:
6082:
6081:
6080:
6078:
6077:
6076:
6052:
6051:
6050:
6049:
6041:
6037:
6029:
6025:
6015:
6002:
5992:
5969:
5968:
5957:
5950:
5937:
5936:
5929:
5913:
5902:
5882:
5881:
5868:
5863:
5858:
5857:
5851:metric prefixes
5846:
5842:
5832:
5828:
5823:
5784:
5775:
5769:
5766:
5758:
5757:
5746:
5722:
5721:
5702:
5701:
5682:
5681:
5626:
5625:
5595:
5590:
5589:
5568:
5563:
5562:
5543:
5542:
5523:
5522:
5503:
5502:
5466:
5455:
5449:
5418:
5413:
5412:
5385:
5384:
5362:
5361:
5326:
5325:
5293:
5273:
5263:
5250:
5236:
5224:
5208:
5173:
5164:
5156:
5155:
5124:
5119:
5118:
5089:
5084:
5083:
5063:
5062:
5043:
5042:
5023:
5022:
5003:
5002:
4959:
4954:
4953:
4919:
4914:
4913:
4888:
4887:
4844:
4839:
4838:
4789:
4788:
4774:
4749:
4748:
4724:
4723:
4704:
4703:
4674:
4662:
4647:
4639:
4614:
4613:
4584:
4563:
4538:
4537:
4521:
4455:
4454:
4426:
4421:
4420:
4392:
4391:
4364:
4359:
4358:
4319:
4318:
4291:
4286:
4285:
4244:
4233:
4232:
4211:
4206:
4205:
4175:
4170:
4169:
4128:
4117:
4116:
4095:
4090:
4089:
4032:
4031:
3972:
3968:
3961:
3957:
3917:
3916:
3881:
3876:
3875:
3864:
3812:
3811:
3786:
3785:
3760:
3759:
3726:
3725:
3695:
3694:
3663:
3662:
3654:
3638:
3579:
3571:
3554:
3553:
3498:
3490:
3473:
3472:
3428:
3423:
3422:
3403:
3402:
3383:
3382:
3331:
3330:
3309:
3279:
3278:
3253:
3252:
3202:
3194:
3178:
3177:
3123:
3115:
3099:
3098:
3051:
3046:
3045:
3025:
3024:
3004:
3003:
2955:
2951:
2944:
2940:
2920:
2919:
2889:
2888:
2866:
2865:
2846:
2816:
2805:
2804:
2774:
2773:
2744:
2743:
2694:
2686:
2669:
2668:
2613:
2605:
2588:
2587:
2549:
2544:
2543:
2490:
2489:
2470:
2469:
2435:
2434:
2415:
2414:
2395:
2394:
2375:
2374:
2355:
2354:
2335:
2334:
2309:
2308:
2289:
2288:
2269:
2268:
2234:
2233:
2202:
2201:
2175:
2170:
2129:
2112:
2111:
2082:
2065:
2064:
2035:
2018:
2017:
1978:
1961:
1960:
1913:
1896:
1895:
1861:
1860:
1829:
1828:
1788:
1787:
1765:
1757:
1740:
1739:
1684:
1676:
1659:
1658:
1626:
1625:
1590:
1589:
1566:
1565:
1537:
1536:
1498:
1497:
1457:
1456:
1437:
1436:
1417:
1416:
1390:
1389:
1357:
1356:
1328:
1327:
1308:
1307:
1288:
1287:
1273:
1232:
1231:
1212:
1211:
1196:
1190:
1139:
1138:
1098:
1097:
1041:
1040:
1015:
1014:
984:
983:
955:
954:
947:
868:
867:
822:
821:
778:
777:
736:
735:
700:
699:
664:
655:
647:
646:
594:
593:
553:
552:
533:
532:
506:
505:
482:
481:
462:
461:
428:
422:
385:
369:
367:Scale variation
320:1:0 – 1:600,000
306:Classification
265:
210:
187:
182:
145:
106:principal scale
62:
58:
39:
28:
23:
22:
15:
12:
11:
5:
6085:
6083:
6075:
6074:
6069:
6064:
6054:
6053:
6048:
6047:
6035:
6023:
5955:
5949:978-1402049606
5948:
5927:
5900:
5865:
5864:
5862:
5859:
5856:
5855:
5825:
5824:
5822:
5819:
5818:
5817:
5810:
5805:
5800:
5795:
5790:
5783:
5780:
5745:
5742:
5729:
5709:
5689:
5678:
5677:
5676:
5675:
5663:
5660:
5657:
5654:
5651:
5648:
5645:
5642:
5639:
5636:
5633:
5608:
5605:
5601:
5598:
5575:
5571:
5550:
5530:
5510:
5499:
5498:
5497:
5496:
5485:
5481:
5475:
5472:
5469:
5464:
5461:
5458:
5452:
5448:
5445:
5442:
5439:
5436:
5433:
5430:
5425:
5421:
5395:
5392:
5372:
5369:
5349:
5346:
5342:
5339:
5336:
5333:
5322:
5321:
5320:
5319:
5308:
5300:
5296:
5292:
5288:
5285:
5280:
5276:
5270:
5266:
5262:
5257:
5253:
5249:
5243:
5239:
5231:
5227:
5223:
5220:
5215:
5211:
5207:
5199:
5196:
5193:
5189:
5185:
5179:
5176:
5170:
5167:
5162:
5159:
5150:
5147:
5144:
5140:
5136:
5131:
5127:
5101:
5096:
5092:
5070:
5050:
5030:
5010:
4990:
4987:
4984:
4981:
4978:
4975:
4972:
4969:
4965:
4962:
4941:
4938:
4935:
4932:
4929:
4925:
4922:
4901:
4898:
4895:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4850:
4847:
4835:
4834:
4833:
4832:
4820:
4817:
4814:
4811:
4805:
4802:
4799:
4795:
4792:
4786:
4783:
4780:
4777:
4771:
4768:
4765:
4762:
4731:
4711:
4700:
4699:
4698:
4697:
4686:
4680:
4677:
4672:
4669:
4665:
4659:
4653:
4650:
4645:
4642:
4636:
4633:
4630:
4627:
4611:
4600:
4594:
4591:
4587:
4582:
4579:
4575:
4572:
4569:
4566:
4560:
4557:
4554:
4551:
4520:
4517:
4504:
4503:
4500:
4484:
4481:
4477:
4473:
4470:
4467:
4463:
4433:
4429:
4405:
4402:
4399:
4379:
4376:
4371:
4367:
4351:
4350:
4338:
4335:
4332:
4329:
4326:
4306:
4303:
4298:
4294:
4273:
4270:
4267:
4263:
4259:
4256:
4251:
4247:
4243:
4240:
4218:
4214:
4202:
4190:
4187:
4182:
4178:
4157:
4154:
4151:
4147:
4143:
4140:
4135:
4131:
4127:
4124:
4102:
4098:
4082:
4075:
4074:
4073:
4072:
4071:
4070:
4059:
4056:
4053:
4050:
4047:
4044:
4040:
4020:
4019:
4008:
4004:
3999:
3993:
3990:
3985:
3980:
3977:
3971:
3967:
3964:
3960:
3956:
3953:
3950:
3947:
3944:
3941:
3936:
3933:
3930:
3927:
3924:
3888:
3884:
3863:
3860:
3825:
3822:
3819:
3799:
3796:
3793:
3773:
3770:
3767:
3739:
3736:
3733:
3708:
3705:
3702:
3682:
3679:
3676:
3673:
3670:
3653:
3650:
3637:
3634:
3619:
3618:
3607:
3604:
3601:
3597:
3589:
3586:
3582:
3577:
3574:
3566:
3562:
3550:
3535:
3532:
3529:
3525:
3517:
3514:
3510:
3507:
3504:
3501:
3496:
3493:
3485:
3481:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3434:
3431:
3410:
3390:
3379:
3378:
3367:
3364:
3361:
3358:
3355:
3352:
3347:
3344:
3341:
3338:
3319:of deformation
3308:
3305:
3292:
3289:
3286:
3266:
3263:
3260:
3245:
3244:
3233:
3230:
3227:
3224:
3220:
3212:
3209:
3205:
3200:
3197:
3189:
3185:
3174:
3163:
3160:
3157:
3154:
3150:
3142:
3139:
3135:
3132:
3129:
3126:
3121:
3118:
3110:
3106:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3057:
3054:
3032:
3011:
3000:
2999:
2987:
2982:
2976:
2973:
2968:
2963:
2960:
2954:
2950:
2947:
2943:
2939:
2936:
2933:
2930:
2927:
2917:
2905:
2902:
2899:
2896:
2873:
2845:
2842:
2823:
2819:
2815:
2812:
2787:
2784:
2781:
2757:
2754:
2751:
2742:minor axis is
2728:
2727:
2716:
2712:
2704:
2701:
2697:
2692:
2689:
2681:
2677:
2665:
2650:
2647:
2644:
2640:
2632:
2629:
2625:
2622:
2619:
2616:
2611:
2608:
2600:
2596:
2571:
2568:
2565:
2562:
2559:
2555:
2552:
2528:
2525:
2521:
2517:
2514:
2511:
2507:
2503:
2500:
2497:
2477:
2457:
2454:
2451:
2448:
2445:
2442:
2422:
2402:
2382:
2362:
2342:
2322:
2319:
2316:
2296:
2276:
2265:
2264:
2253:
2250:
2247:
2244:
2241:
2221:
2218:
2215:
2212:
2209:
2185:of deformation
2174:
2171:
2169:
2166:
2158:
2157:
2156:
2155:
2144:
2141:
2136:
2132:
2128:
2125:
2122:
2119:
2108:
2097:
2094:
2089:
2085:
2081:
2078:
2075:
2072:
2061:
2050:
2047:
2042:
2038:
2034:
2031:
2028:
2025:
2014:
2003:
2000:
1995:
1990:
1985:
1981:
1977:
1974:
1971:
1968:
1957:
1946:
1943:
1938:
1932:
1928:
1925:
1920:
1916:
1912:
1909:
1906:
1903:
1877:
1874:
1871:
1868:
1848:
1845:
1842:
1839:
1836:
1825:
1824:
1823:
1822:
1808:
1804:
1801:
1798:
1794:
1791:
1783:
1775:
1772:
1768:
1763:
1760:
1752:
1748:
1736:
1721:
1718:
1715:
1711:
1703:
1700:
1696:
1693:
1690:
1687:
1682:
1679:
1671:
1667:
1636:
1633:
1613:
1610:
1606:
1603:
1600:
1597:
1573:
1553:
1550:
1547:
1544:
1505:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1444:
1424:
1404:
1401:
1397:
1373:
1370:
1367:
1364:
1355:and longitude
1344:
1341:
1338:
1335:
1315:
1306:and longitude
1295:
1272:
1269:
1239:
1230:and longitude
1219:
1206:of deformation
1192:Main article:
1189:
1186:
1179:
1178:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1071:projection map
1066:
1065:
1054:
1051:
1048:
1028:
1025:
1022:
1011:
1000:
997:
994:
991:
971:
968:
965:
962:
946:
943:
939:isoscale lines
890:
887:
884:
881:
878:
875:
845:
842:
838:
835:
832:
829:
820:is denoted by
818:parallel scale
805:
802:
798:
794:
791:
788:
785:
759:
756:
752:
749:
746:
743:
734:is denoted by
732:meridian scale
719:
716:
713:
710:
707:
690:
689:
688:
687:
676:
670:
667:
661:
658:
653:
650:
641:
638:
635:
631:
627:
624:
621:
617:
614:
610:
607:
604:
601:
566:
563:
560:
540:
513:
489:
480:and longitude
469:
421:
418:
384:
381:
373:map projection
368:
365:
347:
346:
343:
340:
336:
335:
332:
329:
325:
324:
321:
318:
314:
313:
310:
307:
276:
275:
264:
261:
245:Imperial units
209:
206:
205:
204:
203:
202:
199:
196:
186:
183:
181:
178:
144:
141:
56:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6084:
6073:
6070:
6068:
6065:
6063:
6060:
6059:
6057:
6044:
6039:
6036:
6032:
6027:
6024:
6019:
6012:
5997:
5989:
5985:
5981:
5977:
5973:
5966:
5964:
5962:
5960:
5956:
5951:
5945:
5941:
5934:
5932:
5928:
5924:
5923:0-226-76747-7
5920:
5916:
5911:
5909:
5907:
5905:
5901:
5897:
5892:
5886:
5879:
5877:
5875:
5873:
5871:
5867:
5860:
5852:
5841:; strictly, 1
5840:
5836:
5830:
5827:
5820:
5816:
5815:
5811:
5809:
5808:Spatial scale
5806:
5804:
5801:
5799:
5798:Scale (ratio)
5796:
5794:
5791:
5789:
5786:
5785:
5781:
5779:
5772:
5761:
5755:
5751:
5743:
5741:
5727:
5707:
5687:
5661:
5658:
5655:
5652:
5649:
5646:
5643:
5640:
5637:
5634:
5631:
5624:
5623:
5622:
5621:
5620:
5606:
5603:
5599:
5596:
5573:
5569:
5548:
5528:
5508:
5483:
5479:
5473:
5470:
5467:
5462:
5459:
5456:
5450:
5446:
5443:
5440:
5437:
5431:
5423:
5419:
5411:
5410:
5409:
5408:
5407:
5393:
5390:
5370:
5367:
5347:
5344:
5340:
5337:
5334:
5331:
5306:
5298:
5294:
5290:
5286:
5283:
5278:
5274:
5268:
5264:
5260:
5255:
5251:
5247:
5241:
5237:
5229:
5225:
5221:
5218:
5213:
5209:
5205:
5197:
5191:
5183:
5177:
5174:
5168:
5165:
5160:
5157:
5148:
5142:
5134:
5129:
5125:
5117:
5116:
5115:
5114:
5113:
5099:
5094:
5090:
5068:
5048:
5028:
5008:
4988:
4985:
4982:
4979:
4976:
4970:
4963:
4960:
4939:
4936:
4930:
4923:
4920:
4899:
4896:
4893:
4873:
4870:
4867:
4864:
4861:
4855:
4848:
4845:
4818:
4815:
4812:
4809:
4800:
4793:
4790:
4784:
4781:
4778:
4775:
4769:
4766:
4763:
4760:
4747:
4746:
4745:
4744:
4743:
4729:
4709:
4684:
4678:
4675:
4670:
4667:
4663:
4657:
4651:
4648:
4643:
4640:
4634:
4631:
4628:
4625:
4612:
4598:
4592:
4589:
4585:
4580:
4577:
4573:
4570:
4567:
4564:
4558:
4555:
4552:
4549:
4536:
4535:
4534:
4533:
4532:
4525:
4518:
4516:
4508:
4501:
4498:
4497:
4496:
4482:
4479:
4471:
4468:
4465:
4451:
4449:
4431:
4427:
4417:
4403:
4400:
4397:
4377:
4374:
4369:
4365:
4354:
4336:
4333:
4330:
4327:
4324:
4304:
4301:
4296:
4292:
4271:
4268:
4265:
4261:
4257:
4254:
4249:
4245:
4241:
4238:
4216:
4212:
4203:
4188:
4185:
4180:
4176:
4155:
4152:
4149:
4145:
4141:
4138:
4133:
4129:
4125:
4122:
4100:
4096:
4087:
4084:the scale is
4083:
4080:
4079:
4078:
4057:
4054:
4051:
4048:
4045:
4042:
4038:
4029:
4028:
4027:
4026:
4025:
4024:
4023:
4006:
4002:
3997:
3991:
3988:
3983:
3978:
3975:
3969:
3965:
3962:
3958:
3954:
3951:
3948:
3945:
3942:
3939:
3934:
3931:
3928:
3925:
3922:
3915:
3914:
3913:
3907:
3903:
3886:
3882:
3868:
3861:
3855:
3851:
3849:
3845:
3844:Great Britain
3840:
3823:
3820:
3817:
3797:
3794:
3791:
3771:
3768:
3765:
3756:
3753:
3737:
3734:
3731:
3723:
3706:
3703:
3700:
3680:
3677:
3674:
3671:
3668:
3660:
3651:
3649:
3642:
3635:
3633:
3631:
3626:
3624:
3605:
3602:
3599:
3595:
3587:
3584:
3580:
3575:
3572:
3564:
3560:
3551:
3533:
3530:
3527:
3523:
3515:
3512:
3508:
3505:
3502:
3499:
3494:
3491:
3483:
3479:
3470:
3469:
3468:
3454:
3451:
3448:
3445:
3439:
3432:
3429:
3408:
3388:
3365:
3362:
3359:
3356:
3353:
3350:
3345:
3342:
3339:
3336:
3329:
3328:
3327:
3325:
3318:
3313:
3304:
3290:
3287:
3284:
3264:
3261:
3258:
3250:
3231:
3228:
3225:
3222:
3218:
3210:
3207:
3203:
3198:
3195:
3187:
3183:
3175:
3161:
3158:
3155:
3152:
3148:
3140:
3137:
3133:
3130:
3127:
3124:
3119:
3116:
3108:
3104:
3096:
3095:
3094:
3080:
3077:
3074:
3071:
3068:
3062:
3055:
3052:
3030:
3009:
2985:
2980:
2974:
2971:
2966:
2961:
2958:
2952:
2948:
2945:
2941:
2937:
2934:
2931:
2928:
2925:
2918:
2903:
2900:
2897:
2894:
2887:
2886:
2885:
2871:
2863:
2855:
2850:
2843:
2841:
2839:
2821:
2817:
2813:
2810:
2802:
2785:
2782:
2779:
2769:
2755:
2752:
2749:
2740:
2735:
2733:
2714:
2710:
2702:
2699:
2695:
2690:
2687:
2679:
2675:
2666:
2648:
2645:
2642:
2638:
2630:
2627:
2623:
2620:
2617:
2614:
2609:
2606:
2598:
2594:
2585:
2584:
2583:
2569:
2566:
2560:
2553:
2550:
2540:
2523:
2519:
2515:
2512:
2509:
2505:
2501:
2498:
2475:
2452:
2449:
2446:
2443:
2420:
2400:
2380:
2360:
2340:
2320:
2317:
2314:
2294:
2274:
2251:
2248:
2245:
2242:
2239:
2232:
2219:
2216:
2213:
2210:
2207:
2200:
2199:
2198:
2196:
2192:
2184:
2179:
2172:
2167:
2165:
2163:
2142:
2139:
2134:
2130:
2126:
2123:
2120:
2117:
2109:
2095:
2092:
2087:
2083:
2079:
2076:
2073:
2070:
2062:
2048:
2045:
2040:
2036:
2032:
2029:
2026:
2023:
2015:
2001:
1998:
1993:
1988:
1983:
1979:
1975:
1972:
1969:
1966:
1958:
1944:
1941:
1936:
1930:
1926:
1923:
1918:
1914:
1910:
1907:
1904:
1901:
1893:
1892:
1891:
1890:
1889:
1872:
1866:
1846:
1843:
1840:
1837:
1834:
1806:
1799:
1792:
1789:
1781:
1773:
1770:
1766:
1761:
1758:
1750:
1746:
1737:
1719:
1716:
1713:
1709:
1701:
1698:
1694:
1691:
1688:
1685:
1680:
1677:
1669:
1665:
1656:
1655:
1654:
1653:
1652:
1650:
1634:
1631:
1611:
1608:
1604:
1601:
1598:
1595:
1587:
1571:
1551:
1548:
1545:
1542:
1529:
1525:
1523:
1519:
1503:
1483:
1480:
1474:
1471:
1468:
1465:
1442:
1422:
1402:
1399:
1395:
1387:
1371:
1368:
1365:
1362:
1342:
1339:
1336:
1333:
1313:
1293:
1285:
1282:The key to a
1277:
1270:
1268:
1266:
1262:
1258:
1253:
1237:
1217:
1205:
1200:
1195:
1187:
1185:
1182:
1165:
1162:
1156:
1153:
1147:
1144:
1124:
1121:
1115:
1112:
1106:
1103:
1095:
1094:
1093:
1091:
1087:
1083:
1078:
1076:
1072:
1052:
1049:
1046:
1026:
1023:
1020:
1012:
998:
995:
992:
989:
969:
966:
963:
960:
952:
951:
950:
944:
942:
940:
936:
932:
930:
926:
922:
921:orthomorphism
918:
913:
910:
906:
902:
885:
882:
879:
873:
865:
861:
857:
840:
836:
833:
827:
819:
800:
796:
792:
789:
786:
775:
771:
754:
750:
747:
741:
733:
714:
711:
708:
697:
693:
674:
668:
665:
659:
656:
651:
648:
639:
633:
625:
619:
615:
612:
608:
605:
599:
592:
591:
590:
589:
588:
586:
582:
578:
564:
561:
558:
551:. In general
538:
531:
527:
511:
503:
487:
467:
459:
454:
450:
448:
443:
439:
438:
433:
430:As proved by
427:
419:
417:
415:
411:
406:
405:New York City
402:
398:
394:
390:
382:
380:
378:
374:
366:
364:
362:
358:
353:
344:
341:
338:
337:
333:
330:
327:
326:
322:
319:
316:
315:
311:
308:
305:
304:
301:
298:
295:
291:
289:
285:
281:
274:
272:
271:spatial scale
267:
266:
262:
260:
258:
254:
250:
246:
242:
238:
233:
231:
227:
222:
220:
219:
215:
207:
200:
197:
194:
193:
192:
191:
190:
184:
179:
177:
174:
169:
166:
162:
158:
154:
153:counting rods
150:
149:ancient China
142:
140:
138:
132:
130:
126:
123:(also called
122:
117:
115:
111:
107:
104:(also called
103:
102:nominal scale
99:
95:
91:
86:
84:
80:
76:
72:
54:
51:
43:
37:
33:
19:
6038:
6026:
5984:Maxima files
5971:
5939:
5914:
5895:
5884:
5829:
5812:
5747:
5744:Ratio symbol
5679:
5500:
5323:
4836:
4701:
4530:
4513:
4452:
4447:
4418:
4355:
4352:
4085:
4076:
4021:
3911:
3873:
3838:
3757:
3751:
3658:
3655:
3646:
3629:
3627:
3620:
3380:
3322:
3246:
3001:
2859:
2801:great circle
2770:
2736:
2729:
2541:
2266:
2195:Plate Carrée
2194:
2188:
2159:
1826:
1585:
1534:
1521:
1517:
1284:quantitative
1283:
1281:
1209:
1183:
1180:
1089:
1085:
1082:printed maps
1081:
1079:
1074:
1070:
1067:
948:
938:
934:
933:
928:
924:
920:
916:
914:
904:
903:
859:
858:
817:
773:
772:
731:
695:
694:
691:
584:
580:
579:
529:
525:
457:
455:
451:
435:
429:
416:on the map.
386:
370:
360:
354:
350:
328:medium scale
299:
293:
292:
279:
277:
269:Contrast to
268:
257:cartographer
234:
223:
216:
213:
211:
188:
170:
146:
133:
128:
124:
121:scale factor
120:
118:
109:
105:
101:
93:
87:
70:
68:
32:Linear scale
6072:Measurement
6062:Cartography
5891:USGS pages.
5839:equals sign
5776:∶
3722:rhumb lines
935:Definition:
905:Definition:
860:Definition:
774:Definition:
696:Definition:
585:point scale
581:Definition:
357:large-scale
339:small scale
317:large scale
294:Large-scale
280:small-scale
214:(graphical)
180:Terminology
161:plumb lines
125:point scale
55:corresponds
6056:Categories
5861:References
5845:cm=0.00001
5748:While the
4284:for which
1388:of length
424:See also:
284:world maps
218:bar scales
57:to 600,000
5728:φ
5708:α
5688:β
5659:α
5656:
5650:φ
5647:
5638:β
5635:
5619:so that
5574:α
5570:μ
5549:φ
5529:α
5509:β
5474:β
5471:
5463:α
5460:
5447:φ
5444:
5432:φ
5424:α
5420:μ
5391:δ
5371:φ
5368:δ
5348:λ
5345:δ
5332:δ
5295:λ
5291:δ
5287:φ
5284:
5252:φ
5248:δ
5222:δ
5206:δ
5195:→
5146:→
5130:α
5126:μ
5095:α
5091:μ
5069:α
5049:φ
5029:β
5009:α
4989:φ
4986:
4971:φ
4931:φ
4900:β
4894:α
4874:φ
4871:
4856:φ
4816:α
4813:
4801:φ
4785:φ
4782:
4767:β
4764:
4730:α
4710:β
4676:δ
4671:λ
4668:δ
4649:δ
4641:δ
4632:β
4629:
4593:φ
4590:δ
4581:λ
4578:δ
4574:φ
4571:
4556:α
4553:
4469:−
4428:φ
4293:φ
4246:φ
4242:
4231:given by
4213:φ
4177:φ
4130:φ
4126:
4097:φ
4055:φ
4052:
3989:φ
3976:π
3966:
3955:
3935:λ
3883:φ
3818:φ
3798:φ
3795:
3738:φ
3735:
3707:φ
3704:
3681:φ
3678:
3606:φ
3603:
3588:φ
3585:δ
3573:δ
3534:φ
3531:
3516:λ
3513:δ
3509:φ
3506:
3492:δ
3455:φ
3452:
3440:φ
3409:φ
3389:λ
3381:where a,
3366:φ
3363:
3346:λ
3291:φ
3288:
3265:φ
3262:
3229:φ
3226:
3211:φ
3208:δ
3196:δ
3159:φ
3156:
3141:λ
3138:δ
3134:φ
3131:
3117:δ
3081:φ
3078:
3063:φ
3031:φ
3010:λ
3002:where a,
2972:φ
2959:π
2949:
2938:
2904:λ
2822:∘
2811:β
2786:φ
2783:
2756:φ
2753:
2703:φ
2700:δ
2688:δ
2649:φ
2646:
2631:λ
2628:δ
2624:φ
2621:
2607:δ
2561:φ
2516:π
2502:π
2499:−
2476:φ
2453:π
2447:π
2444:−
2421:λ
2401:π
2381:φ
2361:λ
2341:φ
2315:λ
2295:λ
2249:φ
2217:λ
2135:∘
2127:
2088:∘
2080:
2041:∘
2033:
1984:∘
1976:
1919:∘
1911:
1873:φ
1847:φ
1844:
1800:φ
1774:φ
1771:δ
1759:δ
1720:φ
1717:
1702:λ
1699:δ
1695:φ
1692:
1678:δ
1632:δ
1612:λ
1609:δ
1596:δ
1552:λ
1504:λ
1484:λ
1481:δ
1475:φ
1472:
1443:φ
1403:φ
1400:δ
1372:λ
1369:δ
1363:λ
1343:φ
1340:δ
1334:φ
1314:λ
1294:φ
1238:λ
1218:φ
1166:φ
1125:λ
1053:φ
1027:λ
999:φ
970:λ
909:conformal
886:φ
880:λ
864:isotropic
841:φ
834:λ
793:π
787:α
755:φ
748:λ
709:α
637:→
620:α
613:φ
606:λ
600:μ
565:β
562:≠
559:α
539:β
512:α
502:ellipsoid
488:λ
468:φ
414:bar scale
312:Examples
165:compasses
114:bar scale
98:projected
18:Map scale
5996:citation
5837:for the
5782:See also
5767:∶
5600:′
5324:Setting
5169:′
5161:′
4964:′
4924:′
4849:′
4794:′
4201:degrees,
4168:so that
4115:where
3784:. Since
3693:. Since
3433:′
3249:addendum
3056:′
2838:addendum
2732:addendum
2554:′
1793:′
1086:constant
660:′
652:′
401:meridian
397:latitude
5754:Unicode
4886:giving
4156:1.00004
3659:tangent
1080:Actual
1075:printed
530:bearing
526:azimuth
410:inverse
251:to one
241:furlong
237:lexical
228:to two
173:Pei Xiu
143:History
77:is the
5946:
5921:
5847:
5843:
5764:
4483:0.0004
4378:0.9996
4337:1.0004
4272:1.0008
4266:0.9996
4258:1.0004
4150:0.9996
4077:Thus
4046:0.9996
3946:0.9996
3929:0.9996
3772:1.0004
2542:Since
2333:) and
2267:where
1415:where
1252:Tissot
816:, the
730:, the
389:survey
309:Range
253:league
63:
59:
5986:and
5821:Notes
5771:RATIO
5750:colon
5521:from
3623:below
3401:and
3023:and
2002:1.414
1649:below
1586:exact
1518:point
917:small
447:globe
442:plane
432:Gauss
426:Scale
249:pouce
230:miles
90:ratio
83:Earth
79:ratio
73:of a
71:scale
36:Scale
6018:link
6011:help
5944:ISBN
5919:ISBN
5762:2236
5541:and
5383:and
5021:and
4952:and
4722:and
4480:<
4334:<
4328:<
4305:2.29
4189:1.62
3839:very
3824:1.62
3630:same
2860:The
2373:and
2189:The
2143:11.5
2096:5.76
1945:1.15
1564:and
1496:with
929:does
583:the
456:Let
361:less
226:inch
159:'s,
69:The
61:cm=6
5976:doi
5778:).
5653:tan
5644:sec
5632:tan
5468:sin
5457:sin
5441:sec
5275:cos
5188:lim
5139:lim
4983:cos
4868:sec
4810:tan
4779:sec
4761:tan
4756:(c)
4742:is
4626:tan
4621:(b)
4568:cos
4550:tan
4545:(a)
4239:sec
4123:sec
4049:sec
3963:tan
3792:sec
3732:sec
3701:sec
3675:sec
3600:cos
3528:sec
3503:cos
3449:cos
3360:sin
3285:sec
3259:sec
3223:sec
3153:sec
3128:cos
3075:sec
2946:tan
2780:sec
2750:sec
2643:sec
2618:cos
2124:sec
2077:sec
2030:sec
1973:sec
1908:sec
1841:sec
1714:sec
1689:cos
1651:.)
1469:cos
630:lim
434:’s
377:map
131:).
127:or
108:or
75:map
6058::
6000::
5998:}}
5994:{{
5974:,
5958:^
5930:^
5903:^
5869:^
5760:U+
3952:ln
3752:is
3625:.
2935:ln
2818:45
2734:.
2539:.
2164:.
2131:85
2084:80
2037:60
1980:45
1915:30
1522:at
901:.
856:.
770:.
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278:A
235:A
163:,
155:,
6020:)
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6009:(
5990:)
5978::
5952:.
5898:.
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5774:(
5662:.
5641:=
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5604:=
5597:y
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5480:]
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5438:=
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5429:(
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5335:x
5307:.
5299:2
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4977:=
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4297:2
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4262:/
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1999:=
1994:2
1989:=
1970:=
1967:k
1942:=
1937:3
1931:/
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1870:(
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1151:(
1148:=
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1107:=
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1024:=
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996:a
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883:,
877:(
874:k
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790:=
784:(
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751:,
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715:0
712:=
706:(
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616:,
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603:(
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273:.
38:.
20:)
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