U.S. ~eological Survey Profes-sional-Paper 1453 .. ~ :.. ' ... Department of the Interior BRUCE BABBITT Secretary U.S. Geological Survey Gordon P. Eaton, Director First printing 1989 Second printing 1994 Any use of trade names in this publication is for descriptive purposes only and does not imply endorsement by the U.S. Geological Survey Library of Congress Cataloging in Publication Data Snyder, John Parr, 1926- An album of map projections. (U.S. Geological Survey professional paper; 1453) Bibliography: p. Includes index. Supt. of Docs. no.: 119.16:1453 1. Map-projection. I. Voxland, Philip M. II. Title. Ill. Series. GA110.S575 1989 526.8 86-600253 For sale by Superintendent of Documents, U.S. Government Printing Office Washington, DC 20401 United States Government Printing Office : 1989 CONTENTS Preface vii Introduction 1 McBryde·Thomas Flat-Polar Parabolic 72 Glossary 2 Quartic Authalic 74 Guide to selecting map projections 5 McBryde-Thomas Flat-Polar Quartic 76 Distortion diagrams 8 Putnii)S P5 78 Cylindrical projections DeNoyer Semi-Elliptical 80 Mercator 10 Robinson 82 Transverse Mercator 12 Collignon 84 Oblique Mercator 14 Eckert I 86 Lambert Cylindrical Equal-Area 16 Eckert II 88 Behrmann Cylindrical Equal-Area 19 Loximuthal 90 Plate Carree 22 Conic projections Equirectangular 24 Equidistant Conic 92 Cassini 26 Lambert Conformal Conic 95 Oblique Plate Carree 28 Bipolar Oblique Conic Conformal 99 Central Cylindrical 30 Albers Equal-Area Conic 100 Gall 33 Lambert Equal-Area Conic 102 Miller Cylindrical 35 Perspective Conic 104 Pseudocylindrical projections Polyconic 106 Sinusoidal 37 Rectangular Polyconic 110 McBryde·Thomas Flat-Polar Sinusoidal 44 Modified Polyconic, International Map of the World series 111 Eckert V 46 Bonne 112 Winkel! 48 Werner 114 Eckert VI 50 Azimuthal projections McBryde 53 52 Perspective Mollweide 54 Gnomonic 116 Eckert Ill 58 Stereographic 120 Eckert IV 60 Orthographic 124 Wagner IV 62 General Vertical Perspective 128 Hatano Asymmetrical Equal-Area 64 Nonperspective Goode Homolosine 66 Azimuthal Equidistant 132 Boggs Eumorphic 68 Lambert Azimuthal Equal-Area 136 Craster Parabolic 70 Airy 140 iii :-:'" CONTENTS Modified Guyou 188 Two-Point Azimuthal 144 Peirce Quincuncial 190 Two-Point Equidistant 146 Adams Hemisphere in a Square 192 Miller Oblated Stereographic 148 Adams World in a Square 1 194 Wiechel 149 Adams World in a Square II 195 Craig Retroazimuthal 150 Lee Conformal World in a Triangle 196 Hammer Retroazimuthal 152 Conformal World in an Ellipse 198 Littrow 154 Vander Grinten I 200 Berghaus Star 156 Vander Grinten II 202 Aitoff 158 Van der Grinten Ill 203 Hammer 160 _ Van der Grinten IV 204 Briesemeister 162 Armadillo 206 Winkel Tripel 164 GSSO 208 Eckert-Greifendorff 166 Space Oblique Mercator 211 Wagner VII 168 Satellite Tracking 213 Chamberlin Trimetric 170 Appendix: Formulas for projections 216 Ti Ited Perspective · 172 Cylindrical projections 218 Miscellaneous Pseudocylindrical projections 220 Bacon Globular 174· Conic projections 224 Fournier Globular 175 Pseudoconic projections 225 Nicolosi Globular 176 Azimuthal projections Perspective 226 Apian Globular I 177 Nonperspective 228 Ortelius Oval 178 Modified 230 Lagrange 180 Miscellaneous projections 234 Eisenlohr 184 Selected references 240 August Epicycloidal 186 Index 242 v Preface Every map must begin, either consciously or on a standardized basis, so that comparisons are unconsciously, with the choice of a map projec­ more realistic. These maps have been plotted on tion, the method of transforming the spherelike film as lines 0.005 in. wide by means of a photo­ surface being mapped onto a flat plane. Because head computer-driven plotter and World Data this transformation can become complicated, Bank I shoreline files (national boundaries there is a tendency to choose the projection included where appropriate). Central meridians that has been used for a suitable available base and other parameters have been given the same map, even if it does not emphasize the desired values in so far as it was practical. properties. Some smaller maps have been prepared with This album has been prepared to acquaint the Dahlgren (Hershey) shoreline file. The map those in the cartographic profession and other projection software for any given map consisted cartophiles with the wide range of map projec­ of one of three programs. WORLD was developed tions that have been developed during the past for a mainframe computer by Philip Voxland few centuries. Because including all the hundreds at the University of Minnesota. CYLIND and of known projection systems would be both diffi- AZIMUTH were developed for a personal micro­ . cult and expensive, this volume presents some 90 computer and converted to mainframe computer basic projections in over 130 different modifica­ programs at the U. S. Geological Survey (USGS) tions and aspects. Dozens of other projections are by John Snyder, with the guidance of Michael K. mentioned briefly. All the popular projections Linck, Jr., and John F. Waananen. The authors have been included, along with many others thank Joel L. Morrison, past president of the Inter­ that have interesting appearances or features, national Cartographic Association and Assistant even if their usefulness for any serious mapping Division Chief for Research, National Mapping is limited. Division, USGS, for writing the introduction to this The most important question to raise about a book, as well as Tau Rho Alpha (USGS) and map projection concerns distortion. How accu­ Arthur H. Robinson (University of Wisconsin/ rately does the flat projection show the desired Madison) for many constructive comments on the portion of the spherelike Earth (or heavens or manuscript. The authors also thank Christopher J. planetary body)? Since accuracy is desired in Cialek, Kathie R. Fraser, Carolyn S. Hulett, and area, shape, and distance but cannot be achieved Ann Yaeger of the USGS for their assistance. in all three simultaneously and since the region With this reprint, several corrections have been mapped may vary in overall extent and shape, the incorporated. We thank Daniel Strebe for report­ answer is not clear cut. Because this album ing most of the errors in the formulas. emphasizes visual comparisons rather than tables of scale variation, the Tissot indicatrix has been John P. Snyder adopted as the most common means of compar­ Philip M. Voxland ing distortion on different parts of the same pro­ jection and on different projections. Computerized plotting has facilitated the preparation of many outline map projections vii Introduction The puzzling world of map projections has No other collection of world portrayals is fascinated humans for at least 2,500 years. comparable, because of the standard-format Perhaps no single aspect of the cartographic illustrations and the accompanying distortion discipline has held the cartographer's interest diagrams presented in this volume. These il­ so consistently throughout history. How does lustrations will be useful for classroom teaching, one portray on a nonspherical surface the es­ for selecting a projection to fit an output format, sentially spherical shape of the planet on which arid for making visual comparisons between we live? It cannot be done without distortion, projections. ' but the number of ways in which that distortion An appendix contains forward formulas for can be handled is infinite. all but a few highly complex projections to Although the subject is inherently mathemat­ transform from latitude and longitude positions ical and very complex in some instances, it on a sphere to rectangular coordinates. Inverse also is highly visual and readily calls forth formulas and formulas treating the Earth as human reactions ranging from near disbelief to an ellipsoid have not been included. Standardized unquestioned acceptance as truth, from confu­ notation is introduced and used throughout. sion to a clear understanding of information. Finally, it must be noted that this volume Map projections have been devised for prop­ could not have been realistically produced with­ aganda purposes, as evidence in court, for use out the aid of a computer. Although a compen­ in detailed planning, and to attract attention. dium of this nature has been needed for decades Most people are affected more than they (to avoid some of the confusion resulting from realize by map projections. Cognition of the identical projections' being called by different relative areas and shapes on the Earth's surface names, if nothing else), the enormity of manu­ and knowledge (correct or incorrect) of angles, ally compiling the necessary information pre­ sizes, distances, and other relationships among vented its production. The computer's ability to places on the globe are determined by the car­ calculate and plot the portrayals made this tographer's choice of map projections. "An album possible. Album of Map Projections" should enable read­ ers to sharpen and in some cases to correct their knowledge of the distances, directions, Joel L. Morrison and relative areas of the world. Although there has been no dearth of valuable books describing a wide range of map projec­ tions, "An Album of Map Projections" is the first attempt to set forth in a comprehensive manner most of the useful and commonly known map projections. Consistent and concise non­ mathematical textural descriptions are accom­ panied by standardized visual protrayals. De­ scriptions of graticules, scale, distortion char­ acteristics, and aspects are carefully worded. Supplemental