Mercator-15Dec2015.Pdf

THE MERCATOR PROJECTIONS THE NORMAL AND TRANSVERSE MERCATOR PROJECTIONS ON THE SPHERE AND THE ELLIPSOID WITH FULL DERIVATIONS OF ALL FORMULAE PETER OSBORNE EDINBURGH 2013 This article describes the mathematics of the normal and transverse Mercator projections on the sphere and the ellipsoid with full derivations of all formulae. The Transverse Mercator projection is the basis of many maps covering individual countries, such as Australia and Great Britain, as well as the set of UTM projections covering the whole world (other than the polar regions). Such maps are invariably covered by a set of grid lines. It is important to appreciate the following two facts about the Transverse Mercator projection and the grids covering it: 1. Only one grid line runs true north–south. Thus in Britain only the grid line coincident with the central meridian at 2◦W is true: all other meridians deviate from grid lines. The UTM series is a set of 60 distinct Transverse Mercator projections each covering a width of 6◦in latitude: the grid lines run true north–south only on the central meridians at 3◦E, 9◦E, 15◦E, ... 2. The scale on the maps derived from Transverse Mercator projections is not uniform: it is a function of position. For ex- ample the Landranger maps of the Ordnance Survey of Great Britain have a nominal scale of 1:50000: this value is only ex- act on two slightly curved lines almost parallel to the central meridian at 2◦W and distant approximately 180km east and west of it. The scale on the central meridian is constant but it is slightly less than the nominal value. The above facts are unknown to the majority of map users. They are the subject of this article together with the presentation of formulae relating latitude and longitude to grid coordinates. Preface For many years I had been intrigued by the the statement on the (British) Ordnance Survey maps pointing out that the grid lines are not exactly aligned with meridians and parallels: four precise figures give the magnitude of the deviation at each corner of the map sheets. My first retirement project has been to find out exactly how these figures are calculated and this has led to an exploration of all aspects of the Transverse Mercator projection on an ellipsoid of revolution (TME). This projection is also used for the Universal Transverse Mercator series of maps covering the whole of the Earth, except for the polar regions. The formulae for TME are given in many books and web pages but the full derivations are only to be found in original publications which are not readily accessible: therefore I decided to write a short article explaining the derivation of the formulae. Pedagogical reasons soon made it apparent that it would be necessary to start with the normal and transverse Mercator projection on the sphere (NMS and TMS) before going on to discuss the normal and transverse Mercator projection on the ellipsoid (NME and TME). As a result, the length of this document has doubled and redoubled but I have resisted the temptation to cut out details which would be straightforward for a professional but daunting for a ‘lay- man’. The mathematics involved is not difficult (depending on your point of view) but it does require the rudiments of complex analysis for the crucial steps. On the other hand the algebra gets fairly heavy at times; Redfearn(1948) talks of a “a particularly tough spot of work” and Hotine(1946) talks of reversing series by “brute force and algebra”—so be warned. Repeating this may be seen as a perverse undertaking on my part but this edi- tion includes derivations using the computer algebra program Maxima(2009) . To make this article as self-contained as possible I have added a number of appendices covering the required mathematics. My sources for the TME formulae are to be found in Empire Survey Review dating from the nineteen forties to sixties. The actual papers are fairly terse, as is normal for papers by professionals for their peers, and their perusal will certainly not add to the details presented here. Books on mathematical cartography are also fairly thin on the ground, moreover they usually try to cover all types of projections whereas we are concerned only with Mercator projections. The few books that I found to be of assistance are listed in the Literature (L) or ReferencesR but they are supplemented with research papers and web material. I would like to thank Harry Kogon for reading, commenting on and even checking the mathematics outlined in these pages. Any remaining errors (and typographical slips) must be attributed to myself—when you find them please send an email to the address below. Peter Osborne Edinburgh, 2008, 2013 Source code [email protected] Contents 1 Introduction9 1.1 Geodesy and the Figure of the Earth...................9 1.2 Topographic surveying.......................... 10 1.3 Cartography................................ 12 1.4 The criteria for a faithful map projection................. 12 1.5 The representative fraction (RF) and the scale factor........... 13 1.6 Graticules, grids, azimuths and bearings................. 14 1.7 Historical outline............................. 15 1.8 Chapter outlines.............................. 18 2 Normal Mercator on the sphere: NMS 21 2.1 Coordinates and distance on the sphere.................. 21 2.2 Normal (equatorial) cylindrical projections................ 26 2.3 Four examples of normal cylindrical projections............. 29 2.4 The normal Mercator projection..................... 35 2.5 Rhumb lines and loxodromes....................... 37 2.6 Distances on rhumbs and great circles.................. 42 2.7 The secant normal Mercator projection.................. 45 2.8 Summary of NMS............................. 47 3 Transverse Mercator on the sphere: TMS 49 3.1 The derivation of the TMS formulae................... 49 3.2 Features of the TMS projection...................... 53 3.3 Meridian distance, footpoint and footpoint latitude........... 60 3.4 The scale factor for the TMS projection................. 61 3.5 Azimuths and grid bearings in TMS................... 61 3.6 The grid convergence of the TMS projection............... 62 3.7 Conformality of general projections................... 64 3.8 Series expansions for the unmodified TMS................ 65 3.9 Secant TMS................................ 68 4 NMS to TMS by complex variables 71 4.1 Introduction................................ 71 4.2 Closed formulae for the TMS transformation.............. 74 4.3 Transformation to the TMS series.................... 75 4.4 The inverse complex series: an alternative method............ 80 4.5 Scale and convergence in TMS...................... 82 5 The geometry of the ellipsoid 87 5.1 Coordinates on the ellipsoid....................... 87 5.2 The parameters of the ellipsoid...................... 88 5.3 Parameterisation by geodetic latitude................... 88 5.4 Cartesian and geographic coordinates.................. 90 5.5 The reduced or parametric latitude.................... 92 5.6 The curvature of the ellipsoid....................... 93 5.7 Distances on the ellipsoid......................... 96 5.8 The meridian distance on the ellipsoid.................. 98 5.9 Inverse meridian distance......................... 101 5.10 Auxiliary latitudes double projections.................. 102 5.11 The conformal latitude.......................... 104 5.12 The rectifying latitude........................... 106 5.13 The authalic latitude........................... 107 5.14 Ellipsoid: summary............................ 109 6 Normal Mercator on the ellipsoid (NME) 111 6.1 Introduction................................ 111 6.2 The direct transformation for NME.................... 112 6.3 The inverse transformation for NME................... 113 6.4 The scale factor.............................. 115 6.5 Rhumb lines................................ 116 6.6 Modified NME.............................. 116 7 Transverse Mercator on the ellipsoid (TME) 117 7.1 Introduction................................ 117 7.2 Derivation of the Redfearn series..................... 121 7.3 Convergence and scale in TME...................... 126 7.4 Convergence in geographical coordinates................ 128 7.5 Convergence in projection coordinates.................. 129 7.6 Scale factor in geographical coordinates................. 129 7.7 Scale factor in projection coordinates................... 130 7.8 Redfearn’s modified (secant) TME series................. 132 8 Applications of TME 135 8.1 Coordinates, grids and origins...................... 135 8.2 The UTM projection........................... 136 8.3 UTM coordinate systems......................... 137 8.4 The British national grid: NGGB.................... 139 8.5 Scale variation in TME projections.................... 142 8.6 Convergence in the TME projection................... 145 8.7 The accuracy of the TME transformations................ 146 8.8 The truncated TME series......................... 149 8.9 The OSGB series............................. 150 8.10 Concluding remarks............................ 151 A Curvature in 2 and 3 dimensions 153 A.1 Planar curves............................... 153 A.2 Curves in three dimensions........................ 156 A.3 Curvature of surfaces........................... 157 A.4 Meusnier’s theorem............................ 158 A.5 Curvature of normal sections....................... 159 B Lagrange expansions 163 B.1 Introduction................................ 163 B.2 Direct inversion of power series..................... 164 B.3 Lagrange’s

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