74:
2414:
1295:
940:
1311:-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.
1315:
31:
572:
211:
193:
2708:. 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.
2630:
2372:
2441:
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
811:
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
1525:
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
622:
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
139:
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.
890:
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
609:
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
546:
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
1652:
902:
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
1919:
2926:
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
910:
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.
101:: 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
2913:
2448:
533:
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
525:
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
3061:
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.
425:
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°.
2192:
826:
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
595:
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
2164:
947:
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
881:
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.
416:
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°.
558:
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
563:
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
1520:
613:
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
1302:
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
1699:
640:
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.
1208:
903:
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–
1544:
2769:
635:
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
585:
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
2625:{\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}}}
602:
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°.
599:
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.
364:. 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.
2774:
2453:
2197:
1943:
1704:
3462:
592:
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.
840:
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
352:. 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.
2661:
530:
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.
1549:
2367:{\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}}}
837:: 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.)
2667:
1267:
164:
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.
1938:
156:
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
2684:
The projection coordinates resulting from the various developments of the ellipsoidal transverse Mercator are Cartesian coordinates such that the central meridian corresponds to the
1538:
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:
2715:
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
434:
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).
848:-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.
442:
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.)
3214:
2377:
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
975:
152:
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
89:) Mercator projection. They share the same underlying mathematical construction and consequently the transverse Mercator inherits many traits from the normal Mercator:
831:. They are also incorporated into the GEOTRANS coordinate converter made available by the United States National Geospatial-Intelligence Agency’s Office of Geomatics.
2641:
1039:
3018:
3284:
It gives full details of most projections, together with interesting introductory sections, but it does not derive any of the projections from first principles.
1019:
995:
3437:
4006:
17:
380:
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.
1914:{\displaystyle {\begin{aligned}x(\lambda ,\varphi )&={\frac {1}{2}}k_{0}a\ln \left,\\y(\lambda ,\varphi )&=k_{0}a\arctan \left,\end{aligned}}}
462:
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.
800:
series were the first to be implemented, possibly because they were much easier to evaluate on the hand calculators of the mid twentieth century.
828:
58:
1047:
547:
transverse Mercator was adopted, in one form or another, by many nations (and international bodies); in addition it provides the basis for the
135:
Both projections have constant scale on the line of tangency (the equator for the normal Mercator and the central meridian for the transverse).
1647:{\displaystyle {\begin{aligned}\sin \varphi '&=\sin \lambda \cos \varphi ,\\\tan \lambda '&=\sec \lambda \tan \varphi .\end{aligned}}}
2995:
110:
Both projections may be modified to secant forms, which means the scale has been reduced so that the cylinder slices through the model globe.
3414:
2918:
The terms "eastings" and "northings" do not mean strict east and north directions. Grid lines of the transverse projection, other than the
589:
The central regions of the transverse projections on sphere and ellipsoid are indistinguishable on the small-scale projections shown here.
2908:{\displaystyle {\begin{aligned}E&=E_{0}+x(\lambda ,\varphi ),\\N&=N_{0}+y(\lambda ,\varphi )-k_{0}m(\varphi _{0}).\end{aligned}}}
671:(the ratio of the difference and sum of the major and minor axes of the ellipsoid). The coefficients are expressed in terms of latitude (
1395:
4748:
4280:
3786:
3663:
3479:
3391:
R. Kuittinen; T. Sarjakoski; M. Ollikainen; M. Poutanen; R. Nuuros; P. Tätilä; J. Peltola; R. Ruotsalainen; M. Ollikainen (2006).
4366:
4162:
4152:
4072:
3476:
A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss–Krüger Projection
4290:
4285:
4260:
4132:
3796:
3791:
3766:
3218:
3395:[map projections related to the ETRS89 system, level coordinates and map sheet division, Appendix 1: Project formulas]
568:
ellipsoid to the plane, although its principal application is to accurate large-scale mapping "close" to the central meridian.
3616:
3594:
3045:
1924:
The above expressions are given in Lambert and also (without derivations) in Snyder, Maling and Osborne (with full details).
4157:
3738:
619:
77:
Comparison of tangent and secant forms of normal, oblique and transverse Mercator projections with standard parallels in red
2651:
813:
789:
551:
series of projections. The Gauss–Krüger projection is now the most widely used projection in accurate large-scale mapping.
3440:[Calculation to convert coordinates to obtain longitude and latitude, meridian aberration angle and scale factor]
3331:
2673:
841:
408:
on the projection. (On the sphere it depends on both latitude and longitude.) The scale is true on the central meridian.
4167:
3968:
3022:
3359:
57:. The transverse version is widely used in national and international mapping systems around the world, including the
4300:
4252:
3913:
3839:
3758:
3146:
3003:). Published by Wilhelm Engelmann. This is Lambert's paper with additional comments by the editor. Available at the
4783:
4691:
4488:
4415:
4371:
4067:
3445:
3393:"ETRS89—järjestelmään liittyvät karttaprojektiot, tasokoordinaatistot ja karttalehtijako, Liite 1: Projektiokaavat"
2749:, is the distance of the true grid origin north of the false origin. If the true origin of the grid is at latitude
2186:: this may be expressed either in terms of the geographical coordinates or in terms of the projection coordinates:
3243:
3191:
3094:
2719:
to the west (and possibly north or south) of the grid origin: the coordinates relative to the false origin define
2159:{\displaystyle {\begin{aligned}\lambda (x,y)&=\arctan \left,\\\varphi (x,y)&=\arcsin \left.\end{aligned}}}
1227:
4536:
4483:
844:). All such modifications have been eclipsed by the power of modern computers and the development of high order
624:
495:
4773:
4644:
4613:
4036:
3814:
3370:
160:. (The text is also available in a modern English translation.) Lambert did not name his projections; the name
3113:
The EEA recommends the transverse Mercator for conformal pan-European mapping at scales larger than 1:500,000.
4728:
4696:
4546:
4177:
4001:
3834:
3824:
3656:
548:
539:
252: = 0. Meridians 90 degrees east and west of the central meridian project to lines of constant
517:
more generally. Other than just a synonym for the ellipsoidal transverse Mercator map projection, the term
324:
direction. The points on the equator at ninety degrees from the central meridian are projected to infinity.
4686:
4400:
4054:
3963:
1526:
central meridian) have no simple relation to the rotated graticule and they project to complicated curves.
649:
In his 1912 paper, Krüger presented two distinct solutions, distinguished here by the expansion parameter:
3392:
4676:
4626:
4589:
4356:
4049:
3898:
3748:
3101:
2955:
2442:
be expressed either in terms of the geographical coordinates or in terms of the projection coordinates:
82:
4604:
4561:
4405:
4270:
3996:
3829:
3819:
3776:
2403: = 1.0004: the scale factor is within 0.04% of unity over a strip of about 510 km wide.
542:
system, but the central meridians of the Gauss–Krüger zones are only 3° apart, as opposed to 6° in UTM.
65:, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.
4528:
3708:
4541:
3926:
3295:
812:
and examined which terms were necessary to attain an accuracy of 1 mm (ground measurement). The
3635:
The projections used to illustrate this article were prepared using Geocart which is available from
4631:
4571:
4551:
4331:
4275:
4182:
4144:
4109:
3781:
3649:
1041:. For a tangent Normal Mercator projection the (unique) formulae which guarantee conformality are:
121:
3296:"The universal grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS)"
396:
on the projection. (On the sphere it depends on latitude only.) The scale is true on the equator.
3858:
3844:
3688:
3171:
2940:
951:
157:
3167:
3163:
1330:) on the standard graticule can also be identified in terms of angles on the rotated graticule:
372:
Greenland is almost as large as Africa; the actual area is about one fourteenth that of Africa.
4743:
4376:
4351:
4086:
3893:
3683:
1535:
4516:
4471:
3500:
3095:"Short Proceedings of the 1st European Workshop on Reference Grids, Ispra, 27–29 October 2003"
4666:
4456:
4410:
4237:
4214:
4197:
3978:
3908:
3130:
4671:
4566:
4390:
4346:
4341:
4336:
4313:
4308:
4229:
3991:
3931:
3903:
3888:
3883:
3878:
3873:
3720:
3183:
3126:
1024:
560:
491:
103:
98:
4658:
4448:
8:
4621:
4556:
4461:
4438:
4265:
4172:
4044:
4023:
3771:
3729:
3571:"Transverse Mercator Projection - preprint of paper and C++ implementation of algorithms"
2945:
153:
114:
54:
4430:
73:
4493:
4104:
3950:
3809:
3212:
A guide to coordinate systems in Great Britain. This is available as a pdf document at
3131:
1279:
on the right hand side of all these equations: this ensures that the scale is equal to
1004:
980:
605:
The point scale factor is independent of direction at any point so that the shape of a
4581:
3052:
This is an excellent survey of virtually all known projections from antiquity to 1993.
554:
The projection, as developed by Gauss and Krüger, was expressed in terms of low order
4778:
4420:
4361:
4326:
4242:
4219:
4099:
4094:
4013:
3958:
3936:
3612:
3142:
3041:
4206:
3986:
3512:
3463:
A highly accurate world wide algorithm for the transverse Mercator mapping (almost)
1338:(angle M′CO) becomes an effective longitude. (The minus sign is necessary so that (
2413:
3390:
3063:
1294:
894:
The exact solution is a valuable tool in assessing the accuracy of the truncated
149:
3276:
Map Projections—A Working Manual. U.S. Geological Survey Professional Paper 1395
3080:
939:
659:(paragraphs 5 to 8): Formulae for the direct projection, giving the coordinates
256:
through the projected poles. All other meridians project to complicated curves.
3672:
3570:
3529:
1314:
926:) series is accurate to 5 nm within 3900 km of the central meridian.
268: = 0 and parallel circles project to straight lines of constant
94:
62:
42:
3516:
3342:
3154:
3004:
340:
The projection is conformal. The shapes of small elements are well preserved.
332:
The projection is conformal. The shapes of small elements are well preserved.
4767:
30:
3475:
490:
The ellipsoidal form of the transverse Mercator projection was developed by
4681:
1362:) axes are related to the rotated graticule in the same way that the axes (
998:
555:
3281:
2642:
Universal Transverse Mercator coordinate system § Simplified formulae
1389:
by the transformation formulae of the tangent Normal Mercator projection:
1203:{\displaystyle x=a\lambda \,,\qquad y=a\ln \left={\frac {a}{2}}\ln \left.}
885:
732:
are expansions (of orders 5 and 4 respectively) in terms of the longitude
3693:
3019:"Notes and Comments on the Composition of Terrestrial and Celestial Maps"
2981:
2950:
2738:, is the distance of the true grid origin east of the false origin. The
2175:
1214:
389:
125:
404:
The point scale factor is independent of direction. It is a function of
3316:
915:
is accurate to within 1 mm but the Redfearn version of the Krüger–
470:
450:
Convergence (the angle between projected meridians and grid lines with
3465:, in Proc. XXIII Intl. Cartographic Conf. (ICC2007), Moscow, p. 2.1.2.
3415:"Gauss Conformal Projection (Transverse Mercator): Krüger's Formulas"
2174:
In terms of the coordinates with respect to the rotated graticule the
284: = 0 but all other parallels are complicated closed curves.
4738:
3261:. Washington: U.S. Coast and Geodetic Survey Special Publication 251.
2756:
on the central meridian and the scale factor the central meridian is
1373:
The tangent transverse Mercator projection defines the coordinates (
4733:
2978:
Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten
2635:
943:
The normal aspect of a tangent cylindrical projection of the sphere
736:, expressed in radians: the coefficients are expressed in terms of
724:(paragraphs 13 and 14): Formulae giving the projection coordinates
454:
constant) is identically zero. Grid north and true north coincide.
1515:{\displaystyle x'=-a\lambda '\,\qquad y'={\frac {a}{2}}\ln \left.}
1221:, is independent of direction: it is a function of latitude only:
4594:
3641:
2982:
Beyträge zum Gebrauche der Mathematik und deren Anwendung, part 3
929:
644:
667:, are fourth order expansions in terms of the third flattening,
3545:
473:(of constant azimuth on the sphere) project to straight lines.
236: = 0. Other meridians project to straight lines with
3636:
3530:
F. W.J. Olver; D.W. Lozier; R.F. Boisvert; C.W. Clark (2010).
2425:
at a point on the projection is defined by the angle measured
1272:
For the secant version of the projection there is a factor of
571:
210:
192:
167:
2679:
1346:) are related to the rotated graticule in the same way that (
300:
Projected meridians and parallels intersect at right angles.
292:
Projected meridians and parallels intersect at right angles.
3278:. United States Government Printing Office, Washington, D.C.
2672:
Fourth order Redfearn series by concise formulae (example):
200: = ±π, corresponding to approximately 85 degrees).
3597:
Detailed derivations of all formulae quoted in this article
3339:
Canadian Technical Report of Hydrography and Ocean Sciences
3038:
Flattening the Earth: Two Thousand Years of Map Projections
3501:"Transverse Mercator with an accuracy of a few nanometers"
3141:. Vol. 16. Toronto: B. V. Gutsell, York University.
1354:) are related to the standard graticule). The Cartesian (
907:
series are very much better than the Redfearn λ series.
3531:
2763:
then these definitions give eastings and northings by:
886:
Exact Gauss–Krüger and accuracy of the truncated series
2087:
1662:
The direct formulae giving the Cartesian coordinates (
1534:
The angles of the two graticules are related by using
1529:
934:
392:
is independent of direction. It is a function of
2772:
2451:
2195:
1941:
1702:
1547:
1398:
1230:
1050:
1027:
1007:
983:
954:
575:
Ellipsoidal transverse Mercator: a finite projection.
196:
Spherical Normal (equatorial) Mercator (truncated at
3606:
3341:(262). Canadian Hydrographic Service. Archived from
3064:
Schumacher Astronomische Abhandlungen, Altona, no. 3
2666:
Exact (closed form) transverse Mercator projection:
898:
and λ series. For example, the original 1912 Krüger–
3294:Hager, J. W.; Behensky, J. F.; Drew, B. W. (1989).
3238:
Redfearn, J C B (1948). Survey Review, Volume
3084:. Royal Prussian Geodetic Institute, New Series 52.
1289:
248:The central meridian projects to the straight line
232:The central meridian projects to the straight line
4035:
3303:Technical Report TM 8358.2, Defense Mapping Agency
3193:The transverse Mercator projection of the spheroid
2907:
2624:
2429:the projected meridian, which defines true north,
2366:
2158:
1927:
1913:
1646:
1514:
1261:
1202:
1033:
1013:
989:
969:
485:
68:
3293:
3133:Conformal Projections Based on Elliptic Functions
3081:Konforme Abbildung des Erdellipsoids in der Ebene
1657:
856:series have been implemented (to fourth order in
756:are sixth order expansions in terms of the ratio
4765:
3259:Conformal Projections in Geodesy and Cartography
2636:Formulae for the ellipsoidal transverse Mercator
919:series has a maximum error of 1 kilometre.
816:are still the basis of the OSGB map projections.
143:
3498:
3273:
3035:
623:the ellipsoid: inverse series are functions of
3329:
3021:. University of Michigan Press. Archived from
3005:University of Michigan Historical Math Library
1370:) axes are related to the standard graticule.
930:Formulae for the spherical transverse Mercator
645:Implementations of the Gauss–Krüger projection
521:may be used in other slightly different ways:
501:The projection is known by several names: the
27:Adaptation of the standard Mercator projection
3657:
3360:"Projection Cartographique Mercator Traverse"
2996:Ostwalds Klassiker der exakten Wissenschaften
1670:) follow immediately from the above. Setting
1262:{\displaystyle k(\varphi )=\sec \varphi .\,}
699:but with coefficients expressed in terms of
214:Spherical transverse Mercator (truncated at
3121:
3119:
3040:. University of Chicago Press. p. 82.
1334:(angle M′CP) is an effective latitude and −
168:Normal and transverse spherical projections
148:In constructing a map on any projection, a
3664:
3650:
3332:"Wide zone transverse Mercator projection"
2680:Coordinates, grids, eastings and northings
2093:
776:, with coefficients expressed in terms of
280:The equator projects to the straight line
264:The equator projects to the straight line
208:
190:
81:The transverse Mercator projection is the
4749:Map projection of the tri-axial ellipsoid
3866:
3532:"NIST Handbook of Mathematical Functions"
3494:
3492:
3480:Geospatial Information Authority of Japan
3403:(in Finnish). Finnish Geodetic Institute.
3269:
3267:
3177:
3029:
2972:
2970:
1424:
1258:
1063:
3444:(in Chinese). p. 22. Archived from
3232:
3116:
3074:
3072:
3055:
2688:axis and the equator corresponds to the
2412:
1313:
1293:
938:
891:solution is described by Karney (2011).
570:
209:
191:
72:
29:
3590:
3588:
3586:
3584:
3208:
3206:
2987:
14:
4766:
3609:Coordinate Systems and Map Projections
3600:
3489:
3468:
3264:
3196:. (Errata and comments in Volume
3016:
2967:
2662:Transverse Mercator: flattening series
312:direction. The poles lie at infinity.
4717:
4612:
4514:
4130:
3706:
3645:
3455:
3069:
3010:
3581:
3546:"Maxima - A computer algebra system"
3203:
2652:Transverse Mercator: Redfearn series
1932:Inverting the above equations gives
1322:The position of an arbitrary point (
877:Higher order versions of the Krüger–
790:Transverse Mercator: Redfearn series
695:are also fourth order expansions in
4515:
3611:(second ed.). Pergamon Press.
3461:K. E. Engsager and K. Poder, 2007,
3287:
3186:(1945). Survey Review, Volume
3125:
2727:which will always be positive. The
2674:Transverse Mercator: Bowring series
2668:Transverse Mercator: exact solution
2392:is approximately 180 km. When
1530:The relation between the graticules
935:Spherical normal Mercator revisited
842:Transverse Mercator: Bowring series
807:: In 1945, L. P. Lee confirmed the
320:The projection is unbounded in the
308:The projection is unbounded in the
53:) is an adaptation of the standard
24:
3671:
3280:This paper can be downloaded from
2650:Gauss-Kruger series in longitude:
2646:Details of actual implementations
2437:, defining grid north. Therefore,
25:
4795:
3629:
3242:(Part 69), pp 318–322,
3200:(Part 61), pp. 277–278.
3190:(Part 58), pp 142–152.
2396:is approximately 255 km and
503:(ellipsoidal) transverse Mercator
4692:Quadrilateralized spherical cube
4372:Quadrilateralized spherical cube
2993:Albert Wangerin (Editor), 1894.
2976:Lambert, Johann Heinrich. 1772.
2928:
1693:to accommodate secant versions)
1290:Normal and transverse graticules
515:Gauss–Krüger transverse Mercator
494:in 1822 and further analysed by
128:is independent of direction and
34:A transverse Mercator projection
3563:
3538:
3523:
3438:"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"
3430:
3407:
3384:
3352:
3323:
3309:
3251:
1928:Inverse transformation formulae
1425:
1067:
486:Ellipsoidal transverse Mercator
360:Distortion increases with
348:Distortion increases with
69:Standard and transverse aspects
4281:Lambert cylindrical equal-area
3707:
3371:Institut Geographique National
3087:
2895:
2882:
2863:
2851:
2815:
2803:
2700:are defined for all values of
2533:
2521:
2508:
2487:
2471:
2459:
2408:
2301:
2289:
2215:
2203:
2169:
2065:
2053:
1961:
1949:
1847:
1835:
1722:
1710:
1658:Direct transformation formulae
1298:Transverse mercator graticules
1240:
1234:
1213:Conformality implies that the
620:Gauss–Krüger coordinate system
61:. When paired with a suitable
18:Gauss–Krüger coordinate system
13:
1:
4729:Interruption (map projection)
4131:
3607:Maling, Derek Hylton (1992).
3534:. Cambridge University Press.
2961:
748:. The inverse projection for
549:Universal Transverse Mercator
540:universal transverse Mercator
144:Spherical transverse Mercator
59:Universal Transverse Mercator
4718:
4367:Lambert azimuthal equal-area
4163:Guyou hemisphere-in-a-square
4153:Adams hemisphere-in-a-square
3575:geographiclib.sourceforge.io
3245:Transverse Mercator formulae
2384: = 0.9996 so that
1318:Transverse mercator geometry
1021:is a prescribed function of
860:) by the following nations.
687:). The inverse formulae for
496:Johann Heinrich Louis Krüger
113:Both exist in spherical and
7:
3102:European Environment Agency
2956:Oblique Mercator projection
2934:
970:{\displaystyle x=a\lambda }
922:Karney's own 8th-order (in
579:
10:
4800:
2639:
1686:(and restoring factors of
465:
445:
429:
411:
383:
367:
343:
327:
303:
287:
259:
227:
222:in units of Earth radius).
187:
173:
132:shapes are well preserved;
4724:
4713:
4657:
4640:
4603:
4580:
4527:
4523:
4510:
4470:
4447:
4429:
4389:
4322:
4299:
4251:
4228:
4205:
4196:
4143:
4139:
4126:
4085:
4063:
4022:
3977:
3949:
3922:
3857:
3805:
3757:
3728:
3719:
3715:
3702:
3679:
3517:10.1007/s00190-011-0445-3
3156:The Canadian Cartographer
3017:Tobler, Waldo R. (1972).
618:In most applications the
538:system is similar to the
183:
176:
3595:The Mercator Projections
3499:C. F. F. Karney (2011).
3274:Snyder, John P. (1987).
3215:"Welcome to GPS Network"
3139:Cartographica Monographs
3036:Snyder, John P. (1993).
2433:a grid line of constant
2417:The angle of convergence
181:
179:
174:
4784:Cylindrical projections
4168:Lambert conformal conic
3330:N. Stuifbergen (2009).
3257:Thomas, Paul D (1952).
3104:. 2004-06-14. p. 6
2941:List of map projections
2656:Gauss-Kruger series in
158:Johann Heinrich Lambert
4301:Tobler hyperelliptical
3914:Tobler hyperelliptical
3840:Space-oblique Mercator
3637:Mapthematics Geocart 3
2909:
2626:
2421:The convergence angle
2418:
2368:
2182: = sec
2160:
1915:
1648:
1536:spherical trigonometry
1516:
1319:
1299:
1263:
1204:
1035:
1015:
991:
971:
944:
576:
223:
201:
78:
35:
4774:Conformal projections
3550:maxima.sourceforge.io
3317:"Office of Geomatics"
2910:
2627:
2416:
2369:
2161:
1916:
1649:
1517:
1317:
1297:
1264:
1205:
1036:
1034:{\displaystyle \phi }
1016:
992:
972:
942:
574:
213:
195:
120:Both projections are
76:
33:
4677:Cahill–Keyes M-shape
4537:Chamberlin trimetric
3401:Technical Report JHS
3153:Supplement No. 1 to
2929:angle of convergence
2770:
2660:(third flattening):
2449:
2388: = 1 when
2193:
1939:
1700:
1545:
1396:
1228:
1048:
1025:
1005:
981:
952:
683:) and eccentricity (
561:Laurence Patrick Lee
492:Carl Friedrich Gauss
184:Transverse Mercator
85:of the standard (or
4744:Tissot's indicatrix
4645:Central cylindrical
4286:Smyth equal-surface
4188:Transverse Mercator
4037:General perspective
3792:Smyth equal-surface
3744:Transverse Mercator
3474:Kawase, K. (2011):
3078:Krüger, L. (1912).
2946:Mercator projection
2178:factor is given by
162:transverse Mercator
55:Mercator projection
40:transverse Mercator
4697:Waterman butterfly
4547:Miller cylindrical
4178:Peirce quincuncial
4073:Lambert equal-area
3825:Gall stereographic
3505:Journal of Geodesy
3478:, Bulletin of the
2905:
2903:
2622:
2620:
2419:
2364:
2362:
2156:
2154:
2091:
1911:
1909:
1644:
1642:
1512:
1320:
1300:
1259:
1200:
1031:
1011:
987:
967:
945:
577:
390:point scale factor
224:
202:
79:
36:
4761:
4760:
4757:
4756:
4709:
4708:
4705:
4704:
4653:
4652:
4506:
4505:
4502:
4501:
4385:
4384:
4122:
4121:
4118:
4117:
4081:
4080:
3969:Lambert conformal
3945:
3944:
3859:Pseudocylindrical
3853:
3852:
3618:978-0-08-037233-4
3066:, p. 5–30.
3047:978-0-226-76747-5
2608:
2580:
2351:
2277:
2276:
2142:
2114:
2090:
2036:
2008:
1819:
1740:
1503:
1445:
1191:
1143:
1120:
1107:
1014:{\displaystyle y}
990:{\displaystyle a}
805:Lee–Redfearn–OSGB
483:
482:
154:ellipsoidal model
83:transverse aspect
16:(Redirected from
4791:
4715:
4714:
4672:Cahill Butterfly
4610:
4609:
4590:Goode homolosine
4525:
4524:
4512:
4511:
4477:
4476:(Mecca or Qibla)
4357:Goode homolosine
4203:
4202:
4141:
4140:
4128:
4127:
4033:
4032:
4028:
3899:Goode homolosine
3864:
3863:
3749:Oblique Mercator
3726:
3725:
3717:
3716:
3704:
3703:
3666:
3659:
3652:
3643:
3642:
3624:
3622:
3604:
3598:
3592:
3579:
3578:
3567:
3561:
3560:
3558:
3557:
3542:
3536:
3535:
3527:
3521:
3520:
3496:
3487:
3472:
3466:
3459:
3453:
3452:
3450:
3443:
3434:
3428:
3427:
3425:
3424:
3419:
3411:
3405:
3404:
3398:
3388:
3382:
3381:
3379:
3378:
3364:
3356:
3350:
3349:
3347:
3336:
3327:
3321:
3320:
3313:
3307:
3306:
3300:
3291:
3285:
3279:
3271:
3262:
3255:
3249:
3236:
3230:
3229:
3227:
3226:
3217:. Archived from
3210:
3201:
3181:
3175:
3152:
3136:
3123:
3114:
3112:
3110:
3109:
3099:
3091:
3085:
3076:
3067:
3059:
3053:
3051:
3033:
3027:
3026:
3014:
3008:
2991:
2985:
2974:
2914:
2912:
2911:
2906:
2904:
2894:
2893:
2878:
2877:
2844:
2843:
2796:
2795:
2713:true grid origin
2631:
2629:
2628:
2623:
2621:
2614:
2610:
2609:
2607:
2603:
2602:
2589:
2581:
2579:
2575:
2574:
2561:
2373:
2371:
2370:
2365:
2363:
2356:
2352:
2350:
2346:
2345:
2332:
2320:
2319:
2278:
2269:
2268:
2253:
2252:
2237:
2236:
2235:
2226:
2165:
2163:
2162:
2157:
2155:
2148:
2144:
2143:
2141:
2137:
2136:
2123:
2115:
2113:
2109:
2108:
2095:
2092:
2088:
2042:
2038:
2037:
2035:
2031:
2030:
2017:
2009:
2007:
2003:
2002:
1989:
1920:
1918:
1917:
1912:
1910:
1903:
1899:
1866:
1865:
1824:
1820:
1818:
1792:
1766:
1751:
1750:
1741:
1733:
1653:
1651:
1650:
1645:
1643:
1611:
1565:
1521:
1519:
1518:
1513:
1508:
1504:
1502:
1501:
1480:
1479:
1458:
1446:
1438:
1433:
1423:
1406:
1286:on the equator.
1268:
1266:
1265:
1260:
1209:
1207:
1206:
1201:
1196:
1192:
1190:
1173:
1156:
1144:
1136:
1131:
1127:
1126:
1122:
1121:
1113:
1108:
1100:
1040:
1038:
1037:
1032:
1020:
1018:
1017:
1012:
996:
994:
993:
988:
976:
974:
973:
968:
775:
773:
772:
767:
764:
240: constant.
177:Normal Mercator
172:
171:
104:central meridian
21:
4799:
4798:
4794:
4793:
4792:
4790:
4789:
4788:
4764:
4763:
4762:
4753:
4720:
4701:
4649:
4636:
4599:
4576:
4562:Van der Grinten
4519:
4517:By construction
4498:
4475:
4474:
4466:
4443:
4425:
4406:Equirectangular
4392:
4381:
4318:
4295:
4291:Trystan Edwards
4247:
4224:
4192:
4135:
4114:
4087:Pseudoazimuthal
4077:
4059:
4026:
4025:
4018:
3973:
3941:
3937:Winkel I and II
3918:
3849:
3830:Gall isographic
3820:Equirectangular
3801:
3797:Trystan Edwards
3753:
3711:
3698:
3675:
3670:
3632:
3627:
3619:
3605:
3601:
3593:
3582:
3569:
3568:
3564:
3555:
3553:
3544:
3543:
3539:
3528:
3524:
3497:
3490:
3473:
3469:
3460:
3456:
3448:
3441:
3436:
3435:
3431:
3422:
3420:
3417:
3413:
3412:
3408:
3396:
3389:
3385:
3376:
3374:
3367:geodesie.ign.fr
3362:
3358:
3357:
3353:
3345:
3334:
3328:
3324:
3315:
3314:
3310:
3298:
3292:
3288:
3272:
3265:
3256:
3252:
3237:
3233:
3224:
3222:
3213:
3211:
3204:
3182:
3178:
3149:
3124:
3117:
3107:
3105:
3097:
3093:
3092:
3088:
3077:
3070:
3060:
3056:
3048:
3034:
3030:
3015:
3011:
2992:
2988:
2975:
2968:
2964:
2937:
2902:
2901:
2889:
2885:
2873:
2869:
2839:
2835:
2828:
2822:
2821:
2791:
2787:
2780:
2773:
2771:
2768:
2767:
2762:
2755:
2748:
2737:
2682:
2644:
2638:
2619:
2618:
2598:
2594:
2593:
2588:
2570:
2566:
2565:
2560:
2553:
2549:
2536:
2515:
2514:
2474:
2452:
2450:
2447:
2446:
2411:
2402:
2383:
2361:
2360:
2341:
2337:
2336:
2331:
2327:
2315:
2311:
2304:
2283:
2282:
2264:
2260:
2248:
2244:
2231:
2227:
2225:
2218:
2196:
2194:
2191:
2190:
2172:
2153:
2152:
2132:
2128:
2127:
2122:
2104:
2100:
2099:
2094:
2086:
2085:
2081:
2068:
2047:
2046:
2026:
2022:
2021:
2016:
1998:
1994:
1993:
1988:
1981:
1977:
1964:
1942:
1940:
1937:
1936:
1930:
1908:
1907:
1880:
1876:
1861:
1857:
1850:
1829:
1828:
1793:
1767:
1765:
1761:
1746:
1742:
1732:
1725:
1703:
1701:
1698:
1697:
1692:
1660:
1641:
1640:
1612:
1604:
1595:
1594:
1566:
1558:
1548:
1546:
1543:
1542:
1532:
1494:
1481:
1472:
1459:
1457:
1453:
1437:
1426:
1416:
1399:
1397:
1394:
1393:
1381:) in terms of −
1292:
1285:
1278:
1229:
1226:
1225:
1174:
1157:
1155:
1151:
1135:
1112:
1099:
1098:
1094:
1087:
1083:
1049:
1046:
1045:
1026:
1023:
1022:
1006:
1003:
1002:
982:
979:
978:
953:
950:
949:
937:
932:
888:
835:Other countries
814:Redfearn series
768:
765:
760:
759:
757:
679:), major axis (
647:
582:
507:Gauss conformal
488:
170:
146:
71:
28:
23:
22:
15:
12:
11:
5:
4797:
4787:
4786:
4781:
4776:
4759:
4758:
4755:
4754:
4752:
4751:
4746:
4741:
4736:
4731:
4725:
4722:
4721:
4711:
4710:
4707:
4706:
4703:
4702:
4700:
4699:
4694:
4689:
4684:
4679:
4674:
4669:
4663:
4661:
4655:
4654:
4651:
4650:
4648:
4647:
4641:
4638:
4637:
4635:
4634:
4629:
4624:
4618:
4616:
4607:
4601:
4600:
4598:
4597:
4592:
4586:
4584:
4578:
4577:
4575:
4574:
4569:
4564:
4559:
4554:
4549:
4544:
4542:Kavrayskiy VII
4539:
4533:
4531:
4521:
4520:
4508:
4507:
4504:
4503:
4500:
4499:
4497:
4496:
4491:
4486:
4480:
4478:
4472:Retroazimuthal
4468:
4467:
4465:
4464:
4459:
4453:
4451:
4445:
4444:
4442:
4441:
4435:
4433:
4427:
4426:
4424:
4423:
4418:
4413:
4408:
4403:
4397:
4395:
4391:Equidistant in
4387:
4386:
4383:
4382:
4380:
4379:
4374:
4369:
4364:
4359:
4354:
4349:
4344:
4339:
4334:
4329:
4323:
4320:
4319:
4317:
4316:
4311:
4305:
4303:
4297:
4296:
4294:
4293:
4288:
4283:
4278:
4273:
4268:
4263:
4257:
4255:
4249:
4248:
4246:
4245:
4240:
4234:
4232:
4226:
4225:
4223:
4222:
4217:
4211:
4209:
4200:
4194:
4193:
4191:
4190:
4185:
4180:
4175:
4170:
4165:
4160:
4155:
4149:
4147:
4137:
4136:
4124:
4123:
4120:
4119:
4116:
4115:
4113:
4112:
4107:
4102:
4097:
4091:
4089:
4083:
4082:
4079:
4078:
4076:
4075:
4070:
4064:
4061:
4060:
4058:
4057:
4052:
4047:
4041:
4039:
4030:
4020:
4019:
4017:
4016:
4011:
4010:
4009:
4004:
3994:
3989:
3983:
3981:
3975:
3974:
3972:
3971:
3966:
3961:
3955:
3953:
3947:
3946:
3943:
3942:
3940:
3939:
3934:
3929:
3927:Kavrayskiy VII
3923:
3920:
3919:
3917:
3916:
3911:
3906:
3901:
3896:
3891:
3886:
3881:
3876:
3870:
3868:
3861:
3855:
3854:
3851:
3850:
3848:
3847:
3842:
3837:
3832:
3827:
3822:
3817:
3812:
3806:
3803:
3802:
3800:
3799:
3794:
3789:
3784:
3779:
3774:
3769:
3763:
3761:
3755:
3754:
3752:
3751:
3746:
3741:
3735:
3733:
3723:
3713:
3712:
3700:
3699:
3697:
3696:
3691:
3686:
3680:
3677:
3676:
3673:Map projection
3669:
3668:
3661:
3654:
3646:
3640:
3639:
3631:
3630:External links
3628:
3626:
3625:
3617:
3599:
3580:
3562:
3537:
3522:
3488:
3486:, pp 1–13
3467:
3454:
3451:on 2018-05-08.
3429:
3406:
3383:
3373:. January 1995
3351:
3348:on 2016-08-09.
3322:
3308:
3286:
3263:
3250:
3231:
3202:
3176:
3147:
3115:
3086:
3068:
3054:
3046:
3028:
3025:on 2016-03-04.
3009:
2986:
2965:
2963:
2960:
2959:
2958:
2953:
2948:
2943:
2936:
2933:
2916:
2915:
2900:
2897:
2892:
2888:
2884:
2881:
2876:
2872:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2842:
2838:
2834:
2831:
2829:
2827:
2824:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2794:
2790:
2786:
2783:
2781:
2779:
2776:
2775:
2760:
2753:
2746:
2740:false northing
2735:
2681:
2678:
2677:
2676:
2670:
2664:
2654:
2637:
2634:
2633:
2632:
2617:
2613:
2606:
2601:
2597:
2592:
2587:
2584:
2578:
2573:
2569:
2564:
2559:
2556:
2552:
2548:
2545:
2542:
2539:
2537:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2475:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2454:
2410:
2407:
2400:
2381:
2375:
2374:
2359:
2355:
2349:
2344:
2340:
2335:
2330:
2326:
2323:
2318:
2314:
2310:
2307:
2305:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2284:
2281:
2275:
2272:
2267:
2263:
2259:
2256:
2251:
2247:
2243:
2240:
2234:
2230:
2224:
2221:
2219:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2198:
2171:
2168:
2167:
2166:
2151:
2147:
2140:
2135:
2131:
2126:
2121:
2118:
2112:
2107:
2103:
2098:
2084:
2080:
2077:
2074:
2071:
2069:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2048:
2045:
2041:
2034:
2029:
2025:
2020:
2015:
2012:
2006:
2001:
1997:
1992:
1987:
1984:
1980:
1976:
1973:
1970:
1967:
1965:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1944:
1929:
1926:
1922:
1921:
1906:
1902:
1898:
1895:
1892:
1889:
1886:
1883:
1879:
1875:
1872:
1869:
1864:
1860:
1856:
1853:
1851:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1830:
1827:
1823:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1764:
1760:
1757:
1754:
1749:
1745:
1739:
1736:
1731:
1728:
1726:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1705:
1690:
1682: = −
1659:
1656:
1655:
1654:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1613:
1610:
1607:
1603:
1600:
1597:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1567:
1564:
1561:
1557:
1554:
1551:
1550:
1531:
1528:
1523:
1522:
1511:
1507:
1500:
1497:
1493:
1490:
1487:
1484:
1478:
1475:
1471:
1468:
1465:
1462:
1456:
1452:
1449:
1444:
1441:
1436:
1432:
1429:
1422:
1419:
1415:
1412:
1409:
1405:
1402:
1291:
1288:
1283:
1276:
1270:
1269:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1211:
1210:
1199:
1195:
1189:
1186:
1183:
1180:
1177:
1172:
1169:
1166:
1163:
1160:
1154:
1150:
1147:
1142:
1139:
1134:
1130:
1125:
1119:
1116:
1111:
1106:
1103:
1097:
1093:
1090:
1086:
1082:
1079:
1076:
1073:
1070:
1066:
1062:
1059:
1056:
1053:
1030:
1010:
986:
966:
963:
960:
957:
936:
933:
931:
928:
887:
884:
875:
874:
871:
868:
865:
850:
849:
838:
832:
817:
794:
793:
716:
675:), longitude (
646:
643:
616:
615:
611:
603:
600:
597:
593:
590:
587:
581:
578:
544:
543:
531:
513:in Europe; or
487:
484:
481:
480:
478:
476:
474:
468:
464:
463:
460:
457:
455:
448:
444:
443:
440:
437:
435:
432:
428:
427:
423:
420:
418:
414:
410:
409:
402:
399:
397:
386:
382:
381:
378:
375:
373:
370:
366:
365:
358:
355:
353:
346:
342:
341:
338:
335:
333:
330:
326:
325:
318:
315:
313:
306:
302:
301:
298:
295:
293:
290:
286:
285:
278:
275:
273:
262:
258:
257:
246:
243:
241:
230:
226:
225:
218: = ±
207:
205:
203:
189:
186:
185:
182:
180:
178:
175:
169:
166:
145:
142:
137:
136:
133:
124:, so that the
118:
111:
108:
70:
67:
63:geodetic datum
43:map projection
26:
9:
6:
4:
3:
2:
4796:
4785:
4782:
4780:
4777:
4775:
4772:
4771:
4769:
4750:
4747:
4745:
4742:
4740:
4737:
4735:
4732:
4730:
4727:
4726:
4723:
4716:
4712:
4698:
4695:
4693:
4690:
4688:
4685:
4683:
4680:
4678:
4675:
4673:
4670:
4668:
4665:
4664:
4662:
4660:
4656:
4646:
4643:
4642:
4639:
4633:
4632:Stereographic
4630:
4628:
4625:
4623:
4620:
4619:
4617:
4615:
4611:
4608:
4606:
4602:
4596:
4593:
4591:
4588:
4587:
4585:
4583:
4579:
4573:
4572:Winkel tripel
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4552:Natural Earth
4550:
4548:
4545:
4543:
4540:
4538:
4535:
4534:
4532:
4530:
4526:
4522:
4518:
4513:
4509:
4495:
4492:
4490:
4487:
4485:
4482:
4481:
4479:
4473:
4469:
4463:
4460:
4458:
4455:
4454:
4452:
4450:
4446:
4440:
4437:
4436:
4434:
4432:
4428:
4422:
4419:
4417:
4414:
4412:
4409:
4407:
4404:
4402:
4399:
4398:
4396:
4394:
4388:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4332:Briesemeister
4330:
4328:
4325:
4324:
4321:
4315:
4312:
4310:
4307:
4306:
4304:
4302:
4298:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4258:
4256:
4254:
4250:
4244:
4241:
4239:
4236:
4235:
4233:
4231:
4227:
4221:
4218:
4216:
4213:
4212:
4210:
4208:
4204:
4201:
4199:
4195:
4189:
4186:
4184:
4183:Stereographic
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4150:
4148:
4146:
4142:
4138:
4134:
4129:
4125:
4111:
4110:Winkel tripel
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4092:
4090:
4088:
4084:
4074:
4071:
4069:
4066:
4065:
4062:
4056:
4055:Stereographic
4053:
4051:
4048:
4046:
4043:
4042:
4040:
4038:
4034:
4031:
4029:
4021:
4015:
4012:
4008:
4005:
4003:
4000:
3999:
3998:
3995:
3993:
3990:
3988:
3985:
3984:
3982:
3980:
3979:Pseudoconical
3976:
3970:
3967:
3965:
3962:
3960:
3957:
3956:
3954:
3952:
3948:
3938:
3935:
3933:
3930:
3928:
3925:
3924:
3921:
3915:
3912:
3910:
3907:
3905:
3902:
3900:
3897:
3895:
3892:
3890:
3887:
3885:
3882:
3880:
3877:
3875:
3872:
3871:
3869:
3865:
3862:
3860:
3856:
3846:
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3807:
3804:
3798:
3795:
3793:
3790:
3788:
3785:
3783:
3780:
3778:
3775:
3773:
3770:
3768:
3765:
3764:
3762:
3760:
3756:
3750:
3747:
3745:
3742:
3740:
3737:
3736:
3734:
3731:
3727:
3724:
3722:
3718:
3714:
3710:
3705:
3701:
3695:
3692:
3690:
3687:
3685:
3682:
3681:
3678:
3674:
3667:
3662:
3660:
3655:
3653:
3648:
3647:
3644:
3638:
3634:
3633:
3620:
3614:
3610:
3603:
3596:
3591:
3589:
3587:
3585:
3576:
3572:
3566:
3551:
3547:
3541:
3533:
3526:
3518:
3514:
3510:
3506:
3502:
3495:
3493:
3485:
3481:
3477:
3471:
3464:
3458:
3447:
3439:
3433:
3416:
3410:
3402:
3394:
3387:
3372:
3369:(in French).
3368:
3361:
3355:
3344:
3340:
3333:
3326:
3318:
3312:
3304:
3297:
3290:
3283:
3277:
3270:
3268:
3260:
3254:
3247:
3246:
3241:
3235:
3221:on 2012-02-11
3220:
3216:
3209:
3207:
3199:
3195:
3194:
3189:
3185:
3180:
3173:
3169:
3165:
3161:
3160:
3157:
3150:
3148:0-919870-16-3
3144:
3140:
3135:
3134:
3128:
3122:
3120:
3103:
3096:
3090:
3083:
3082:
3075:
3073:
3065:
3058:
3049:
3043:
3039:
3032:
3024:
3020:
3013:
3006:
3002:
2998:
2997:
2990:
2983:
2979:
2973:
2971:
2966:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2938:
2932:
2930:
2925:
2921:
2898:
2890:
2886:
2879:
2874:
2870:
2866:
2860:
2857:
2854:
2848:
2845:
2840:
2836:
2832:
2830:
2825:
2818:
2812:
2809:
2806:
2800:
2797:
2792:
2788:
2784:
2782:
2777:
2766:
2765:
2764:
2759:
2752:
2745:
2741:
2734:
2730:
2729:false easting
2726:
2722:
2718:
2714:
2709:
2707:
2703:
2699:
2695:
2691:
2687:
2675:
2671:
2669:
2665:
2663:
2659:
2655:
2653:
2649:
2648:
2647:
2643:
2615:
2611:
2604:
2599:
2595:
2590:
2585:
2582:
2576:
2571:
2567:
2562:
2557:
2554:
2550:
2546:
2543:
2540:
2538:
2530:
2527:
2524:
2518:
2511:
2505:
2502:
2499:
2496:
2493:
2490:
2484:
2481:
2478:
2476:
2468:
2465:
2462:
2456:
2445:
2444:
2443:
2440:
2436:
2432:
2428:
2424:
2415:
2406:
2404:
2399:
2395:
2391:
2387:
2380:
2357:
2353:
2347:
2342:
2338:
2333:
2328:
2324:
2321:
2316:
2312:
2308:
2306:
2298:
2295:
2292:
2286:
2279:
2273:
2270:
2265:
2261:
2257:
2254:
2249:
2245:
2241:
2238:
2232:
2228:
2222:
2220:
2212:
2209:
2206:
2200:
2189:
2188:
2187:
2185:
2181:
2177:
2149:
2145:
2138:
2133:
2129:
2124:
2119:
2116:
2110:
2105:
2101:
2096:
2082:
2078:
2075:
2072:
2070:
2062:
2059:
2056:
2050:
2043:
2039:
2032:
2027:
2023:
2018:
2013:
2010:
2004:
1999:
1995:
1990:
1985:
1982:
1978:
1974:
1971:
1968:
1966:
1958:
1955:
1952:
1946:
1935:
1934:
1933:
1925:
1904:
1900:
1896:
1893:
1890:
1887:
1884:
1881:
1877:
1873:
1870:
1867:
1862:
1858:
1854:
1852:
1844:
1841:
1838:
1832:
1825:
1821:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1762:
1758:
1755:
1752:
1747:
1743:
1737:
1734:
1729:
1727:
1719:
1716:
1713:
1707:
1696:
1695:
1694:
1689:
1685:
1681:
1677:
1674: =
1673:
1669:
1665:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1614:
1608:
1605:
1601:
1598:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1568:
1562:
1559:
1555:
1552:
1541:
1540:
1539:
1537:
1527:
1509:
1505:
1498:
1495:
1491:
1488:
1485:
1482:
1476:
1473:
1469:
1466:
1463:
1460:
1454:
1450:
1447:
1442:
1439:
1434:
1430:
1427:
1420:
1417:
1413:
1410:
1407:
1403:
1400:
1392:
1391:
1390:
1388:
1384:
1380:
1376:
1371:
1369:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1316:
1312:
1310:
1306:
1296:
1287:
1282:
1275:
1255:
1252:
1249:
1246:
1243:
1237:
1231:
1224:
1223:
1222:
1220:
1216:
1197:
1193:
1187:
1184:
1181:
1178:
1175:
1170:
1167:
1164:
1161:
1158:
1152:
1148:
1145:
1140:
1137:
1132:
1128:
1123:
1117:
1114:
1109:
1104:
1101:
1095:
1091:
1088:
1084:
1080:
1077:
1074:
1071:
1068:
1064:
1060:
1057:
1054:
1051:
1044:
1043:
1042:
1028:
1008:
1000:
984:
964:
961:
958:
955:
941:
927:
925:
920:
918:
914:
908:
906:
901:
897:
892:
883:
880:
872:
869:
866:
863:
862:
861:
859:
855:
847:
843:
839:
836:
833:
830:
825:
821:
818:
815:
810:
806:
803:
802:
801:
799:
791:
787:
783:
779:
771:
763:
755:
751:
747:
743:
739:
735:
731:
727:
723:
722:
717:
714:
710:
706:
702:
698:
694:
690:
686:
682:
678:
674:
670:
666:
662:
658:
657:
654:Krüger–
652:
651:
650:
642:
639:
634:
630:
626:
621:
612:
608:
604:
601:
598:
594:
591:
588:
584:
583:
573:
569:
567:
562:
557:
552:
550:
541:
537:
532:
529:
524:
523:
522:
520:
516:
512:
508:
504:
499:
497:
493:
479:
477:
475:
472:
469:
466:
461:
458:
456:
453:
449:
446:
441:
438:
436:
433:
430:
424:
421:
419:
415:
412:
407:
403:
400:
398:
395:
391:
387:
384:
379:
376:
374:
371:
368:
363:
359:
356:
354:
351:
347:
344:
339:
336:
334:
331:
328:
323:
319:
316:
314:
311:
307:
304:
299:
296:
294:
291:
288:
283:
279:
276:
274:
271:
267:
263:
260:
255:
251:
247:
244:
242:
239:
235:
231:
228:
221:
217:
212:
206:
204:
199:
194:
188:
165:
163:
159:
155:
151:
141:
134:
131:
127:
123:
119:
116:
112:
109:
106:
105:
100:
96:
92:
91:
90:
88:
84:
75:
66:
64:
60:
56:
52:
48:
44:
41:
32:
19:
4627:Orthographic
4187:
4158:Gauss–Krüger
4050:Orthographic
3845:Web Mercator
3743:
3739:Gauss–Krüger
3608:
3602:
3574:
3565:
3554:. Retrieved
3549:
3540:
3525:
3508:
3504:
3483:
3470:
3457:
3446:the original
3432:
3421:. Retrieved
3409:
3400:
3386:
3375:. Retrieved
3366:
3354:
3343:the original
3338:
3325:
3311:
3302:
3289:
3275:
3258:
3253:
3244:
3239:
3234:
3223:. Retrieved
3219:the original
3197:
3192:
3187:
3179:
3158:
3155:
3138:
3132:
3106:. Retrieved
3089:
3079:
3057:
3037:
3031:
3023:the original
3012:
3000:
2994:
2989:
2984:, section 6)
2977:
2923:
2919:
2917:
2757:
2750:
2743:
2739:
2732:
2728:
2724:
2720:
2717:false origin
2716:
2712:
2710:
2705:
2701:
2697:
2693:
2689:
2685:
2683:
2657:
2645:
2438:
2434:
2430:
2426:
2422:
2420:
2405:
2397:
2393:
2389:
2385:
2378:
2376:
2183:
2179:
2173:
1931:
1923:
1687:
1683:
1679:
1675:
1671:
1667:
1663:
1661:
1533:
1524:
1386:
1382:
1378:
1374:
1372:
1367:
1363:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1321:
1308:
1304:
1301:
1280:
1273:
1271:
1218:
1212:
999:Earth radius
946:
923:
921:
916:
912:
909:
904:
899:
895:
893:
889:
878:
876:
857:
853:
851:
845:
834:
823:
819:
808:
804:
797:
795:
785:
781:
777:
769:
761:
753:
749:
745:
741:
737:
733:
729:
725:
720:
718:
712:
708:
704:
700:
696:
692:
688:
684:
680:
676:
672:
668:
664:
660:
655:
653:
648:
637:
632:
628:
625:eccentricity
617:
606:
565:
556:power series
553:
545:
536:Gauss–Krüger
535:
528:Gauss–Krüger
527:
519:Gauss–Krüger
518:
514:
511:Gauss–Krüger
510:
506:
502:
500:
489:
451:
405:
393:
361:
349:
321:
309:
281:
269:
265:
253:
249:
237:
233:
219:
215:
197:
161:
147:
138:
129:
102:
86:
80:
50:
46:
39:
37:
4605:Perspective
4393:some aspect
4377:Strebe 1995
4352:Equal Earth
4271:Gall–Peters
4253:Cylindrical
4068:Equidistant
3964:Equidistant
3894:Equal Earth
3777:Gall–Peters
3721:Cylindrical
3511:: 475–485.
3282:USGS pages.
2951:Scale (map)
2692:axis. Both
2409:Convergence
2176:point scale
2170:Point scale
1215:point scale
852:The Krüger–
796:The Krüger–
505:in the US;
471:Rhumb lines
126:point scale
115:ellipsoidal
99:cylindrical
95:projections
4768:Categories
4667:AuthaGraph
4659:Polyhedral
4529:Compromise
4457:Loximuthal
4449:Loxodromic
4411:Sinusoidal
4261:Balthasart
4238:Sinusoidal
4215:Sinusoidal
4198:Equal-area
3909:Sinusoidal
3867:Equal-area
3767:Balthasart
3759:Equal-area
3732:-conformal
3709:By surface
3556:2024-07-27
3423:2024-07-27
3377:2024-07-27
3225:2012-01-11
3184:Lee, L. P.
3127:Lee, L. P.
3108:2009-08-27
2962:References
2640:See also:
820:Thomas–UTM
596:bisected.)
4739:Longitude
4567:Wagner VI
4416:Two-point
4347:Eckert VI
4342:Eckert IV
4337:Eckert II
4314:Mollweide
4309:Collignon
4276:Hobo–Dyer
4230:Bottomley
4145:Conformal
4133:By metric
4024:Azimuthal
3997:Polyconic
3992:Bottomley
3932:Wagner VI
3904:Mollweide
3889:Eckert VI
3884:Eckert IV
3879:Eckert II
3874:Collignon
3782:Hobo–Dyer
2887:φ
2867:−
2861:φ
2855:λ
2813:φ
2807:λ
2725:northings
2586:
2558:
2547:
2519:γ
2506:φ
2503:
2497:λ
2494:
2485:
2469:φ
2463:λ
2457:γ
2325:
2274:φ
2271:
2258:λ
2255:
2242:−
2213:φ
2207:λ
2120:
2079:
2051:φ
2014:
1986:
1975:
1947:λ
1897:φ
1894:
1888:λ
1885:
1874:
1845:φ
1839:λ
1816:φ
1813:
1807:λ
1804:
1798:−
1790:φ
1787:
1781:λ
1778:
1759:
1720:φ
1714:λ
1635:φ
1632:
1626:λ
1623:
1606:λ
1602:
1589:φ
1586:
1580:λ
1577:
1560:φ
1556:
1496:φ
1492:
1486:−
1474:φ
1470:
1451:
1418:λ
1411:−
1253:φ
1250:
1238:φ
1188:φ
1185:
1179:−
1171:φ
1168:
1149:
1115:φ
1102:π
1092:
1081:
1061:λ
1029:ϕ
965:λ
627:and both
610:meridian.
498:in 1912.
122:conformal
117:versions.
4779:Geocodes
4734:Latitude
4719:See also
4682:Dymaxion
4622:Gnomonic
4557:Robinson
4462:Mercator
4439:Gnomonic
4431:Gnomonic
4266:Behrmann
4173:Mercator
4045:Gnomonic
4027:(planar)
4002:American
3772:Behrmann
3730:Mercator
3129:(1976).
2935:See also
2721:eastings
1609:′
1563:′
1499:′
1477:′
1431:′
1421:′
1404:′
580:Features
4595:HEALPix
4494:Littrow
4105:Wiechel
4007:Chinese
3951:Conical
3815:Central
3810:Cassini
3787:Lambert
3684:History
3319:. 2024.
3172:107–114
997:is the
977:(where
911:Krüger–
867:Finland
788:. (See
774:
758:
719:Krüger–
4614:Planar
4582:Hybrid
4489:Hammer
4421:Werner
4362:Hammer
4327:Albers
4243:Werner
4220:Werner
4100:Hammer
4095:Aitoff
4014:Werner
3959:Albers
3835:Miller
3694:Portal
3615:
3552:. 2009
3170:, and
3168:92–101
3162:. pp.
3145:
3044:
2544:arctan
2482:arctan
2076:arcsin
1972:arctan
1871:arctan
1307:- and
1001:) and
870:Sweden
864:France
822:: The
586:globe.
150:sphere
87:Normal
4484:Craig
4401:Conic
4207:Bonne
3987:Bonne
3449:(PDF)
3442:(PDF)
3418:(PDF)
3397:(PDF)
3363:(PDF)
3346:(PDF)
3335:(PDF)
3299:(PDF)
3098:(PDF)
2980:. In
873:Japan
607:small
566:whole
130:local
93:Both
4687:ISEA
3689:List
3613:ISBN
3164:1–14
3143:ISBN
3042:ISBN
2922:and
2723:and
2711:The
2704:and
2696:and
2555:tanh
2427:from
2322:cosh
2089:sech
1983:sinh
1678:and
1385:and
784:and
752:and
744:and
728:and
711:and
691:and
663:and
631:and
388:The
97:are
38:The
3513:doi
2583:tan
2500:sin
2491:tan
2262:cos
2246:sin
2117:sin
2011:sec
1891:tan
1882:sec
1810:cos
1801:sin
1784:cos
1775:sin
1629:tan
1620:sec
1599:tan
1583:cos
1574:sin
1553:sin
1489:sin
1467:sin
1247:sec
1182:sin
1165:sin
1089:tan
829:UTM
614:up.
509:or
51:TMP
4770::
3583:^
3573:.
3548:.
3509:85
3507:.
3503:.
3491:^
3484:59
3482:,
3399:.
3365:.
3337:.
3301:.
3266:^
3205:^
3166:,
3159:13
3137:.
3118:^
3100:.
3071:^
3001:54
2969:^
2931:.
2742:,
2731:,
2431:to
2184:φ′
1756:ln
1684:x′
1676:y′
1448:ln
1387:φ′
1383:λ′
1379:y′
1375:x′
1360:y′
1356:x′
1344:λ′
1340:φ′
1336:λ′
1332:φ′
1217:,
1146:ln
1078:ln
792:.)
780:,
740:,
707:,
703:,
467:•
459:•
447:•
439:•
431:•
422:•
413:•
401:•
385:•
377:•
369:•
357:•
345:•
337:•
329:•
317:•
305:•
297:•
289:•
277:•
272:.
261:•
245:•
229:•
49:,
47:TM
3665:e
3658:t
3651:v
3623:.
3621:.
3577:.
3559:.
3519:.
3515::
3426:.
3380:.
3305:.
3248:.
3240:9
3228:.
3198:8
3188:8
3174:.
3151:.
3111:.
3050:.
3007:.
2999:(
2924:y
2920:x
2899:.
2896:)
2891:0
2883:(
2880:m
2875:0
2871:k
2864:)
2858:,
2852:(
2849:y
2846:+
2841:0
2837:N
2833:=
2826:N
2819:,
2816:)
2810:,
2804:(
2801:x
2798:+
2793:0
2789:E
2785:=
2778:E
2761:0
2758:k
2754:0
2751:φ
2747:0
2744:N
2736:0
2733:E
2706:ϕ
2702:λ
2698:y
2694:x
2690:y
2686:x
2658:n
2616:.
2612:)
2605:a
2600:0
2596:k
2591:y
2577:a
2572:0
2568:k
2563:x
2551:(
2541:=
2534:)
2531:y
2528:,
2525:x
2522:(
2512:,
2509:)
2488:(
2479:=
2472:)
2466:,
2460:(
2439:γ
2435:x
2423:γ
2401:0
2398:k
2394:x
2390:x
2386:k
2382:0
2379:k
2358:.
2354:)
2348:a
2343:0
2339:k
2334:x
2329:(
2317:0
2313:k
2309:=
2302:)
2299:y
2296:,
2293:x
2290:(
2287:k
2280:,
2266:2
2250:2
2239:1
2233:0
2229:k
2223:=
2216:)
2210:,
2204:(
2201:k
2180:k
2150:.
2146:]
2139:a
2134:0
2130:k
2125:y
2111:a
2106:0
2102:k
2097:x
2083:[
2073:=
2066:)
2063:y
2060:,
2057:x
2054:(
2044:,
2040:]
2033:a
2028:0
2024:k
2019:y
2005:a
2000:0
1996:k
1991:x
1979:[
1969:=
1962:)
1959:y
1956:,
1953:x
1950:(
1905:,
1901:]
1878:[
1868:a
1863:0
1859:k
1855:=
1848:)
1842:,
1836:(
1833:y
1826:,
1822:]
1795:1
1772:+
1769:1
1763:[
1753:a
1748:0
1744:k
1738:2
1735:1
1730:=
1723:)
1717:,
1711:(
1708:x
1691:0
1688:k
1680:y
1672:x
1668:y
1666:,
1664:x
1638:.
1617:=
1592:,
1571:=
1510:.
1506:]
1483:1
1464:+
1461:1
1455:[
1443:2
1440:a
1435:=
1428:y
1414:a
1408:=
1401:x
1377:,
1368:y
1366:,
1364:x
1358:,
1352:λ
1350:,
1348:φ
1342:,
1328:λ
1326:,
1324:φ
1309:y
1305:x
1284:0
1281:k
1277:0
1274:k
1256:.
1244:=
1241:)
1235:(
1232:k
1219:k
1198:.
1194:]
1176:1
1162:+
1159:1
1153:[
1141:2
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1133:=
1129:]
1124:)
1118:2
1110:+
1105:4
1096:(
1085:[
1075:a
1072:=
1069:y
1065:,
1058:a
1055:=
1052:x
1009:y
985:a
962:a
959:=
956:x
924:n
917:λ
913:n
905:n
900:n
896:n
879:n
858:n
854:n
846:n
824:λ
809:λ
798:λ
786:e
782:a
778:y
770:a
766:/
762:x
754:λ
750:φ
746:e
742:a
738:φ
734:λ
730:y
726:x
721:λ
715:.
713:e
709:a
705:y
701:x
697:n
693:λ
689:φ
685:e
681:a
677:λ
673:φ
669:n
665:y
661:x
656:n
638:x
633:y
629:x
452:x
406:x
394:y
362:x
350:y
322:x
310:y
282:y
270:y
266:y
254:y
250:x
238:x
234:x
220:π
216:x
198:y
107:.
45:(
20:)
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