Map ProjectionsTheory and Applications

Frederick Pearson, II

• Publisher: CRC Press
• Year: 2018
• Print ISBN: 9780849368882
• eBook ISBN: 9781351433693
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

About the Author: 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, geodi

Chapter 1

INTRODUCTION

A map is a visual representation of a part, or all of the surface of the Earth. It is expected that the various terrestrial features shown on the map will be approximately in their true relationship. The success of the representation depends on the map projection chosen to produce the map.

Map projection is the orderly transfer of positions of places on the surface of the Earth to corresponding points on a flat sheet of paper, a map. Since the surface of a sphere cannot be laid flat on a plane without distortion, the process of transformation requires a degree of approximation and simplification.³ This first chapter lays the groundwork for this subject by detailing, in a qualitative way, the basic problem and introducing the nomenclature of maps. Succeeding chapters consider the mathematical techniques and the simplifications required to obtain manageable solutions.

All projections introduce distortions in the map. The types of distortion are considered in terms of length, angle, shape, and area. This first chapter discusses the qualitative aspects of the problem, while Chapter 7 deals with it quantitatively.

An examination of the geometric shapes on a globe representing the Earth indicates basic figures. If these figures are maintained on a map, a great step is taken in limiting the adverse effects of distortion. Scale factor refers to the relative length of a line on the model for the Earth, and the same line as represented on the map. The concept of scale factor is instrumental in reducing Earth-sized lengths to map-sized lengths.

Map projections may be classified in a number of ways. The principal one is by the features preserved from distortion by the mapping technique. Other methods of classification depend on the plotting surface employed, the location of the points of contact of this surface with the Earth, the orientation of the plotting surface with respect to the direction of the polar axis of the Earth, and whether the plotting surface is tangent or secant to the Earth. Finally, maps can be classified according to whether or not a map can be drawn by purely graphical means.

This chapter also considers the plotting equations derived in Chapters 4, 5, and 6 in a general way. These equations are the basis of the orderly transformation from the model of the Earth to the map. The results of evaluating these equations are often given in plotting tables. The conventions incorporated in the plotting tables of this text are discussed.

I. INTRODUCTION TO THE PROBLEM²

Map projection requires the transformation of positions from a curved surface, the Earth, onto a plane surface, the map, in an orderly fashion. The problem occurs because of the difference in the surfaces involved.

The model of the Earth is taken as either a sphere or a spheroid (Chapter 3). These curved surfaces have two finite radii of curvature. The map is a plane surface, and a plane is characterized by two infinite radii of curvature. As is shown in Chapter 2, it is impossible to tranform from a surface of two finite radii of curvature to a surface of two infinite radii of curvature without introducing some distortion. The sphere and the spheroid are called nondevelopable surfaces. This refers to the inability of these surfaces to be developed (i.e., laid flat) onto a plane in a distortion-free manner.

Intermediate between the nondevelopable sphere and spheroid and the plane are surfaces with one finite and one infinite radius of curvature. The examples of this type of surface of interest to mapping are the cylinder and the cone. These surfaces are called developable. Both the cylinder and the cone can be cut and then developed (essentially unrolled along the finite radius of curvature) to form a plane. This development introduces no distortion, and, thus, these figures may be used as intermediate plotting surfaces between the sphere and spheroid and the plane. However, in any transformation from the sphere or spheroid to the developable surface some damage has already been done. The transformation from the nondevelopable to the developable surface invariably introduces some degree of distortion.

Distortion is inevitable, and a map with ideal properties is never attained. Consider the following properties of an ideal map:

1. Areas on the map maintain correct proportion to areas on the Earth.

2. Distances on the map must remain in true scale.

3. Directions and angles on the map must remain true.

4. Shapes on the map must be the same as on the Earth.

The best a cartographer can hope for is the realization of one or two of these ideal properties in a single map projection. The other ideal properties will then be subject to distortions, but hopefully, only to a controlled extent. There is always some compromise involved.

The projections of Chapters 4, 5, and 6 attempt to achieve one or more of these ideal properties at the expense of the others. The best projection is chosen for a specific application. Distortion in the other features is then considered to be tolerable.

II. BASIC GEOMETRIC SHAPES

A series of basic geometric shapes can be found on a gore cut from the model of the Earth. A gore is the tapered shape obtained by slicing the model of the Earth through its poles. This is represented in Figure 1. Consider the top half of the gore. The lower third of the half-gore can be represented on a flat sheet of paper by a rectangular shape. The top third of the half-gore is easily represented on the map by a triangular shape. The middle third of the half-gore suggests a trapezoidal shape.

FIGURE 1. Basic cartographic shapes.

One of the easiest ways to minimize distortions is to choose a projection whose basic geometrical figures closely represent the shape of the corresponding figure on the model of the Earth. As is shown in Chapters...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• 1. Introduction
• 2. Mathematical Fundamentals
• 3. Figure of the Earth
• 4. Equal Area Projections
• 5. Conformal Projections
• 6. Conventional Projections
• 7. Theory of Distortions
• 8. Mapping Applications
• 9. Computer Applications
• 10. Uses of Map Projections
• Index