Map Projections Map Projections A Reference Manual LEV M. BUGAYEVSKIY JOHN P. SNYDER CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business Published in 1995 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1995 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 1 International Standard Book Number-10: 0-7484-0304-3 (Softcover) International Standard Book Number-13: 978-0-7484-0304-2 (Softcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress Visit the Taylor & Francis Web site at http ://www. taylorandfrancis.com and the CRC Press Web site at http ://www.crcp ress.com Contents Symbols ix Preface xi Introduction xiii 1 General theory of map projections 1 1.1 Coordinate systems used in mathematical cartography 1 1.2 Definition of map projections: equations for meridians and parallels; the map graticule; conditions for transformation 13 1.3 Elements for transforming an infinitestimal spheroidal (or spherical) quadrangle onto a plane 15 1.4 Scale 17 1.5 Conditions for conformal, equal-area, and equidistant transformation of an ellipsoidal (or spherical) surface onto a plane 20 1.6 Distortion on map projections 22 1.7 Transformation of one type of surface onto other types: that of the ellipsoid of revolution onto the surface of a sphere 30 1.8 Classification of map projections 41 2 Map projections with straight parallels 49 2.1 Cylindrical projections 49 2.2 Pseudocylindrical projections 64 3 Map projections with parallels in the shape of concentric circles 89 3.1 Conic projections 89 3.2 Azimuthal projections 101 3.3 Perspective azimuthal projections 109 3.4 Pseudoconic projections 123 3.5 Pseudoazimuthal projections 129 3.6 Retroazimuthal projections 132 4 Map projections with parallels in the shape of non-concentric circles 135 4.1 General formulas for polyconic projections 135 4.2 Polyconic projections in a general sense 135 v vi Map projections: a reference manual 4.3 Polyconic projections in a narrow sense 149 4.4 Characteristics of polyconic projections 153 5 Projections for topographic and named-quadrangle maps; projections used in geodesy 155 5.1 Topographic map projections 155 5.2 Projections used for maps at scales of 1 : 1000 000 and 1 : 2500000 165 5.3 Conformal projections of the ellipsoid used in geodesy 166 6 Map projection research 171 6.1 Direct and inverse problems of mathematical cartography involved in the theory of direct transformation of surfaces onto a plane 171 6.2 Equations for inverse transformation 173 6.3 Map projection research by solving the direct problem of mathematical cartography 174 6.4 Map projection research by solving the inverse problem of mathematical cartography 185 7 Best and ideal map projections; projections satisfying given conditions of representation 193 7.1 General conditions for the best and ideal projections 193 7.2 Chebyshev projections 195 7.3 Conformal projections with adaptable isocols 198 7.4 Conformal projections using elliptic functions 206 7.5 Quasiconformal transformation of flat regions; classes of equal-area projections closest to conformality 208 7.6 Projections with orthogonal map graticules 211 7.7 Euler projections 212 7.8 Two-point azimuthal projection 214 7.9 Two-point equidistant projection 215 7.10 Projections for anamorphous maps 217 7.11 Isometric coordinates and conformal cylindrical projections of the triaxial ellipsoid 219 7.12 Map projections for maps on globes 222 7.13 Mapping geodetic lines, loxodromes, small circles, and trajectory lines of artificial satellites of the Earth 222 8 Numerical methods in mathematical cartography 229 8.1 Interpolation and extrapolation 230 8.2 Numerical differentiation 231 8.3 Numerical integration 232 8.4 Approximation 233 9 Choice and identification of map projections 235 9.1 Theoretical fundamentals of choosing a map projection 235 9.2 Distortion characteristics on projections of the former USSR, continents, oceans, hemispheres, and world maps 237 Contents vii 9.3 Projections for published maps 240 9.4 Approximate hierarchy of requirements for map projections with varying perception and assessment of cartographic information 240 9.5 Distortion requirements on various types of small-scale maps 240 9.6 Identification of a map projection from the shape of its graticule of meridians and parallels 246 10 Problems and directions of automation in obtaining and applying map projections 249 10.1 Calculation of map projections on a computer 249 10.2 Transformation of the map projection of a base map into a given projection 251 10.3 Computerized selection of map projections 259 10.4 Research on map projections in an automated environment to meet given requirements 261 10.5 Identification of map projections in an automated environment 261 10.6 Automated identification and reduction of measurements from maps using their mathematical basis 262 10.7 Automated plotting of elements of the mathematical basis 262 11 Summary of miscellaneous projections, with references 265 References 279 Appendix 1 295 Appendix 2 299 Appendix 3 304 Appendix 4 307 Appendix 5 312 Appendix 6 315 Appendix 7 317 Appendix 8 319 Index 321 Symbols The following symbols are used frequently enough to merit general listing. Occasionally, some are used for other purposes. Symbols used temporarily are not listed here. a as linear value: (1) equatorial radius or semimajor axis of the ellipsoid of revolution, or (2) maximum scale factor at point on map. As angle: azimuth in spherical (or spheroidal) polar coordinate system. b (1) for ellipsoid of revolution, the polar radius or semiminor axis, or (2) minimum scale factor at point on map. For triaxial ellipsoid, the equa- torial radius at a right angle to the semimajor axis a. c for triaxial ellipsoid, the polar radius or semiaxis. e (1) for ellipsoid of revolution, the (first) eccentricity of the ellipse defined by dimensions a and b, where e = [(a2 — b2)/a2~]1/2; for triaxial ellipsoid, the eccentricity of the ellipse forming the equator; or (2) a Gaussian coefficient (see section 1.3.1). e' for ellipsoid of revolution, the second eccentricity, where e' = ej (i-e2)1/2. f (1) function of the following parenthetical argument such as <j>, or (2) a Gaussian coefficient (see section 1.3.1). h (1) elevation of a given point above the surface of the reference ellip- soid, or (2) a Gaussian coefficient (see section 1.3.2). In natural logarithm, or logarithm to base e, where e = 2.718 28 M radius of curvature of meridian at a given point on the ellipsoid (see section 1.1.1 and Appendix 2). m linear scale factor along meridian (see section 1.4.1). N radius of curvature of ellipsoid surface in a plane orthogonal to the meridian (see section 1.1.1 and Appendix 2). n linear scale factor along parallel of latitude (see section 1.4.1). n' an auxiliary function of ellipsoid semiaxes a and b, namely n' = (a ~ b)/(a + b). p area scale factor. q isometric latitude. r radius of parallel of latitude (/>, where r = N cos </>. S area of portion of surface of ellipsoid or sphere. s linear distance along surface of sphere or ellipsoid. M, v coordinate system for intermediate use. v distortion variable (see sections 1.6.4 and 1.6.5). ix Map projections: a reference manual three-dimensional axes with center at a given point on the surface of the sphere or ellipsoid, the Y-axis in the meridian plane pointing to the North Pole, the Z-axis coinciding with the normal to the ellipsoid surface, and the X-axis positive east from this point, three-dimensional axes for the sphere or ellipsoid, with the center at the center of the ellipsoid, and the XG-, YG-, and ZG-axes increasing in the direction of the Greenwich meridian in the equatorial plane, the meridian 90°E in the equatorial plane, and the North Pole, respec- tively. rectangular coordinate: distance to the right of the vertical line (Y- axis) passing through the origin or center of a projection (if negative, it is distance to the left). partial derivative dx/dcj); similarly for yh etc. second derivative d2x/d(f)2, etc. rectangular coordinate: distance above the horizontal line (X-axis) passing through the origin or center of a projection (if negative, it is distance below). (1) spherical angle from selected point on surface of sphere or ellipsoid to some other point on surface, as viewed from center, or (2) rectangu- lar coordinate in direction of Z-axis. convergence of meridian (deviation from direction of Y-axis on map), azimuth in plane polar coordinates. deviation of graticule intersection from right angle on map.