Map Projections: Theory and Applications

385 Pages·2018·17.6 MB·English

by  Frederick Pearson II

Map Projections:
Theory and Applications
Frederick Pearson, II, M.S., P.D.D.
Systems Engineer
Combat Systems
Naval Surface Warfare Center
Dahlgren, Virginia
Lecturer
Department of Civil Engineering
Virginia Polytechnic Institute
Falls Church, Virginia
0
CRC Press
Taylor & Francis Group
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
Library of Congress Cataloging-in-Publication Data
Pearson, Frederick, 1936-
Map projections :theory and applications I Frederick Pearson D.
p. em.
Includes bibliographical references.
ISBN 13: 978-0-8493-6888-2
1. Map-projection. I. Title.
GAllO.P425 1990
526' .8--dc20 89-23986
ClP
This book contains infonnation obtained from authentic and highly regarded sources. Reprinted
material is quoted with permission, and sources are indicated. A wide variety of references are
listed. Reasonable efforts have been made to publish reliable data and infonnation, but the author
and the publisher cannot assume responsibility for the validity of all materials or for the con
sequences of their use.
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Direct all inquiries to CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Raton, Florida
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© 1990 by CRC Press, Inc.
No claim to original U.S. Government works
International Standard Book Number 0-8493-6888-X
Library of Congress Card Number 89-23986
This book is dedicated to my beloved Shirley.
PREFACE
This present volume is the logical step forward from two previous works
by the author. The first was Map Projection Equations, a Naval Surface
Warfare Center Technical Report. This was an attempt to collect the maximum
number of existing map projections in a single source, using a standardized
notation. Then came Map Projection Methods, in which an attempt was made
at modernization of the method of presentation. After teaching from this text
for three summers in the Geodetic Engineering program at Virginia Poly
technic Institute at Falls Church the author could see that further improvement
was necessary. Based on questions and suggestions from the students in these
classes, the present volume is a further attempt to move from the classical
and theoretical to the modem and practical. In the present volume, projections
of mainly historical value have been excluded. Projections of use in photo
grammetry, remote sensing, and target tracking applications have been em
phasized more. The presentation of the projections that are the core of most
cartographic work has been clarified.
In Chapter 1, the problems involved in map projection theory and practice
are introduced. The concept of an ideal map is explored, and the deviations
from the ideal and their causes are discussed. The basic nomenclature of map
projections is defined.
Chapter 2 begins the mathematical consideration of map projection tech
niques and applications. In this chapter, the mathematical groundwork is laid
down. The mathematical level of the student needed to understand this text
is calculus and trigonometry. More advanced topics will be summarized and
developed in this chapter.
The chapter begins with a discussion of the differential geometry of an
arbitrary curve in space. This is followed by consideration of an arbitrary
spatial surface. Of more interest to map projections is the differential geometry
of surfaces of revolution. The form of the basic transformation matrices are
developed in detail. A number of mathematical entities of importance in
mapping are discussed. These are the convention for azimuth, constant of the
cone, convergence of the meridians, and the basic coordinate systems for the
Earth and the map. The important spherical trigonometric formulae for ro
tations are introduced.
In Chapter 3, the surfaces of revolution of immediate interest to mapping
are considered. These are the sphere and the spheroid taken as useful models
of the figure of the Earth. Formulas are developed for linear and angular
measurement on the spheroid and sphere.
Chapters 4, 5, and 6 are dedicated to the derivation of the various pro
jections themselves. In these chapters are the equal area, conformal, and
conventional projections of major use today. In every case, the result is a set
of direct transformation equations in which the Cartesian plotting coordinates
may be obtained from geographic coordinates. In selected cases, the inverse
transformation from Cartesian to geographic coordinates is also included.
Distortion is evaluated mathematically in Chapter 7. Numerical estimates
of distortion can be obtained from the equations developed in this chapter.
Chapter 8 includes extended examples of the use of map projection for
mulae in the solution of modem problems.
Since it should be obvious for the complexity of the equations in Chapters
4, 5, and 6, that no worker in the field of mapping would want to routinely
evaluate them with a pocket calculator, the computerization of the equations
is considered in Chapter 9. The equations for the direct and inverse trans
formations, as well as grid systems, are treated.
A general discussion on the uses and choices in the field of map projections
rounds out the text in Chapter 10.
This volume is intended to be the text in a one semester course in the
mathematical aspects of mapping and cartography. The level of mathematics
is consistent with upper level undergraduate and graduate courses.
Special thanks go to John P. Snyder, formerly of the U.S. Geological
Survey, who, over the years, has consistently challenged me to produce a
better presentation of the material. I also thank him for his review of the
present volume and his many helpful suggestions.
A word of thanks is due to many people. These include Dr. L. Meirovitch
of Virginia Polytechnic Institute for his encouragement of my first attempt,
Dr. J. Junkins of Texas A. M., for my second attempt, and Dr. W. G. Rich
and Glenn Bolick of the Naval Surface Warfare Center, for the present volume.
Thanks to B. McNamara of the U.S. Geological Survey for permitting the
use of his mapping program to generate many of the tables, and to the author's
wife, Shirley, for typing the manuscript.
THE AUTHOR
Mr. Frederick Pearson, M.S., M.S., P.D.D., received a B.S. in History
from St. Louis University, St. Louis, Missouri in 1961, and M.S. in As
tronomy from Yale University, New Haven, Connecticut in 1964, a Diploma
in Business Administration from the Hamilton Institute in 1968, and M.S. in
Engineering Mechanics from Virginia Polytechnic Institute in 1977, and a
P.D.D. in Civil Engineering from the University of Wisconsin, Madison,
Wisconsin, in 1981.
Mr. Pearson has experience in applied science and engineering. He de
veloped star charts and satellite trajectory programs and designed a celestial
navigation device for the Aeronautical Chart and Information Center. At
Emerson Electric Company, he conducted studies in orbital analysis and
satellite rendezvous, for which he won the AIAA Young Professional Scientist
Award in 1968. Mr. Pearson has also done vibration analyses and design
work on spin-stabilized missiles. For TRW systems, he conducted mission
planning and devised emergency back-up procedures for the Apollo missions.
At McDonnell-Douglas, he worked on the design of the control systems of
the HARPOON Missile and the guidance for the Space Shuttle.
Mr. Pearson is employed at the Naval Surface Warfare Center where his
work has ranged from the analysis of satellite orbits to the design of machine
elements and structures. He is presently a Systems Engineer in the AEGIS
Program, working on Infrared Sensors.
Mr. Pearson has taught Map Projections at Air Force Cartography School,
and Astrodynamics and Orbit Determination in a TRW Employee Training
Program. Also, he has taught Map Projections for the Civil Engineering
Department of Virginia Polytechnic Institute.
Mr. Pearson's publications include: Map Projection Equations, NSWC,
1977; Map Projection Methods, Sigma, 1984; and Map Projection Software,
Sigma, 1984.
TABLE OF CONTENTS
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I. Introduction to the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. Basic Geometric Shapes .................................... 2
III. Distortion ................................................... 3
IV. Scale ....................................................... 5
A. Example 1 .......................................... 6
B. Example 2 .......................................... 6
V. Feature Preserved in Projection ............................. 6
VI. Projection Surface .......................................... 8
VII. Orientation of the Azimuthal Plane ......................... 9
VIII. Orientation of a Cone or Cylinder .......................... 9
IX. Tangency and Secancy .................................... 10
X. Projection Techniques ..................................... 10
XI. Plotting Equations ......................................... 12
XII. Plotting Tables ............................................ 14
References ........................................................ 15
2. Mathematical Fundamentals .................................... 17
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
II. Coordinate Systems and Azimuth ......................... 17
III. Grid Systems .............................................. 20
IV. Differential Geometry of Space Curves .................... 20
A. Example 1 ......................... , ............... 27
B. Example 2 ......................................... 29
V. Differential Geometry of a General Surface ............... 29
VI. First Fundamental Form ................................... 31
A. Example 3 ......................................... 34
B. Example 4 ......................................... 34
C. Example 5 ......................................... 34
D. Example 6 ......................................... 34
E. Example 7 ......................................... 35
VII. Second Fundamental Form ................................ 35
VIII. Surfaces of Revolution .................................... 43
IX. Developable Surfaces ...................................... 48
X. Transformation Matrices ................................... 50
A. Example 8 ......................................... 55
XI. Mathematical Definition of Equality of Area and
Conformality .............................................. 56
XII. Rotation of Coordinate System ............................ 56
XIII. Convergency of the Meridians ............................. 59
A. Example 9 ......................................... 61
XIV. Constant of the Cone and Slant Height .................... 62
A. Example 10 ........................................ 64
B. Example 11 ........................................ 65
References ........................................................ 65
3. Figure of the Earth .............................................. 67
I. Introduction ............................................... 67
II. Geodetic Considerations ................................... 67
A. Example 1 ......................................... 69
III. Geometry of the Ellipse ................................... 69
A. Example 2 ......................................... 72
IV. The Spheroid as a Model of the Earth ..................... 72
A. Example 3 ......................................... 86
B. Example 4 ......................................... 86
C. Example 5 ......................................... 86
V. The Spherical Model of the Earth ......................... 87
A. Example 6 ......................................... 92
B. Example 7 ......................................... 92
C. Example 8 ......................................... 93
D. Example 9 ......................................... 93
E. Example 10 ........................................ 93
F. Example 11 ........................................ 94
VI. The Triaxial Ellipsoid ..................................... 94
References ........................................................ 96
4. Equal Area Projections .......................................... 97
I. Introduction ............................................... 97
II. General Procedure ......................................... 97
III. The Authalic Sphere ....................................... 99
A. Example 1 ........................................ 102
B. Example 2 ........................................ 102
IV. Albers, One Standard Parallel ............................ 103
A. Example 3 ........................................ 108
B. Example 4 ........................................ 109
V. Albers, Two Standard Parallels ........................... 110
A. Example 5 ........................................ 116
VI. Bonne .................................................... 118
A. Example 6 ........................................ 121
VII. Azimuthal ................................................ 123
A. Example 7 ........................................ 127
VIII. Cylindrical ............................................... 129
A. Example 8 ........................................ 132
B. Example 9 ........................................ 133
IX. Sinusoidal ................................................ 133
A. Example 10 ....................................... 137
B. Example 11 ....................................... 137
X. Mollweide ................................................ 138
A. Example 12 ....................................... 141
B. Example 13 ....................................... 143
XI. Parabolic ................................................. 144
A. Example 14 ....................................... 148
B. Example 15 ....................................... 148
XII. Hammer-Aitoff ........................................... 149
A. Example 16 ....................................... 151
XIII. Boggs Eumorphic ........................................ 152
XIV. Eckert IV ................................................. 154
XV. Interrupted Projections ................................... 156
A. Example 17 ....................................... 158
References ....................................................... 159
5. Conformal Projections .......................................... 161
I. Introduction .............................................. 161
II. General Procedures ....................................... 161
III. Conformal Sphere ........................................ 162
A. Example 1 ........................................ 168
IV. Lambert Conformal, One Standard Parallel. .............. 168
A. Example 2 ........................................ 174
V. Lambert Conformal, Two Standard Parallels ............. 175
VI. Stereographic ............................................. 182
A. Example 3 ........................................ 187
B. Example 4 ........................................ 189
VII. Mercator .................................................. 189
A. Example 5 ........................................ 196
B. Example 6 ........................................ 198
C. Example 7 ........................................ 199
D. Example 8 ........................................ 199
E. Example 9 ........................................ 200
VIII. State Plane Coordinates .................................. 201
IX. Military Grid Systems .................................... 207
A. Example 10 ....................................... 211
References ....................................................... 213
6. Conventional Projections ....................................... 215
I. Introduction .............................................. 215
II. Summary of Procedures .................................. 215
III. Gnomonic ................................................ 216
A. Example 1 ........................................ 224