Map Projections

Frederick Pearson has extensive experience in teaching map projection at the Air Force Cartography School and Virginia Polytechnic Institute. He developed star charts, satellite trajectory programs, and a celestial navigation device for the Aeronautical Chart and Information Center. He is an expert in orbital analysis of satellites, and control and guidance systems. At McDonnell-Douglas, he worked on the guidance system for the space shuttle. This text develops the plotting equations for the major map projections. The emphasis is on obtaining usable algorithms for computed aided plotting and CRT display. The problem of map projection is stated, and the basic terminology is introduced. The required fundamental mathematics is reviewed, and transformation theory is developed. Theories from differential geometry are particularized for the transformation from a sphere or spheroid as the model of the earth onto a selected plotting surface.The most current parameters to describe the figure of the earth are given. Formulas are included to calculate meridian length, parallel length, geodetic and geocentric latitude, azimuth, and distances on the sphere or spheroid. Equal area, conformal, and conventional projection transformations are derived. All result in direct transformation from geographic to cartesian coordinates. For selected projections, inverse transformations from cartesian to geographic coordinates are given.Since the avoidance of distortion is important, the theory of distortion is explored. Formulas are developed to give a quantitative estimate of linear, area, and angular distortions. Extended examples are given for several mapping problems of interest. Computer applications, and efficient algorithms are presented. This book is an appropriate text for a course in the mathematical aspects of mapping and cartography. Map projections are of interest to workers in many fields. Some of these are mathematicians, engineers, surveyors, geodicests, geographers, astronomers, and military intelligence analysts and strategists.

INTRODUCTION. Introduction to the Problem. Basic Geometric Shapes. Distortion. Scale. Feature Preserved in Projections. Projection Surface. Orientation of the Azimuthal Plane. Orientation of a Cone of Cylinder. Tangency or Secancy. Projection Technique. Plotting Equations. Plotting Tables. MATHEMATICAL FUNDAMENTALS. Coordinate Systems and Azimuth. Grid Systems. Differential Geometry of Space Curves. Differential Geometry of a General Surface. First Fundamental Form. Second Fundamental Form. Surfaces of Revolution. Developable Surfaces. Transformation Matrices. Definition of Equality of Area and Conformality. Rotation of Coordinate Systems. Convergency of the Meridians. Constant of the Cone and Slant Height. FIGURE OF THE EARTH. Geodetic Considerations. Geometry of the Elipse. The Spheroid as a Model of the Earth. The Spherical Model of the Earth. The Triaxial Ellipsoid. EQUAL AREA PROJECTIONS. General Procedures. The Authalic Sphere. Albers, One Standard Parallel. Albers, Two Standard Parallels. Bonne. Azimuthal. Cylindrical. Sinusoidal. Mollweide. Parabolic. Hammer-Aitoff. Boggs Eumorphic. Eckert IV. Interrupted Projections. CONFORMAL PROJECTIONS. General Procedures. Conformal Sphere. Lambert Conformal, One Standard Parallel. Lambert Conformal, Two Standard Parallels. Stereographic. Mercator. State Plane Coordinates. Military Grid Systems. CONVENTIONAL PROJECTIONS. Summary of Procedures. Gnomonic. Azimuthal Equidistant. Orthographic. Simple Conic, One Standard Parallel. Simple Conic, Two Standard Parallels. Conical Perspective. Polyconic. Perspective Cylindrical. Plate Carree'. Carte Parallelogrammatique. Miller. Globular. Aerial Perspective. Van der Grinten. Cassini. Robinson. THEORY OF DISTORTIONS. Qualitative View of Distortion. Quantization of Distortion. Distortions from Euclidean Geometry. Distortions from Different Geometry. Distortions in Equal Area Projections. Distortions in Conventional Projections. MAPPING APPLICATIONS. Map Projections in the Southern Hemisphere. Distortion in the Transformation from the Spheroid to the Authalic Sphere. Distances on the Loxodrome. Tracking System Displays. Differential Distances about a Position. COMPUTER APPLICATIONS. Direct Transformation Subroutines. Inverse Transformation Subroutines. Calling Program for Subroutines. State Plane Coordinates. UTM Grids. Computer Graphics. USES OF MAP PROJECTIONS. Fidelity to Features on the Earth. Characteristics of Parallels and Meridians. Considerations in the Choice of a Projection. Recommended Areas of Coverage. Recommended Set of Map Projections. Conclusion.