The Normal and Transverse Mercator Projections on the Sphere and the Ellipsoid with Full Derivations of All Formulae

TMS transformations from NMS by rotation of the graticule. Four global TMS projections. Meridian distance, footpoint and footpoint latitude. Scale factors. Relation between azimuth and grid bearing. Grid convergence. Conformality, the Cauchy–Riemann conditions and isotropy of scale. Series expansions for the TMS transformation formulae. Secant TMS. 3.1 The derivation of the TMS formulae In Chapter 2 we constructed the normal Mercator projection (NMS). The strength of NMS is its conformality, preserving local angles exactly and preserving shapes in “small” regions (orthomorphism). Furthermore, meridians project to grid lines and conformality implies that rhumb lines project to constant grid bearings, thereby guaranteeing the continuing usefulness of NMS as an aid to navigation. As a topographic map of the globe NMS has shortcomings in that the projection greatly distorts shapes as one approaches the poles—because of the rapid change of scale with latitude. However, the (unmodified) NMS is exactly to scale on the equator and is very accurate within a narrow strip of about three degrees centred on the equator (extending to five degrees for the secant NMS). It is this accuracy near the equator that we wish to exploit by constructing a projection which takes a complete meridian great circle as a ‘kind of equator’ and uses ‘NMS on its side’ to achieve a conformal and accurate projection within a narrow band adjoining the chosen meridian. This is the transverse Mercator projection (TMS) first demonstrated by Lambert (1772). The crucial point is that if we have a projection which is very accurate close to one meridian then a set of such projections will provide accurate coverage of the whole sphere. The secant versions of the transverse Mercator projection on the ellipsoid (TME), are of great importance. One such projection may be used for map projections of countries which have a predominantly north-south orientation, for example the Ordnance Survey of Great Britain; see also OSGB (1999). The Universal Transverse Mercator set of projections cover the entire sphere (between the latitudes of 84◦N and 80◦S) using 60 zones of width 6◦ in longitude centred on meridians at 3◦, 9◦, 15◦, . . . (UTM, 1989). Chapter 3. Transverse Mercator on the sphere: TMS 50 In unmodified NMS the equator has unit scale because we project onto a cylinder tangential to the sphere at the equator, (Figure 2.3) Therefore, for TMS we seek a projection onto a cylinder which is tangential to the sphere on some chosen meridian or strictly, a pair of meridians such as the great circle formed by meridians at Greenwich and 180◦E: the geometry is shown in Figure 3.1a. This will guarantee that the scale is unity on the meridian: the problem is to construct the functions x(λ ,φ) and y(λ ,φ) such that the projection is also conformal.