63:
1300:-axes defined on the figure are related to the equator and central meridian exactly as they are for the normal projection. In the figure on the right a rotated graticule is related to the transverse cylinder in the same way that the normal cylinder is related to the standard graticule. The 'equator', 'poles' (E and W) and 'meridians' of the rotated graticule are identified with the chosen central meridian, points on the equator 90 degrees east and west of the central meridian, and great circles through those points.
2403:
1284:
929:
1304:
20:
2697:. The projection does not define a grid: the grid is an independent construct which could be defined arbitrarily. In practice the national implementations, and UTM, do use grids aligned with the Cartesian axes of the projection, but they are of finite extent, with origins which need not coincide with the intersection of the central meridian with the equator.
561:
200:
182:
2619:
2361:
2430:
is positive in the quadrant north of the equator and east of the central meridian and also in the quadrant south of the equator and west of the central meridian. The convergence must be added to a grid bearing to obtain a bearing from true north. For the secant transverse
Mercator the convergence may
800:
expansions of Krüger and proposed their adoption by the OSGB but
Redfearn (1948) pointed out that they were not accurate because of (a) the relatively high latitudes of Great Britain and (b) the great width of the area mapped, over 10 degrees of longitude. Redfearn extended the series to eighth order
1514:
This transformation projects the central meridian to a straight line of finite length and at the same time projects the great circles through E and W (which include the equator) to infinite straight lines perpendicular to the central meridian. The true parallels and meridians (other than equator and
611:
is applied to a narrow strip near the central meridians where the differences between the spherical and ellipsoidal versions are small, but nevertheless important in accurate mapping. Direct series for scale, convergence and distortion are functions of eccentricity and both latitude and longitude on
128:
Since the central meridian of the transverse
Mercator can be chosen at will, it may be used to construct highly accurate maps (of narrow width) anywhere on the globe. The secant, ellipsoidal form of the transverse Mercator is the most widely applied of all projections for accurate large-scale maps.
879:
An exact solution by E. H. Thompson is described by L. P. Lee. It is constructed in terms of elliptic functions (defined in chapters 19 and 22 of the NIST handbook) which can be calculated to arbitrary accuracy using algebraic computing systems such as Maxima. Such an implementation of the exact
598:
region is reasonably well preserved. The necessary condition is that the magnitude of scale factor must not vary too much over the region concerned. Note that while South
America is distorted greatly the island of Ceylon is small enough to be reasonably shaped although it is far from the central
535:
The projection is conformal with a constant scale on the central meridian. (There are other conformal generalisations of the transverse
Mercator from the sphere to the ellipsoid but only Gauss-Krüger has a constant scale on the central meridian.) Throughout the twentieth century the Gauss–Krüger
1641:
891:
series compares very favourably with the exact values: they differ by less than 0.31 μm within 1000 km of the central meridian and by less than 1 mm out to 6000 km. On the other hand, the difference of the
Redfearn series used by GEOTRANS and the exact solution is less than
1908:
2915:
axes, do not run north-south or east-west as defined by parallels and meridians. This is evident from the global projections shown above. Near the central meridian the differences are small but measurable. The difference between the north-south grid lines and the true meridians is the
899:
The
Redfearn series becomes much worse as the zone widens. Karney discusses Greenland as an instructive example. The long thin landmass is centred on 42W and, at its broadest point, is no more than 750 km from that meridian while the span in longitude reaches almost 50 degrees.
90:: for the normal Mercator, the axis of the cylinder coincides with the polar axis and the line of tangency with the equator. For the transverse Mercator, the axis of the cylinder lies in the equatorial plane, and the line of tangency is any chosen meridian, thereby designated the
2902:
2437:
522:
The term is also used for a particular set of transverse
Mercator projections used in narrow zones in Europe and South America, at least in Germany, Turkey, Austria, Slovenia, Croatia, Bosnia-Herzegovina, Serbia, Montenegro, North Macedonia, Finland and Argentina. This
514:
Sometimes, the term is used for a particular computational method for transverse
Mercator: that is, how to convert between latitude/longitude and projected coordinates. There is no simple closed formula to do so when the earth is modelled as an ellipsoid. But the
3050:
Gauss, Karl
Friedrich, 1825. "Allgemeine Auflösung der Aufgabe: die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden, daß die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird" Preisarbeit der Kopenhagener Akademie 1822.
414:
The projection is reasonably accurate near the central meridian. Scale at an angular distance of 5° (in longitude) away from the central meridian is less than 0.4% greater than scale at the central meridian, and is about 1.54% at an angular distance of 10°.
2181:
815:
expansions of Krüger were also confirmed by Paul Thomas in 1952: they are readily available in Snyder. His projection formulae, completely equivalent to those presented by Redfearn, were adopted by the United States Defence Mapping Agency as the basis for the
584:
The equator bisects Africa, crosses South America and then continues onto the complete outer boundary of the projection; the top and bottom edges and the right and left edges must be identified (i.e. they represent the same lines on the globe). (Indonesia is
2153:
936:
The normal cylindrical projections are described in relation to a cylinder tangential at the equator with axis along the polar axis of the sphere. The cylindrical projections are constructed so that all points on a meridian are projected to points with
870:
series have been implemented to seventh order by Engsager and Poder and to tenth order by Kawase. Apart from a series expansion for the transformation between latitude and conformal latitude, Karney has implemented the series to thirtieth order.
405:
The projection is reasonably accurate near the equator. Scale at an angular distance of 5° (in latitude) away from the equator is less than 0.4% greater than scale at the equator, and is about 1.54% greater at an angular distance of 10°.
547:
which were assumed to diverge in the east-west direction, exactly as in the spherical version. This was proved to be untrue by British cartographer E. H. Thompson, whose unpublished exact (closed form) version of the projection, reported by
552:
in 1976, showed that the ellipsoidal projection is finite (below). This is the most striking difference between the spherical and ellipsoidal versions of the transverse Mercator projection: Gauss–Krüger gives a reasonable projection of the
1509:
602:
The choice of central meridian greatly affects the appearance of the projection. If 90°W is chosen then the whole of the Americas is reasonable. If 145°E is chosen the Far East is good and Australia is oriented with north
1291:
The figure on the left shows how a transverse cylinder is related to the conventional graticule on the sphere. It is tangential to some arbitrarily chosen meridian and its axis is perpendicular to that of the sphere. The
1688:
629:
constant grid lines is no longer zero (except on the equator) so that a grid bearing must be corrected to obtain an azimuth from true north. The difference is small, but not negligible, particularly at high latitudes.
1197:
892:
1 mm out to a longitude difference of 3 degrees, corresponding to a distance of 334 km from the central meridian at the equator but a mere 35 km at the northern limit of an UTM zone. Thus the Krüger–
1533:
2758:
624:
on the projection. In the secant version the lines of true scale on the projection are no longer parallel to central meridian; they curve slightly. The convergence angle between projected meridians and the
574:
Near the central meridian (Greenwich in the above example) the projection has low distortion and the shapes of Africa, western Europe, the British Isles, Greenland, and Antarctica compare favourably with a
2614:{\displaystyle {\begin{aligned}\gamma (\lambda ,\varphi )&=\arctan(\tan \lambda \sin \varphi ),\\\gamma (x,y)&=\arctan \left(\tanh {\frac {x}{k_{0}a}}\tan {\frac {y}{k_{0}a}}\right).\end{aligned}}}
591:
The map is conformal. Lines intersecting at any specified angle on the ellipsoid project into lines intersecting at the same angle on the projection. In particular parallels and meridians intersect at 90°.
588:
Distortion increases towards the right and left boundaries of the projection but it does not increase to infinity. Note the Galapagos Islands where the 90° west meridian meets the equator at bottom left.
353:. The projection is not suited for world maps. Distortion is small near the central meridian and the projection (particularly in its ellipsoidal form) is suitable for accurate mapping of narrow regions.
2763:
2442:
2186:
1932:
1693:
3451:
581:
The meridians at 90° east and west of the chosen central meridian project to horizontal lines through the poles. The more distant hemisphere is projected above the north pole and below the south pole.
829:
Many variants of the Redfearn series have been proposed but only those adopted by national cartographic agencies are of importance. For an example of modifications which do not have this status see
341:. The projection is not suited for world maps. Distortion is small near the equator and the projection (particularly in its ellipsoidal form) is suitable for accurate mapping of equatorial regions.
519:
method gives the same results as other methods, at least if you are sufficiently near the central meridian: less than 100 degrees of longitude, say. Further away, some methods become inaccurate.
1538:
2356:{\displaystyle {\begin{aligned}k(\lambda ,\varphi )&={\frac {k_{0}}{\sqrt {1-\sin ^{2}\lambda \cos ^{2}\varphi }}},\\k(x,y)&=k_{0}\cosh \left({\frac {x}{k_{0}a}}\right).\end{aligned}}}
826:: The Redfearn series are the basis for geodetic mapping in many countries: Australia, Germany, Canada, South Africa to name but a few. (A list is given in Appendix A.1 of Stuifbergen 2009.)
1256:
153:
dates from the second half of the nineteenth century. The principal properties of the transverse projection are here presented in comparison with the properties of the normal projection.
1927:
145:
must be chosen if greater accuracy is required; see next section. The spherical form of the transverse Mercator projection was one of the seven new projections presented, in 1772, by
2673:
The projection coordinates resulting from the various developments of the ellipsoidal transverse Mercator are Cartesian coordinates such that the central meridian corresponds to the
1527:
on the spherical triangle NM′P defined by the true meridian through the origin, OM′N, the true meridian through an arbitrary point, MPN, and the great circle WM′PE. The results are:
2704:
is always taken on the central meridian so that grid coordinates will be negative west of the central meridian. To avoid such negative grid coordinates, standard practice defines a
423:
In the secant version the scale is reduced on the equator and it is true on two lines parallel to the projected equator (and corresponding to two parallel circles on the sphere).
3995:
837:-series outlined below. The precise Redfearn series, although of low order, cannot be disregarded as they are still enshrined in the quasi-legal definitions of OSGB and UTM etc.
431:
In the secant version the scale is reduced on the central meridian and it is true on two lines parallel to the projected central meridian. (The two lines are not meridians.)
3203:
2366:
The second expression shows that the scale factor is simply a function of the distance from the central meridian of the projection. A typical value of the scale factor is
964:
141:
is normally chosen to model the Earth when the extent of the mapped region exceeds a few hundred kilometers in length in both dimensions. For maps of smaller regions, an
78:) Mercator projection. They share the same underlying mathematical construction and consequently the transverse Mercator inherits many traits from the normal Mercator:
820:. They are also incorporated into the GEOTRANS coordinate converter made available by the United States National Geospatial-Intelligence Agency’s Office of Geomatics.
2630:
1028:
817:
47:
3007:
3273:
It gives full details of most projections, together with interesting introductory sections, but it does not derive any of the projections from first principles.
2984:
1008:
984:
3426:
369:
When Greenland and Africa are both near the central meridian, their shapes are good and the ratio of the areas is a good approximation to actual values.
4737:
4269:
3775:
3652:
3468:
1903:{\displaystyle {\begin{aligned}x(\lambda ,\varphi )&={\frac {1}{2}}k_{0}a\ln \left,\\y(\lambda ,\varphi )&=k_{0}a\arctan \left,\end{aligned}}}
451:
Convergence is zero on the equator and non-zero everywhere else. It increases as the poles are approached. Grid north and true north do not coincide.
4355:
4151:
4141:
4061:
789:
series were the first to be implemented, possibly because they were much easier to evaluate on the hand calculators of the mid twentieth century.
1036:
536:
transverse Mercator was adopted, in one form or another, by many nations (and international bodies); in addition it provides the basis for the
124:
Both projections have constant scale on the line of tangency (the equator for the normal Mercator and the central meridian for the transverse).
4146:
3727:
1636:{\displaystyle {\begin{aligned}\sin \varphi '&=\sin \lambda \cos \varphi ,\\\tan \lambda '&=\sec \lambda \tan \varphi .\end{aligned}}}
608:
3605:
3034:
99:
Both projections may be modified to secant forms, which means the scale has been reduced so that the cylinder slices through the model globe.
3403:
2907:
The terms "eastings" and "northings" do not mean strict east and north directions. Grid lines of the transverse projection, other than the
2640:
802:
778:
578:
The central regions of the transverse projections on sphere and ellipsoid are indistinguishable on the small-scale projections shown here.
2897:{\displaystyle {\begin{aligned}E&=E_{0}+x(\lambda ,\varphi ),\\N&=N_{0}+y(\lambda ,\varphi )-k_{0}m(\varphi _{0}).\end{aligned}}}
660:(the ratio of the difference and sum of the major and minor axes of the ellipsoid). The coefficients are expressed in terms of latitude (
2662:
1384:
830:
4156:
3957:
4289:
4279:
4274:
4249:
4241:
3902:
3828:
3785:
3780:
3755:
3747:
3135:
3380:
R. Kuittinen; T. Sarjakoski; M. Ollikainen; M. Poutanen; R. Nuuros; P. Tätilä; J. Peltola; R. Ruotsalainen; M. Ollikainen (2006).
4777:
4680:
4477:
4404:
4360:
4056:
3465:
A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss–Krüger Projection
3207:
4525:
4472:
3384:[map projections related to the ETRS89 system, level coordinates and map sheet division, Appendix 1: Project formulas]
557:
ellipsoid to the plane, although its principal application is to accurate large-scale mapping "close" to the central meridian.
484:
4767:
4633:
4602:
4025:
3803:
3583:
3359:
1913:
The above expressions are given in Lambert and also (without derivations) in Snyder, Maling and Osborne (with full details).
2650:
66:
Comparison of tangent and secant forms of normal, oblique and transverse Mercator projections with standard parallels in red
540:
series of projections. The Gauss–Krüger projection is now the most widely used projection in accurate large-scale mapping.
4717:
4685:
4535:
4166:
3990:
3823:
3813:
3645:
3429:[Calculation to convert coordinates to obtain longitude and latitude, meridian aberration angle and scale factor]
537:
528:
3320:
2656:
397:
on the projection. (On the sphere it depends on both latitude and longitude.) The scale is true on the central meridian.
4675:
4389:
4043:
3952:
4665:
3011:
4615:
4578:
4345:
4038:
3887:
3737:
3348:
3090:
2944:
46:. The transverse version is widely used in national and international mapping systems around the world, including the
4259:
3765:
2992:). Published by Wilhelm Engelmann. This is Lambert's paper with additional comments by the editor. Available at the
4550:
4394:
3985:
3818:
3808:
3434:
3382:"ETRS89—järjestelmään liittyvät karttaprojektiot, tasokoordinaatistot ja karttalehtijako, Liite 1: Projektiokaavat"
2738:, is the distance of the true grid origin north of the false origin. If the true origin of the grid is at latitude
2175:: this may be expressed either in terms of the geographical coordinates or in terms of the projection coordinates:
613:
3232:
3180:
3083:
2708:
to the west (and possibly north or south) of the grid origin: the coordinates relative to the false origin define
2148:{\displaystyle {\begin{aligned}\lambda (x,y)&=\arctan \left,\\\varphi (x,y)&=\arcsin \left.\end{aligned}}}
1216:
4530:
3915:
4264:
3770:
833:). All such modifications have been eclipsed by the power of modern computers and the development of high order
4762:
4620:
4560:
4540:
4171:
4133:
4098:
3638:
149:. (The text is also available in a modern English translation.) Lambert did not name his projections; the name
110:
3102:
The EEA recommends the transverse Mercator for conformal pan-European mapping at scales larger than 1:500,000.
241: = 0. Meridians 90 degrees east and west of the central meridian project to lines of constant
3833:
3677:
2929:
506:
more generally. Other than just a synonym for the ellipsoidal transverse Mercator map projection, the term
313:
direction. The points on the equator at ninety degrees from the central meridian are projected to infinity.
146:
1515:
central meridian) have no simple relation to the rotated graticule and they project to complicated curves.
638:
In his 1912 paper, Krüger presented two distinct solutions, distinguished here by the expansion parameter:
62:
4732:
4365:
4340:
3882:
3672:
3381:
1524:
4655:
4445:
4399:
4226:
4203:
4186:
3897:
2431:
be expressed either in terms of the geographical coordinates or in terms of the projection coordinates:
2392: = 1.0004: the scale factor is within 0.04% of unity over a strip of about 510 km wide.
531:
system, but the central meridians of the Gauss–Krüger zones are only 3° apart, as opposed to 6° in UTM.
54:, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.
4660:
4555:
4335:
4330:
4325:
4302:
4297:
4218:
3980:
3920:
3892:
3877:
3872:
3867:
3862:
3284:
3172:
3115:
801:
and examined which terms were necessary to attain an accuracy of 1 mm (ground measurement). The
549:
480:
92:
3624:
The projections used to illustrate this article were prepared using Geocart which is available from
4610:
4545:
4450:
4427:
4254:
4161:
4033:
3760:
3718:
2934:
2402:
1030:. For a tangent Normal Mercator projection the (unique) formulae which guarantee conformality are:
142:
103:
43:
3285:"The universal grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS)"
1283:
385:
on the projection. (On the sphere it depends on latitude only.) The scale is true on the equator.
4482:
4093:
3798:
3160:
940:
3156:
3152:
1319:) on the standard graticule can also be identified in terms of angles on the rotated graticule:
928:
361:
Greenland is almost as large as Africa; the actual area is about one fourteenth that of Africa.
1303:
4772:
4409:
4350:
4320:
4315:
4231:
4208:
4088:
4083:
4002:
3947:
3925:
3601:
3489:
3131:
3084:"Short Proceedings of the 1st European Workshop on Reference Grids, Ispra, 27–29 October 2003"
3030:
4195:
3975:
3501:
3119:
19:
1013:
138:
3560:"Transverse Mercator Projection - preprint of paper and C++ implementation of algorithms"
4647:
4593:
4570:
4517:
4505:
4460:
4437:
4419:
4379:
4121:
4075:
4012:
3967:
3939:
3847:
3709:
3697:
3661:
3201:
A guide to coordinate systems in Great Britain. This is available as a pdf document at
3120:
1268:
on the right hand side of all these equations: this ensures that the scale is equal to
993:
969:
594:
The point scale factor is independent of direction at any point so that the shape of a
87:
83:
71:
51:
31:
3041:
This is an excellent survey of virtually all known projections from antiquity to 1993.
543:
The projection, as developed by Gauss and Krüger, was expressed in terms of low order
4756:
4670:
3452:
A highly accurate world wide algorithm for the transverse Mercator mapping (almost)
1327:(angle M′CO) becomes an effective longitude. (The minus sign is necessary so that (
987:
544:
3052:
883:
The exact solution is a valuable tool in assessing the accuracy of the truncated
3682:
3265:
Map Projections—A Working Manual. U.S. Geological Survey Professional Paper 1395
3069:
2939:
2164:
1203:
648:(paragraphs 5 to 8): Formulae for the direct projection, giving the coordinates
378:
245:
through the projected poles. All other meridians project to complicated curves.
114:
3559:
915:) series is accurate to 5 nm within 3900 km of the central meridian.
257: = 0 and parallel circles project to straight lines of constant
3505:
3331:
3143:
2993:
459:
329:
The projection is conformal. The shapes of small elements are well preserved.
321:
The projection is conformal. The shapes of small elements are well preserved.
4727:
3464:
479:
The ellipsoidal form of the transverse Mercator projection was developed by
1351:) axes are related to the rotated graticule in the same way that the axes (
3270:
2631:
Universal Transverse Mercator coordinate system § Simplified formulae
1378:
by the transformation formulae of the tangent Normal Mercator projection:
1192:{\displaystyle x=a\lambda \,,\qquad y=a\ln \left={\frac {a}{2}}\ln \left.}
721:
are expansions (of orders 5 and 4 respectively) in terms of the longitude
4722:
3008:"Notes and Comments on the Composition of Terrestrial and Celestial Maps"
2970:
2727:, is the distance of the true grid origin east of the false origin. The
560:
393:
The point scale factor is independent of direction. It is a function of
199:
181:
4583:
3305:
904:
is accurate to within 1 mm but the Redfearn version of the Krüger–
439:
Convergence (the angle between projected meridians and grid lines with
3454:, in Proc. XXIII Intl. Cartographic Conf. (ICC2007), Moscow, p. 2.1.2.
3404:"Gauss Conformal Projection (Transverse Mercator): Krüger's Formulas"
2163:
In terms of the coordinates with respect to the rotated graticule the
273: = 0 but all other parallels are complicated closed curves.
3250:. Washington: U.S. Coast and Geodetic Survey Special Publication 251.
2745:
on the central meridian and the scale factor the central meridian is
1362:
The tangent transverse Mercator projection defines the coordinates (
2967:
Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten
932:
The normal aspect of a tangent cylindrical projection of the sphere
725:, expressed in radians: the coefficients are expressed in terms of
713:(paragraphs 13 and 14): Formulae giving the projection coordinates
443:
constant) is identically zero. Grid north and true north coincide.
2401:
1504:{\displaystyle x'=-a\lambda '\,\qquad y'={\frac {a}{2}}\ln \left.}
1302:
1282:
1210:, is independent of direction: it is a function of latitude only:
927:
559:
198:
180:
61:
18:
3630:
2971:
Beyträge zum Gebrauche der Mathematik und deren Anwendung, part 3
656:, are fourth order expansions in terms of the third flattening,
4706:
4503:
4119:
3695:
3634:
3534:
462:(of constant azimuth on the sphere) project to straight lines.
225: = 0. Other meridians project to straight lines with
3625:
3519:
F. W.J. Olver; D.W. Lozier; R.F. Boisvert; C.W. Clark (2010).
2414:
at a point on the projection is defined by the angle measured
1261:
For the secant version of the projection there is a factor of
1335:) are related to the rotated graticule in the same way that (
289:
Projected meridians and parallels intersect at right angles.
281:
Projected meridians and parallels intersect at right angles.
3267:. United States Government Printing Office, Washington, D.C.
2661:
Fourth order Redfearn series by concise formulae (example):
189: = ±π, corresponding to approximately 85 degrees).
3586:
Detailed derivations of all formulae quoted in this article
3328:
Canadian Technical Report of Hydrography and Ocean Sciences
3027:
Flattening the Earth: Two Thousand Years of Map Projections
3490:"Transverse Mercator with an accuracy of a few nanometers"
3130:. Vol. 16. Toronto: B. V. Gutsell, York University.
1343:) are related to the standard graticule). The Cartesian (
896:
series are very much better than the Redfearn λ series.
3520:
2752:
then these definitions give eastings and northings by:
875:
Exact Gauss–Krüger and accuracy of the truncated series
2076:
1651:
The direct formulae giving the Cartesian coordinates (
1523:
The angles of the two graticules are related by using
381:
is independent of direction. It is a function of
2761:
2440:
2184:
1930:
1691:
1536:
1387:
1219:
1039:
1016:
996:
972:
943:
564:
Ellipsoidal transverse Mercator: a finite projection.
185:
Spherical Normal (equatorial) Mercator (truncated at
3330:(262). Canadian Hydrographic Service. Archived from
3053:
Schumacher Astronomische Abhandlungen, Altona, no. 3
2655:
Exact (closed form) transverse Mercator projection:
887:
and λ series. For example, the original 1912 Krüger–
4646:
4601:
4592:
4569:
4516:
4459:
4436:
4418:
4378:
4288:
4240:
4217:
4194:
4185:
4132:
4074:
4024:
4011:
3966:
3938:
3855:
3846:
3746:
3717:
3708:
3283:Hager, J. W.; Behensky, J. F.; Drew, B. W. (1989).
3227:
Redfearn, J C B (1948). Survey Review, Volume
3073:. Royal Prussian Geodetic Institute, New Series 52.
237:The central meridian projects to the straight line
221:The central meridian projects to the straight line
3292:Technical Report TM 8358.2, Defense Mapping Agency
3182:The transverse Mercator projection of the spheroid
2896:
2613:
2418:the projected meridian, which defines true north,
2355:
2147:
1902:
1635:
1503:
1250:
1191:
1022:
1002:
978:
958:
3122:Conformal Projections Based on Elliptic Functions
3070:Konforme Abbildung des Erdellipsoids in der Ebene
845:series have been implemented (to fourth order in
745:are sixth order expansions in terms of the ratio
3248:Conformal Projections in Geodesy and Cartography
2625:Formulae for the ellipsoidal transverse Mercator
908:series has a maximum error of 1 kilometre.
805:are still the basis of the OSGB map projections.
612:the ellipsoid: inverse series are functions of
3010:. University of Michigan Press. Archived from
2994:University of Michigan Historical Math Library
1359:) axes are related to the standard graticule.
919:Formulae for the spherical transverse Mercator
634:Implementations of the Gauss–Krüger projection
510:may be used in other slightly different ways:
490:The projection is known by several names: the
16:Adaptation of the standard Mercator projection
3646:
3349:"Projection Cartographique Mercator Traverse"
2985:Ostwalds Klassiker der exakten Wissenschaften
1659:) follow immediately from the above. Setting
8:
1251:{\displaystyle k(\varphi )=\sec \varphi .\,}
688:but with coefficients expressed in terms of
203:Spherical transverse Mercator (truncated at
3110:
3108:
3029:. University of Chicago Press. p. 82.
1323:(angle M′CP) is an effective latitude and −
157:Normal and transverse spherical projections
137:In constructing a map on any projection, a
4703:
4598:
4513:
4500:
4191:
4129:
4116:
4021:
3852:
3714:
3705:
3692:
3653:
3639:
3631:
3321:"Wide zone transverse Mercator projection"
2669:Coordinates, grids, eastings and northings
2082:
765:, with coefficients expressed in terms of
269:The equator projects to the straight line
253:The equator projects to the straight line
197:
179:
70:The transverse Mercator projection is the
4738:Map projection of the tri-axial ellipsoid
3521:"NIST Handbook of Mathematical Functions"
3483:
3481:
3469:Geospatial Information Authority of Japan
3392:(in Finnish). Finnish Geodetic Institute.
3258:
3256:
2961:
2959:
2878:
2862:
2828:
2780:
2762:
2760:
2587:
2577:
2559:
2549:
2441:
2439:
2330:
2320:
2304:
2253:
2237:
2220:
2214:
2185:
2183:
2121:
2111:
2093:
2083:
2075:
2015:
2005:
1987:
1977:
1931:
1929:
1850:
1754:
1735:
1721:
1692:
1690:
1537:
1535:
1446:
1426:
1413:
1386:
1247:
1218:
1144:
1124:
1101:
1088:
1052:
1038:
1015:
995:
971:
942:
3433:(in Chinese). p. 22. Archived from
3063:
3061:
2677:axis and the equator corresponds to the
880:solution is described by Karney (2011).
3579:
3577:
3575:
3573:
3197:
3195:
2955:
3598:Coordinate Systems and Map Projections
3185:. (Errata and comments in Volume
2651:Transverse Mercator: flattening series
301:direction. The poles lie at infinity.
7:
3535:"Maxima - A computer algebra system"
2641:Transverse Mercator: Redfearn series
1921:Inverting the above equations gives
1311:The position of an arbitrary point (
866:Higher order versions of the Krüger–
779:Transverse Mercator: Redfearn series
684:are also fourth order expansions in
3600:(second ed.). Pergamon Press.
3450:K. E. Engsager and K. Poder, 2007,
3175:(1945). Survey Review, Volume
2716:which will always be positive. The
2663:Transverse Mercator: Bowring series
2657:Transverse Mercator: exact solution
2381:is approximately 180 km. When
1519:The relation between the graticules
924:Spherical normal Mercator revisited
831:Transverse Mercator: Bowring series
796:: In 1945, L. P. Lee confirmed the
309:The projection is unbounded in the
297:The projection is unbounded in the
42:) is an adaptation of the standard
3269:This paper can be downloaded from
2639:Gauss-Kruger series in longitude:
2635:Details of actual implementations
2426:, defining grid north. Therefore,
14:
3231:(Part 69), pp 318–322,
3189:(Part 61), pp. 277–278.
3179:(Part 58), pp 142–152.
2385:is approximately 255 km and
492:(ellipsoidal) transverse Mercator
4681:Quadrilateralized spherical cube
4361:Quadrilateralized spherical cube
2982:Albert Wangerin (Editor), 1894.
2965:Lambert, Johann Heinrich. 1772.
2917:
1682:to accommodate secant versions)
1279:Normal and transverse graticules
504:Gauss–Krüger transverse Mercator
483:in 1822 and further analysed by
117:is independent of direction and
23:A transverse Mercator projection
3427:"座標を変換して経緯度、子午線収差角及び縮尺係数を求める計算"
1917:Inverse transformation formulae
1414:
1056:
475:Ellipsoidal transverse Mercator
349:Distortion increases with
337:Distortion increases with
58:Standard and transverse aspects
4270:Lambert cylindrical equal-area
3360:Institut Geographique National
2884:
2871:
2852:
2840:
2804:
2792:
2689:are defined for all values of
2522:
2510:
2497:
2476:
2460:
2448:
2290:
2278:
2204:
2192:
2054:
2042:
1950:
1938:
1836:
1824:
1711:
1699:
1647:Direct transformation formulae
1287:Transverse mercator graticules
1229:
1223:
1202:Conformality implies that the
609:Gauss–Krüger coordinate system
50:. When paired with a suitable
1:
4718:Interruption (map projection)
3596:Maling, Derek Hylton (1992).
3523:. Cambridge University Press.
737:. The inverse projection for
538:Universal Transverse Mercator
529:universal transverse Mercator
133:Spherical transverse Mercator
48:Universal Transverse Mercator
4356:Lambert azimuthal equal-area
4152:Guyou hemisphere-in-a-square
4142:Adams hemisphere-in-a-square
3564:geographiclib.sourceforge.io
3234:Transverse Mercator formulae
2373: = 0.9996 so that
1307:Transverse mercator geometry
1010:is a prescribed function of
849:) by the following nations.
676:). The inverse formulae for
485:Johann Heinrich Louis Krüger
102:Both exist in spherical and
3091:European Environment Agency
2945:Oblique Mercator projection
959:{\displaystyle x=a\lambda }
911:Karney's own 8th-order (in
4794:
2628:
1675:(and restoring factors of
454:
434:
418:
400:
372:
356:
332:
316:
292:
276:
248:
216:
211:in units of Earth radius).
176:
162:
121:shapes are well preserved;
4713:
4702:
4629:
4512:
4499:
4311:
4128:
4115:
4052:
3911:
3794:
3704:
3691:
3668:
3506:10.1007/s00190-011-0445-3
3145:The Canadian Cartographer
3006:Tobler, Waldo R. (1972).
607:In most applications the
527:system is similar to the
172:
165:
3584:The Mercator Projections
3488:C. F. F. Karney (2011).
3263:Snyder, John P. (1987).
3204:"Welcome to GPS Network"
3128:Cartographica Monographs
3025:Snyder, John P. (1993).
2422:a grid line of constant
2406:The angle of convergence
170:
168:
163:
4778:Cylindrical projections
4157:Lambert conformal conic
3319:N. Stuifbergen (2009).
3246:Thomas, Paul D (1952).
3093:. 2004-06-14. p. 6
2930:List of map projections
2645:Gauss-Kruger series in
147:Johann Heinrich Lambert
4290:Tobler hyperelliptical
3903:Tobler hyperelliptical
3829:Space-oblique Mercator
3626:Mapthematics Geocart 3
2898:
2615:
2410:The convergence angle
2407:
2357:
2171: = sec
2149:
1904:
1637:
1525:spherical trigonometry
1505:
1308:
1288:
1252:
1193:
1024:
1004:
980:
960:
933:
565:
212:
190:
67:
24:
4768:Conformal projections
3539:maxima.sourceforge.io
3306:"Office of Geomatics"
2899:
2616:
2405:
2358:
2150:
1905:
1638:
1506:
1306:
1286:
1253:
1194:
1025:
1023:{\displaystyle \phi }
1005:
981:
961:
931:
563:
202:
184:
109:Both projections are
65:
22:
4666:Cahill–Keyes M-shape
4526:Chamberlin trimetric
3390:Technical Report JHS
3142:Supplement No. 1 to
2918:angle of convergence
2759:
2649:(third flattening):
2438:
2377: = 1 when
2182:
1928:
1689:
1534:
1385:
1217:
1037:
1014:
994:
970:
941:
672:) and eccentricity (
550:Laurence Patrick Lee
481:Carl Friedrich Gauss
173:Transverse Mercator
74:of the standard (or
4733:Tissot's indicatrix
4634:Central cylindrical
4275:Smyth equal-surface
4177:Transverse Mercator
4026:General perspective
3781:Smyth equal-surface
3733:Transverse Mercator
3463:Kawase, K. (2011):
3067:Krüger, L. (1912).
2935:Mercator projection
2167:factor is given by
151:transverse Mercator
44:Mercator projection
29:transverse Mercator
4686:Waterman butterfly
4536:Miller cylindrical
4167:Peirce quincuncial
4062:Lambert equal-area
3814:Gall stereographic
3494:Journal of Geodesy
3467:, Bulletin of the
2894:
2892:
2611:
2609:
2408:
2353:
2351:
2145:
2143:
2080:
1900:
1898:
1633:
1631:
1501:
1309:
1289:
1248:
1189:
1020:
1000:
976:
956:
934:
566:
379:point scale factor
213:
191:
68:
25:
4750:
4749:
4746:
4745:
4698:
4697:
4694:
4693:
4642:
4641:
4495:
4494:
4491:
4490:
4374:
4373:
4111:
4110:
4107:
4106:
4070:
4069:
3958:Lambert conformal
3934:
3933:
3848:Pseudocylindrical
3842:
3841:
3607:978-0-08-037233-4
3055:, p. 5–30.
3036:978-0-226-76747-5
2597:
2569:
2340:
2266:
2265:
2131:
2103:
2079:
2025:
1997:
1808:
1729:
1492:
1434:
1180:
1132:
1109:
1096:
1003:{\displaystyle y}
979:{\displaystyle a}
794:Lee–Redfearn–OSGB
472:
471:
143:ellipsoidal model
72:transverse aspect
4785:
4704:
4661:Cahill Butterfly
4599:
4579:Goode homolosine
4514:
4501:
4466:
4465:(Mecca or Qibla)
4346:Goode homolosine
4192:
4130:
4117:
4022:
4017:
3888:Goode homolosine
3853:
3738:Oblique Mercator
3715:
3706:
3693:
3655:
3648:
3641:
3632:
3613:
3611:
3593:
3587:
3581:
3568:
3567:
3556:
3550:
3549:
3547:
3546:
3531:
3525:
3524:
3516:
3510:
3509:
3485:
3476:
3461:
3455:
3448:
3442:
3441:
3439:
3432:
3423:
3417:
3416:
3414:
3413:
3408:
3400:
3394:
3393:
3387:
3377:
3371:
3370:
3368:
3367:
3353:
3345:
3339:
3338:
3336:
3325:
3316:
3310:
3309:
3302:
3296:
3295:
3289:
3280:
3274:
3268:
3260:
3251:
3244:
3238:
3225:
3219:
3218:
3216:
3215:
3206:. Archived from
3199:
3190:
3170:
3164:
3141:
3125:
3112:
3103:
3101:
3099:
3098:
3088:
3080:
3074:
3065:
3056:
3048:
3042:
3040:
3022:
3016:
3015:
3003:
2997:
2980:
2974:
2963:
2903:
2901:
2900:
2895:
2893:
2883:
2882:
2867:
2866:
2833:
2832:
2785:
2784:
2702:true grid origin
2620:
2618:
2617:
2612:
2610:
2603:
2599:
2598:
2596:
2592:
2591:
2578:
2570:
2568:
2564:
2563:
2550:
2362:
2360:
2359:
2354:
2352:
2345:
2341:
2339:
2335:
2334:
2321:
2309:
2308:
2267:
2258:
2257:
2242:
2241:
2226:
2225:
2224:
2215:
2154:
2152:
2151:
2146:
2144:
2137:
2133:
2132:
2130:
2126:
2125:
2112:
2104:
2102:
2098:
2097:
2084:
2081:
2077:
2031:
2027:
2026:
2024:
2020:
2019:
2006:
1998:
1996:
1992:
1991:
1978:
1909:
1907:
1906:
1901:
1899:
1892:
1888:
1855:
1854:
1813:
1809:
1807:
1781:
1755:
1740:
1739:
1730:
1722:
1642:
1640:
1639:
1634:
1632:
1600:
1554:
1510:
1508:
1507:
1502:
1497:
1493:
1491:
1490:
1469:
1468:
1447:
1435:
1427:
1422:
1412:
1395:
1275:on the equator.
1257:
1255:
1254:
1249:
1198:
1196:
1195:
1190:
1185:
1181:
1179:
1162:
1145:
1133:
1125:
1120:
1116:
1115:
1111:
1110:
1102:
1097:
1089:
1029:
1027:
1026:
1021:
1009:
1007:
1006:
1001:
985:
983:
982:
977:
965:
963:
962:
957:
764:
762:
761:
756:
753:
229: constant.
166:Normal Mercator
161:
160:
93:central meridian
4793:
4792:
4788:
4787:
4786:
4784:
4783:
4782:
4763:Map projections
4753:
4752:
4751:
4742:
4709:
4690:
4638:
4625:
4588:
4565:
4551:Van der Grinten
4508:
4506:By construction
4487:
4464:
4463:
4455:
4432:
4414:
4395:Equirectangular
4381:
4370:
4307:
4284:
4280:Trystan Edwards
4236:
4213:
4181:
4124:
4103:
4076:Pseudoazimuthal
4066:
4048:
4015:
4014:
4007:
3962:
3930:
3926:Winkel I and II
3907:
3838:
3819:Gall isographic
3809:Equirectangular
3790:
3786:Trystan Edwards
3742:
3700:
3687:
3664:
3659:
3621:
3616:
3608:
3595:
3594:
3590:
3582:
3571:
3558:
3557:
3553:
3544:
3542:
3533:
3532:
3528:
3518:
3517:
3513:
3487:
3486:
3479:
3462:
3458:
3449:
3445:
3437:
3430:
3425:
3424:
3420:
3411:
3409:
3406:
3402:
3401:
3397:
3385:
3379:
3378:
3374:
3365:
3363:
3356:geodesie.ign.fr
3351:
3347:
3346:
3342:
3334:
3323:
3318:
3317:
3313:
3304:
3303:
3299:
3287:
3282:
3281:
3277:
3262:
3261:
3254:
3245:
3241:
3226:
3222:
3213:
3211:
3202:
3200:
3193:
3171:
3167:
3138:
3114:
3113:
3106:
3096:
3094:
3086:
3082:
3081:
3077:
3066:
3059:
3049:
3045:
3037:
3024:
3023:
3019:
3005:
3004:
3000:
2981:
2977:
2964:
2957:
2953:
2926:
2891:
2890:
2874:
2858:
2824:
2817:
2811:
2810:
2776:
2769:
2757:
2756:
2751:
2744:
2737:
2726:
2671:
2633:
2627:
2608:
2607:
2583:
2582:
2555:
2554:
2542:
2538:
2525:
2504:
2503:
2463:
2436:
2435:
2400:
2391:
2372:
2350:
2349:
2326:
2325:
2316:
2300:
2293:
2272:
2271:
2249:
2233:
2216:
2207:
2180:
2179:
2161:
2142:
2141:
2117:
2116:
2089:
2088:
2074:
2070:
2057:
2036:
2035:
2011:
2010:
1983:
1982:
1970:
1966:
1953:
1926:
1925:
1919:
1897:
1896:
1869:
1865:
1846:
1839:
1818:
1817:
1782:
1756:
1750:
1731:
1714:
1687:
1686:
1681:
1649:
1630:
1629:
1601:
1593:
1584:
1583:
1555:
1547:
1532:
1531:
1521:
1483:
1470:
1461:
1448:
1442:
1415:
1405:
1388:
1383:
1382:
1370:) in terms of −
1281:
1274:
1267:
1215:
1214:
1163:
1146:
1140:
1087:
1083:
1076:
1072:
1035:
1034:
1012:
1011:
992:
991:
968:
967:
939:
938:
926:
921:
877:
824:Other countries
803:Redfearn series
757:
754:
749:
748:
746:
668:), major axis (
636:
571:
496:Gauss conformal
477:
159:
135:
60:
17:
12:
11:
5:
4791:
4789:
4781:
4780:
4775:
4770:
4765:
4755:
4754:
4748:
4747:
4744:
4743:
4741:
4740:
4735:
4730:
4725:
4720:
4714:
4711:
4710:
4707:
4700:
4699:
4696:
4695:
4692:
4691:
4689:
4688:
4683:
4678:
4673:
4668:
4663:
4658:
4652:
4650:
4644:
4643:
4640:
4639:
4637:
4636:
4630:
4627:
4626:
4624:
4623:
4618:
4613:
4607:
4605:
4596:
4590:
4589:
4587:
4586:
4581:
4575:
4573:
4567:
4566:
4564:
4563:
4558:
4553:
4548:
4543:
4538:
4533:
4531:Kavrayskiy VII
4528:
4522:
4520:
4510:
4509:
4504:
4497:
4496:
4493:
4492:
4489:
4488:
4486:
4485:
4480:
4475:
4469:
4467:
4461:Retroazimuthal
4457:
4456:
4454:
4453:
4448:
4442:
4440:
4434:
4433:
4431:
4430:
4424:
4422:
4416:
4415:
4413:
4412:
4407:
4402:
4397:
4392:
4386:
4384:
4380:Equidistant in
4376:
4375:
4372:
4371:
4369:
4368:
4363:
4358:
4353:
4348:
4343:
4338:
4333:
4328:
4323:
4318:
4312:
4309:
4308:
4306:
4305:
4300:
4294:
4292:
4286:
4285:
4283:
4282:
4277:
4272:
4267:
4262:
4257:
4252:
4246:
4244:
4238:
4237:
4235:
4234:
4229:
4223:
4221:
4215:
4214:
4212:
4211:
4206:
4200:
4198:
4189:
4183:
4182:
4180:
4179:
4174:
4169:
4164:
4159:
4154:
4149:
4144:
4138:
4136:
4126:
4125:
4120:
4113:
4112:
4109:
4108:
4105:
4104:
4102:
4101:
4096:
4091:
4086:
4080:
4078:
4072:
4071:
4068:
4067:
4065:
4064:
4059:
4053:
4050:
4049:
4047:
4046:
4041:
4036:
4030:
4028:
4019:
4009:
4008:
4006:
4005:
4000:
3999:
3998:
3993:
3983:
3978:
3972:
3970:
3964:
3963:
3961:
3960:
3955:
3950:
3944:
3942:
3936:
3935:
3932:
3931:
3929:
3928:
3923:
3918:
3916:Kavrayskiy VII
3912:
3909:
3908:
3906:
3905:
3900:
3895:
3890:
3885:
3880:
3875:
3870:
3865:
3859:
3857:
3850:
3844:
3843:
3840:
3839:
3837:
3836:
3831:
3826:
3821:
3816:
3811:
3806:
3801:
3795:
3792:
3791:
3789:
3788:
3783:
3778:
3773:
3768:
3763:
3758:
3752:
3750:
3744:
3743:
3741:
3740:
3735:
3730:
3724:
3722:
3712:
3702:
3701:
3696:
3689:
3688:
3686:
3685:
3680:
3675:
3669:
3666:
3665:
3662:Map projection
3660:
3658:
3657:
3650:
3643:
3635:
3629:
3628:
3620:
3619:External links
3617:
3615:
3614:
3606:
3588:
3569:
3551:
3526:
3511:
3477:
3475:, pp 1–13
3456:
3443:
3440:on 2018-05-08.
3418:
3395:
3372:
3362:. January 1995
3340:
3337:on 2016-08-09.
3311:
3297:
3275:
3252:
3239:
3220:
3191:
3165:
3136:
3104:
3075:
3057:
3043:
3035:
3017:
3014:on 2016-03-04.
2998:
2975:
2954:
2952:
2949:
2948:
2947:
2942:
2937:
2932:
2925:
2922:
2905:
2904:
2889:
2886:
2881:
2877:
2873:
2870:
2865:
2861:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2831:
2827:
2823:
2820:
2818:
2816:
2813:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2783:
2779:
2775:
2772:
2770:
2768:
2765:
2764:
2749:
2742:
2735:
2729:false northing
2724:
2670:
2667:
2666:
2665:
2659:
2653:
2643:
2626:
2623:
2622:
2621:
2606:
2602:
2595:
2590:
2586:
2581:
2576:
2573:
2567:
2562:
2558:
2553:
2548:
2545:
2541:
2537:
2534:
2531:
2528:
2526:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2464:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2443:
2399:
2396:
2389:
2370:
2364:
2363:
2348:
2344:
2338:
2333:
2329:
2324:
2319:
2315:
2312:
2307:
2303:
2299:
2296:
2294:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2273:
2270:
2264:
2261:
2256:
2252:
2248:
2245:
2240:
2236:
2232:
2229:
2223:
2219:
2213:
2210:
2208:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2187:
2160:
2157:
2156:
2155:
2140:
2136:
2129:
2124:
2120:
2115:
2110:
2107:
2101:
2096:
2092:
2087:
2073:
2069:
2066:
2063:
2060:
2058:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2037:
2034:
2030:
2023:
2018:
2014:
2009:
2004:
2001:
1995:
1990:
1986:
1981:
1976:
1973:
1969:
1965:
1962:
1959:
1956:
1954:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1933:
1918:
1915:
1911:
1910:
1895:
1891:
1887:
1884:
1881:
1878:
1875:
1872:
1868:
1864:
1861:
1858:
1853:
1849:
1845:
1842:
1840:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1819:
1816:
1812:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1753:
1749:
1746:
1743:
1738:
1734:
1728:
1725:
1720:
1717:
1715:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1694:
1679:
1671: = −
1648:
1645:
1644:
1643:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1602:
1599:
1596:
1592:
1589:
1586:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1556:
1553:
1550:
1546:
1543:
1540:
1539:
1520:
1517:
1512:
1511:
1500:
1496:
1489:
1486:
1482:
1479:
1476:
1473:
1467:
1464:
1460:
1457:
1454:
1451:
1445:
1441:
1438:
1433:
1430:
1425:
1421:
1418:
1411:
1408:
1404:
1401:
1398:
1394:
1391:
1280:
1277:
1272:
1265:
1259:
1258:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1200:
1199:
1188:
1184:
1178:
1175:
1172:
1169:
1166:
1161:
1158:
1155:
1152:
1149:
1143:
1139:
1136:
1131:
1128:
1123:
1119:
1114:
1108:
1105:
1100:
1095:
1092:
1086:
1082:
1079:
1075:
1071:
1068:
1065:
1062:
1059:
1055:
1051:
1048:
1045:
1042:
1019:
999:
975:
955:
952:
949:
946:
925:
922:
920:
917:
876:
873:
864:
863:
860:
857:
854:
839:
838:
827:
821:
806:
783:
782:
705:
664:), longitude (
635:
632:
605:
604:
600:
592:
589:
586:
582:
579:
576:
570:
567:
533:
532:
520:
502:in Europe; or
476:
473:
470:
469:
467:
465:
463:
457:
453:
452:
449:
446:
444:
437:
433:
432:
429:
426:
424:
421:
417:
416:
412:
409:
407:
403:
399:
398:
391:
388:
386:
375:
371:
370:
367:
364:
362:
359:
355:
354:
347:
344:
342:
335:
331:
330:
327:
324:
322:
319:
315:
314:
307:
304:
302:
295:
291:
290:
287:
284:
282:
279:
275:
274:
267:
264:
262:
251:
247:
246:
235:
232:
230:
219:
215:
214:
207: = ±
196:
194:
192:
178:
175:
174:
171:
169:
167:
164:
158:
155:
134:
131:
126:
125:
122:
113:, so that the
107:
100:
97:
59:
56:
52:geodetic datum
32:map projection
15:
13:
10:
9:
6:
4:
3:
2:
4790:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4760:
4758:
4739:
4736:
4734:
4731:
4729:
4726:
4724:
4721:
4719:
4716:
4715:
4712:
4705:
4701:
4687:
4684:
4682:
4679:
4677:
4674:
4672:
4669:
4667:
4664:
4662:
4659:
4657:
4654:
4653:
4651:
4649:
4645:
4635:
4632:
4631:
4628:
4622:
4621:Stereographic
4619:
4617:
4614:
4612:
4609:
4608:
4606:
4604:
4600:
4597:
4595:
4591:
4585:
4582:
4580:
4577:
4576:
4574:
4572:
4568:
4562:
4561:Winkel tripel
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4542:
4541:Natural Earth
4539:
4537:
4534:
4532:
4529:
4527:
4524:
4523:
4521:
4519:
4515:
4511:
4507:
4502:
4498:
4484:
4481:
4479:
4476:
4474:
4471:
4470:
4468:
4462:
4458:
4452:
4449:
4447:
4444:
4443:
4441:
4439:
4435:
4429:
4426:
4425:
4423:
4421:
4417:
4411:
4408:
4406:
4403:
4401:
4398:
4396:
4393:
4391:
4388:
4387:
4385:
4383:
4377:
4367:
4364:
4362:
4359:
4357:
4354:
4352:
4349:
4347:
4344:
4342:
4339:
4337:
4334:
4332:
4329:
4327:
4324:
4322:
4321:Briesemeister
4319:
4317:
4314:
4313:
4310:
4304:
4301:
4299:
4296:
4295:
4293:
4291:
4287:
4281:
4278:
4276:
4273:
4271:
4268:
4266:
4263:
4261:
4258:
4256:
4253:
4251:
4248:
4247:
4245:
4243:
4239:
4233:
4230:
4228:
4225:
4224:
4222:
4220:
4216:
4210:
4207:
4205:
4202:
4201:
4199:
4197:
4193:
4190:
4188:
4184:
4178:
4175:
4173:
4172:Stereographic
4170:
4168:
4165:
4163:
4160:
4158:
4155:
4153:
4150:
4148:
4145:
4143:
4140:
4139:
4137:
4135:
4131:
4127:
4123:
4118:
4114:
4100:
4099:Winkel tripel
4097:
4095:
4092:
4090:
4087:
4085:
4082:
4081:
4079:
4077:
4073:
4063:
4060:
4058:
4055:
4054:
4051:
4045:
4044:Stereographic
4042:
4040:
4037:
4035:
4032:
4031:
4029:
4027:
4023:
4020:
4018:
4010:
4004:
4001:
3997:
3994:
3992:
3989:
3988:
3987:
3984:
3982:
3979:
3977:
3974:
3973:
3971:
3969:
3968:Pseudoconical
3965:
3959:
3956:
3954:
3951:
3949:
3946:
3945:
3943:
3941:
3937:
3927:
3924:
3922:
3919:
3917:
3914:
3913:
3910:
3904:
3901:
3899:
3896:
3894:
3891:
3889:
3886:
3884:
3881:
3879:
3876:
3874:
3871:
3869:
3866:
3864:
3861:
3860:
3858:
3854:
3851:
3849:
3845:
3835:
3832:
3830:
3827:
3825:
3822:
3820:
3817:
3815:
3812:
3810:
3807:
3805:
3802:
3800:
3797:
3796:
3793:
3787:
3784:
3782:
3779:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3759:
3757:
3754:
3753:
3751:
3749:
3745:
3739:
3736:
3734:
3731:
3729:
3726:
3725:
3723:
3720:
3716:
3713:
3711:
3707:
3703:
3699:
3694:
3690:
3684:
3681:
3679:
3676:
3674:
3671:
3670:
3667:
3663:
3656:
3651:
3649:
3644:
3642:
3637:
3636:
3633:
3627:
3623:
3622:
3618:
3609:
3603:
3599:
3592:
3589:
3585:
3580:
3578:
3576:
3574:
3570:
3565:
3561:
3555:
3552:
3540:
3536:
3530:
3527:
3522:
3515:
3512:
3507:
3503:
3499:
3495:
3491:
3484:
3482:
3478:
3474:
3470:
3466:
3460:
3457:
3453:
3447:
3444:
3436:
3428:
3422:
3419:
3405:
3399:
3396:
3391:
3383:
3376:
3373:
3361:
3358:(in French).
3357:
3350:
3344:
3341:
3333:
3329:
3322:
3315:
3312:
3307:
3301:
3298:
3293:
3286:
3279:
3276:
3272:
3266:
3259:
3257:
3253:
3249:
3243:
3240:
3236:
3235:
3230:
3224:
3221:
3210:on 2012-02-11
3209:
3205:
3198:
3196:
3192:
3188:
3184:
3183:
3178:
3174:
3169:
3166:
3162:
3158:
3154:
3150:
3149:
3146:
3139:
3137:0-919870-16-3
3133:
3129:
3124:
3123:
3117:
3111:
3109:
3105:
3092:
3085:
3079:
3076:
3072:
3071:
3064:
3062:
3058:
3054:
3047:
3044:
3038:
3032:
3028:
3021:
3018:
3013:
3009:
3002:
2999:
2995:
2991:
2987:
2986:
2979:
2976:
2972:
2968:
2962:
2960:
2956:
2950:
2946:
2943:
2941:
2938:
2936:
2933:
2931:
2928:
2927:
2923:
2921:
2919:
2914:
2910:
2887:
2879:
2875:
2868:
2863:
2859:
2855:
2849:
2846:
2843:
2837:
2834:
2829:
2825:
2821:
2819:
2814:
2807:
2801:
2798:
2795:
2789:
2786:
2781:
2777:
2773:
2771:
2766:
2755:
2754:
2753:
2748:
2741:
2734:
2730:
2723:
2719:
2718:false easting
2715:
2711:
2707:
2703:
2698:
2696:
2692:
2688:
2684:
2680:
2676:
2668:
2664:
2660:
2658:
2654:
2652:
2648:
2644:
2642:
2638:
2637:
2636:
2632:
2624:
2604:
2600:
2593:
2588:
2584:
2579:
2574:
2571:
2565:
2560:
2556:
2551:
2546:
2543:
2539:
2535:
2532:
2529:
2527:
2519:
2516:
2513:
2507:
2500:
2494:
2491:
2488:
2485:
2482:
2479:
2473:
2470:
2467:
2465:
2457:
2454:
2451:
2445:
2434:
2433:
2432:
2429:
2425:
2421:
2417:
2413:
2404:
2397:
2395:
2393:
2388:
2384:
2380:
2376:
2369:
2346:
2342:
2336:
2331:
2327:
2322:
2317:
2313:
2310:
2305:
2301:
2297:
2295:
2287:
2284:
2281:
2275:
2268:
2262:
2259:
2254:
2250:
2246:
2243:
2238:
2234:
2230:
2227:
2221:
2217:
2211:
2209:
2201:
2198:
2195:
2189:
2178:
2177:
2176:
2174:
2170:
2166:
2158:
2138:
2134:
2127:
2122:
2118:
2113:
2108:
2105:
2099:
2094:
2090:
2085:
2071:
2067:
2064:
2061:
2059:
2051:
2048:
2045:
2039:
2032:
2028:
2021:
2016:
2012:
2007:
2002:
1999:
1993:
1988:
1984:
1979:
1974:
1971:
1967:
1963:
1960:
1957:
1955:
1947:
1944:
1941:
1935:
1924:
1923:
1922:
1916:
1914:
1893:
1889:
1885:
1882:
1879:
1876:
1873:
1870:
1866:
1862:
1859:
1856:
1851:
1847:
1843:
1841:
1833:
1830:
1827:
1821:
1814:
1810:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1751:
1747:
1744:
1741:
1736:
1732:
1726:
1723:
1718:
1716:
1708:
1705:
1702:
1696:
1685:
1684:
1683:
1678:
1674:
1670:
1666:
1663: =
1662:
1658:
1654:
1646:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1603:
1597:
1594:
1590:
1587:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1557:
1551:
1548:
1544:
1541:
1530:
1529:
1528:
1526:
1518:
1516:
1498:
1494:
1487:
1484:
1480:
1477:
1474:
1471:
1465:
1462:
1458:
1455:
1452:
1449:
1443:
1439:
1436:
1431:
1428:
1423:
1419:
1416:
1409:
1406:
1402:
1399:
1396:
1392:
1389:
1381:
1380:
1379:
1377:
1373:
1369:
1365:
1360:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1322:
1318:
1314:
1305:
1301:
1299:
1295:
1285:
1278:
1276:
1271:
1264:
1244:
1241:
1238:
1235:
1232:
1226:
1220:
1213:
1212:
1211:
1209:
1205:
1186:
1182:
1176:
1173:
1170:
1167:
1164:
1159:
1156:
1153:
1150:
1147:
1141:
1137:
1134:
1129:
1126:
1121:
1117:
1112:
1106:
1103:
1098:
1093:
1090:
1084:
1080:
1077:
1073:
1069:
1066:
1063:
1060:
1057:
1053:
1049:
1046:
1043:
1040:
1033:
1032:
1031:
1017:
997:
989:
973:
953:
950:
947:
944:
930:
923:
918:
916:
914:
909:
907:
903:
897:
895:
890:
886:
881:
874:
872:
869:
861:
858:
855:
852:
851:
850:
848:
844:
836:
832:
828:
825:
822:
819:
814:
810:
807:
804:
799:
795:
792:
791:
790:
788:
780:
776:
772:
768:
760:
752:
744:
740:
736:
732:
728:
724:
720:
716:
712:
711:
706:
703:
699:
695:
691:
687:
683:
679:
675:
671:
667:
663:
659:
655:
651:
647:
646:
643:Krüger–
641:
640:
639:
633:
631:
628:
623:
619:
615:
610:
601:
597:
593:
590:
587:
583:
580:
577:
573:
572:
568:
562:
558:
556:
551:
546:
541:
539:
530:
526:
521:
518:
513:
512:
511:
509:
505:
501:
497:
493:
488:
486:
482:
474:
468:
466:
464:
461:
458:
455:
450:
447:
445:
442:
438:
435:
430:
427:
425:
422:
419:
413:
410:
408:
404:
401:
396:
392:
389:
387:
384:
380:
376:
373:
368:
365:
363:
360:
357:
352:
348:
345:
343:
340:
336:
333:
328:
325:
323:
320:
317:
312:
308:
305:
303:
300:
296:
293:
288:
285:
283:
280:
277:
272:
268:
265:
263:
260:
256:
252:
249:
244:
240:
236:
233:
231:
228:
224:
220:
217:
210:
206:
201:
195:
193:
188:
183:
177:
156:
154:
152:
148:
144:
140:
132:
130:
123:
120:
116:
112:
108:
105:
101:
98:
95:
94:
89:
85:
81:
80:
79:
77:
73:
64:
57:
55:
53:
49:
45:
41:
37:
33:
30:
21:
4616:Orthographic
4176:
4147:Gauss–Krüger
4039:Orthographic
3834:Web Mercator
3732:
3728:Gauss–Krüger
3597:
3591:
3563:
3554:
3543:. Retrieved
3538:
3529:
3514:
3497:
3493:
3472:
3459:
3446:
3435:the original
3421:
3410:. Retrieved
3398:
3389:
3375:
3364:. Retrieved
3355:
3343:
3332:the original
3327:
3314:
3300:
3291:
3278:
3264:
3247:
3242:
3233:
3228:
3223:
3212:. Retrieved
3208:the original
3186:
3181:
3176:
3168:
3147:
3144:
3127:
3121:
3095:. Retrieved
3078:
3068:
3046:
3026:
3020:
3012:the original
3001:
2989:
2983:
2978:
2973:, section 6)
2966:
2912:
2908:
2906:
2746:
2739:
2732:
2728:
2721:
2717:
2713:
2709:
2706:false origin
2705:
2701:
2699:
2694:
2690:
2686:
2682:
2678:
2674:
2672:
2646:
2634:
2427:
2423:
2419:
2415:
2411:
2409:
2394:
2386:
2382:
2378:
2374:
2367:
2365:
2172:
2168:
2162:
1920:
1912:
1676:
1672:
1668:
1664:
1660:
1656:
1652:
1650:
1522:
1513:
1375:
1371:
1367:
1363:
1361:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1312:
1310:
1297:
1293:
1290:
1269:
1262:
1260:
1207:
1201:
988:Earth radius
935:
912:
910:
905:
901:
898:
893:
888:
884:
882:
878:
867:
865:
846:
842:
840:
834:
823:
812:
808:
797:
793:
786:
784:
774:
770:
766:
758:
750:
742:
738:
734:
730:
726:
722:
718:
714:
709:
707:
701:
697:
693:
689:
685:
681:
677:
673:
669:
665:
661:
657:
653:
649:
644:
642:
637:
626:
621:
617:
614:eccentricity
606:
595:
554:
545:power series
542:
534:
525:Gauss–Krüger
524:
517:Gauss–Krüger
516:
508:Gauss–Krüger
507:
503:
500:Gauss–Krüger
499:
495:
491:
489:
478:
440:
394:
382:
350:
338:
310:
298:
270:
258:
254:
242:
238:
226:
222:
208:
204:
186:
150:
136:
127:
118:
91:
75:
69:
39:
35:
28:
26:
4594:Perspective
4382:some aspect
4366:Strebe 1995
4341:Equal Earth
4260:Gall–Peters
4242:Cylindrical
4057:Equidistant
3953:Equidistant
3883:Equal Earth
3766:Gall–Peters
3710:Cylindrical
3500:: 475–485.
3271:USGS pages.
2940:Scale (map)
2681:axis. Both
2398:Convergence
2165:point scale
2159:Point scale
1204:point scale
841:The Krüger–
785:The Krüger–
494:in the US;
460:Rhumb lines
115:point scale
104:ellipsoidal
88:cylindrical
84:projections
4757:Categories
4656:AuthaGraph
4648:Polyhedral
4518:Compromise
4446:Loximuthal
4438:Loxodromic
4400:Sinusoidal
4250:Balthasart
4227:Sinusoidal
4204:Sinusoidal
4187:Equal-area
3898:Sinusoidal
3856:Equal-area
3756:Balthasart
3748:Equal-area
3721:-conformal
3698:By surface
3545:2024-07-27
3412:2024-07-27
3366:2024-07-27
3214:2012-01-11
3173:Lee, L. P.
3116:Lee, L. P.
3097:2009-08-27
2951:References
2629:See also:
809:Thomas–UTM
585:bisected.)
4728:Longitude
4556:Wagner VI
4405:Two-point
4336:Eckert VI
4331:Eckert IV
4326:Eckert II
4303:Mollweide
4298:Collignon
4265:Hobo–Dyer
4219:Bottomley
4134:Conformal
4122:By metric
4013:Azimuthal
3986:Polyconic
3981:Bottomley
3921:Wagner VI
3893:Mollweide
3878:Eckert VI
3873:Eckert IV
3868:Eckert II
3863:Collignon
3771:Hobo–Dyer
2876:φ
2856:−
2850:φ
2844:λ
2802:φ
2796:λ
2714:northings
2575:
2547:
2536:
2508:γ
2495:φ
2492:
2486:λ
2483:
2474:
2458:φ
2452:λ
2446:γ
2314:
2263:φ
2260:
2247:λ
2244:
2231:−
2202:φ
2196:λ
2109:
2068:
2040:φ
2003:
1975:
1964:
1936:λ
1886:φ
1883:
1877:λ
1874:
1863:
1834:φ
1828:λ
1805:φ
1802:
1796:λ
1793:
1787:−
1779:φ
1776:
1770:λ
1767:
1748:
1709:φ
1703:λ
1624:φ
1621:
1615:λ
1612:
1595:λ
1591:
1578:φ
1575:
1569:λ
1566:
1549:φ
1545:
1485:φ
1481:
1475:−
1463:φ
1459:
1440:
1407:λ
1400:−
1242:φ
1239:
1227:φ
1177:φ
1174:
1168:−
1160:φ
1157:
1138:
1104:φ
1091:π
1081:
1070:
1050:λ
1018:ϕ
954:λ
616:and both
599:meridian.
487:in 1912.
111:conformal
106:versions.
4773:Geocodes
4723:Latitude
4708:See also
4671:Dymaxion
4611:Gnomonic
4546:Robinson
4451:Mercator
4428:Gnomonic
4420:Gnomonic
4255:Behrmann
4162:Mercator
4034:Gnomonic
4016:(planar)
3991:American
3761:Behrmann
3719:Mercator
3118:(1976).
2924:See also
2710:eastings
1598:′
1552:′
1488:′
1466:′
1420:′
1410:′
1393:′
569:Features
4584:HEALPix
4483:Littrow
4094:Wiechel
3996:Chinese
3940:Conical
3804:Central
3799:Cassini
3776:Lambert
3673:History
3308:. 2024.
3161:107–114
986:is the
966:(where
900:Krüger–
856:Finland
777:. (See
763:
747:
708:Krüger–
4603:Planar
4571:Hybrid
4478:Hammer
4410:Werner
4351:Hammer
4316:Albers
4232:Werner
4209:Werner
4089:Hammer
4084:Aitoff
4003:Werner
3948:Albers
3824:Miller
3683:Portal
3604:
3541:. 2009
3159:, and
3157:92–101
3151:. pp.
3134:
3033:
2533:arctan
2471:arctan
2065:arcsin
1961:arctan
1860:arctan
1296:- and
990:) and
859:Sweden
853:France
811:: The
575:globe.
139:sphere
76:Normal
4473:Craig
4390:Conic
4196:Bonne
3976:Bonne
3438:(PDF)
3431:(PDF)
3407:(PDF)
3386:(PDF)
3352:(PDF)
3335:(PDF)
3324:(PDF)
3288:(PDF)
3087:(PDF)
2969:. In
862:Japan
596:small
555:whole
119:local
82:Both
4676:ISEA
3678:List
3602:ISBN
3153:1–14
3132:ISBN
3031:ISBN
2911:and
2712:and
2700:The
2693:and
2685:and
2544:tanh
2416:from
2311:cosh
2078:sech
1972:sinh
1667:and
1374:and
773:and
741:and
733:and
717:and
700:and
680:and
652:and
620:and
377:The
86:are
27:The
3502:doi
2572:tan
2489:sin
2480:tan
2251:cos
2235:sin
2106:sin
2000:sec
1880:tan
1871:sec
1799:cos
1790:sin
1773:cos
1764:sin
1618:tan
1609:sec
1588:tan
1572:cos
1563:sin
1542:sin
1478:sin
1456:sin
1236:sec
1171:sin
1154:sin
1078:tan
818:UTM
603:up.
498:or
40:TMP
4759::
3572:^
3562:.
3537:.
3498:85
3496:.
3492:.
3480:^
3473:59
3471:,
3388:.
3354:.
3326:.
3290:.
3255:^
3194:^
3155:,
3148:13
3126:.
3107:^
3089:.
3060:^
2990:54
2958:^
2920:.
2731:,
2720:,
2420:to
2173:φ′
1745:ln
1673:x′
1665:y′
1437:ln
1376:φ′
1372:λ′
1368:y′
1364:x′
1349:y′
1345:x′
1333:λ′
1329:φ′
1325:λ′
1321:φ′
1206:,
1135:ln
1067:ln
781:.)
769:,
729:,
696:,
692:,
456:•
448:•
436:•
428:•
420:•
411:•
402:•
390:•
374:•
366:•
358:•
346:•
334:•
326:•
318:•
306:•
294:•
286:•
278:•
266:•
261:.
250:•
234:•
218:•
38:,
36:TM
3654:e
3647:t
3640:v
3612:.
3610:.
3566:.
3548:.
3508:.
3504::
3415:.
3369:.
3294:.
3237:.
3229:9
3217:.
3187:8
3177:8
3163:.
3140:.
3100:.
3039:.
2996:.
2988:(
2913:y
2909:x
2888:.
2885:)
2880:0
2872:(
2869:m
2864:0
2860:k
2853:)
2847:,
2841:(
2838:y
2835:+
2830:0
2826:N
2822:=
2815:N
2808:,
2805:)
2799:,
2793:(
2790:x
2787:+
2782:0
2778:E
2774:=
2767:E
2750:0
2747:k
2743:0
2740:φ
2736:0
2733:N
2725:0
2722:E
2695:ϕ
2691:λ
2687:y
2683:x
2679:y
2675:x
2647:n
2605:.
2601:)
2594:a
2589:0
2585:k
2580:y
2566:a
2561:0
2557:k
2552:x
2540:(
2530:=
2523:)
2520:y
2517:,
2514:x
2511:(
2501:,
2498:)
2477:(
2468:=
2461:)
2455:,
2449:(
2428:γ
2424:x
2412:γ
2390:0
2387:k
2383:x
2379:x
2375:k
2371:0
2368:k
2347:.
2343:)
2337:a
2332:0
2328:k
2323:x
2318:(
2306:0
2302:k
2298:=
2291:)
2288:y
2285:,
2282:x
2279:(
2276:k
2269:,
2255:2
2239:2
2228:1
2222:0
2218:k
2212:=
2205:)
2199:,
2193:(
2190:k
2169:k
2139:.
2135:]
2128:a
2123:0
2119:k
2114:y
2100:a
2095:0
2091:k
2086:x
2072:[
2062:=
2055:)
2052:y
2049:,
2046:x
2043:(
2033:,
2029:]
2022:a
2017:0
2013:k
2008:y
1994:a
1989:0
1985:k
1980:x
1968:[
1958:=
1951:)
1948:y
1945:,
1942:x
1939:(
1894:,
1890:]
1867:[
1857:a
1852:0
1848:k
1844:=
1837:)
1831:,
1825:(
1822:y
1815:,
1811:]
1784:1
1761:+
1758:1
1752:[
1742:a
1737:0
1733:k
1727:2
1724:1
1719:=
1712:)
1706:,
1700:(
1697:x
1680:0
1677:k
1669:y
1661:x
1657:y
1655:,
1653:x
1627:.
1606:=
1581:,
1560:=
1499:.
1495:]
1472:1
1453:+
1450:1
1444:[
1432:2
1429:a
1424:=
1417:y
1403:a
1397:=
1390:x
1366:,
1357:y
1355:,
1353:x
1347:,
1341:λ
1339:,
1337:φ
1331:,
1317:λ
1315:,
1313:φ
1298:y
1294:x
1273:0
1270:k
1266:0
1263:k
1245:.
1233:=
1230:)
1224:(
1221:k
1208:k
1187:.
1183:]
1165:1
1151:+
1148:1
1142:[
1130:2
1127:a
1122:=
1118:]
1113:)
1107:2
1099:+
1094:4
1085:(
1074:[
1064:a
1061:=
1058:y
1054:,
1047:a
1044:=
1041:x
998:y
974:a
951:a
948:=
945:x
913:n
906:λ
902:n
894:n
889:n
885:n
868:n
847:n
843:n
835:n
813:λ
798:λ
787:λ
775:e
771:a
767:y
759:a
755:/
751:x
743:λ
739:φ
735:e
731:a
727:φ
723:λ
719:y
715:x
710:λ
704:.
702:e
698:a
694:y
690:x
686:n
682:λ
678:φ
674:e
670:a
666:λ
662:φ
658:n
654:y
650:x
645:n
627:x
622:y
618:x
441:x
395:x
383:y
351:x
339:y
311:x
299:y
271:y
259:y
255:y
243:y
239:x
227:x
223:x
209:π
205:x
187:y
96:.
34:(
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