Transverse Mercator with an accuracy of a few nanometers

Implementations of two algorithms for the transverse Mercator projection are described; these achieve accuracies close to machine precision. One is based on the exact equations of Thompson and Lee and the other uses an extension of Krueger's series for the projection to higher order. The exact method provides an accuracy of 9 nm over the entire ellipsoid, while the errors in the series method are less than 5 nm within 3900 km of the central meridian. In each case, the meridian convergence and scale are also computed with similar accuracy. The speed of the series method is competitive with other less accurate algorithms and the exact method is about 5 times slower.Comment: LaTeX, 10 pages, 3 figures. Includes some revisions. Supplementary material is available at http://geographiclib.sourceforge.net/tm.htm

Algorithms for geodesics

Algorithms for the computation of geodesics on an ellipsoid of revolution are given. These provide accurate, robust, and fast solutions to the direct and inverse geodesic problems and they allow differential and integral properties of geodesics to be computed.Comment: LaTex, 12 pages, 8 figures. Version 2 corrects some errors and adds numerical examples. Supplementary material is available at http://geographiclib.sourceforge.net/geod.htm

Mapping the Deep Blue Oceans

The ocean terrain spanning the globe is vast and complex—far from an immense flat plain of mud. To map these depths accurately and wisely, we must understand how cartographic abstraction and generalization work both in analog cartography and digital GIS. This chapter explores abstraction practices such as selection and exaggeration with respect to mapping the oceans, showing significant continuity in such practices across cartography and contemporary GIS. The role of measurement and abstraction—as well as of political and economic power, and sexual and personal bias—in these sciences is illustrated by the biographies of Marie Tharp and Bruce Heezen, whose mapping of the Mid-Atlantic Ridge precipitated a paradigm shift in geology

Simple and highly accurate formulas for the computation of Transverse

A conformal approximation to the Transverse Mercator (TM) map projection, global in longitude lambda and isometric latitude q, is constructed. New formulas for the point scale factor and grid convergence are also shown. Assuming that the true values of the TM coordinates are given by conveniently truncated Gauss-Kruger series expansions, we use the maximum norm of the absolute error to measure globally the accuracy of the approximation. For a Universal Transverse Mercator (UTM) zone the accuracy equals 0.21 mm, whereas for the region of the ellipsoid bounded by the meridians +/- 20A degrees the accuracy is equal to 0.3 mm. Our approach is based on a four-term perturbation series approximation to the radius r(q) of the parallel q, with a maximum absolute deviation of 0.43 mm. The small parameter of the power series expansion is the square of the eccentricity of the ellipsoid. This closed approximation to r(q) is obtained by solving a regularly perturbed Cauchy problem with the Poincar, method of the small parameter

The accuracy of human population maps for public health application

ObjectivesHuman population totals are used for generating burden of disease estimates at global, continental and national scales to help guide priority setting in international health financing. These exercises should be aware of the accuracy of the demographic information used.MethodsThe analysis presented in this paper tests the accuracy of five large-area, public-domain human population distribution data maps against high spatial resolution population census data enumerated in Kenya in 1999. We illustrate the epidemiological significance, by assessing the impact of using these different human population surfaces in determining populations at risk of various levels of climate suitability for malaria transmission. We also describe how areal weighting, pycnophylactic interpolation and accessibility potential interpolation techniques can be used to generate novel human population distribution surfaces from local census information and evaluate to what accuracy this can be achieved.ResultsWe demonstrate which human population distribution surface performed best and which population interpolation techniques generated the most accurate bespoke distributions. Despite various levels of modelling complexity, the accuracy achieved by the different surfaces was primarily determined by the spatial resolution of the input population data. The simplest technique of areal weighting performed best. Conclusions Differences in estimates of populations at risk of malaria in Kenya of over 1 million persons can be generated by the choice of surface, highlighting the importance of these considerations in deriving per capita health metrics in public health. Despite focussing on Kenya the results of these analyses have general application and are discussed in this wider context