OPTIMIZATION OF GEOGRAPHIC MAP PROJECTIONS t FOR CANADIAN TERRITORY by Kresho Frankich Dipl. Ing., Geodetic University of Zagreb, 1959 M.A.Sc., University of British Columbia, 1974 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Special Arrangements c Kresho Frankich 1982 Simon Fraser University November, 1982 All rights reserved. This thesis may not be reproduced in whole or in part, by photocopy or other means, without permission of the author. APPROVAL Name : Kresho F'rankich Degree: Doctor of Philosophy Title of Thesis: Optimization of Geographic Map Projection for Canadian Territory Examining Committee : Chairperson: Dr. Thomas Poiker - Dr. Robert Russell Dr. Arthur Roberts Dr. Hwarlfi rakiwsky External Exf miner 1 The University of Calgary March 3, 1953 Date Approved: PARTIAL COPYRIGHT LICENSE I hereby grant to Simon Fraser University the right to lend my thesis, project or extended essay (the title of which is shown below) to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. I further agree that permission for multiple copying of this work for scholarly purposes may be granted by me or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without my written permission. Title of Thesis/Project/Extended Essay Author: (signature) ( name iii I ABSTRACT One of the main tasks of mathematical cartography is to determine a projection of a mapped territory in such a way that the resulting deformations of angles, areas and distances are objectively minimized. Since the transformation process will generally change the original distances it is appropriate to adopt the deformation of distances as the basic parameter for the evaluation of map projections. As the qualitative measure of map projections the author decided to use the Airy- Kavraiskii criterion where A is the area of the mapping domain, a and b semi-axes of the indicatrix of Tissot, and the integration is extended to the whole domain. Until now all optimization of map projec- tions were referred to domains with analytically defined boundaries, for example, a spherical trapezoid, spherical cap or a hemisphere, and for those map projections in which the analytical evaluation of the integral was possible. The author expands the optimization process to irregular domains with boundaries consisting of a series of discrete points. The minimization of the criterion leads to a least square adjustment problem. I The main purpose of the project was to develop a uniform method to optimize the standard and most frequently used mapping systems in geography for Canadian territory. The scope of optimization was enlarged by the inclusion of the optimization of modified equiareal projections as well as the determination of the Chebyshev conformal projections. Almost all small scale maps of the territory of Canada have been based on the normal aspect of the Lambert Conformal Conic projection with standard parallels at latitudes of 49' and 77". Every optimized projection in the research yielded a smaller value for the Airy-Kavraiskii criterion. Thus, it was proven that any standard map project ion with properly selected metagraticule and constant parameters of the projection is much better than the official projection. The best result was achieved with the optimized equidistant projection. Since the projection equations for the equidistant conic projection are very simple and the projection gives the best result with respect to the Airy-Kavraiskii criterion, the author highly recommends its application for small scale maps of Canada. Dedicated to JASNA, who encouraged the author more than anybody but, at the same time, suffered from his research more than anybody. ACKNOWLEDGEMENT There are many people who to a smaller or larger extent influenced the author and thus contributed to this research. The author received his introduction to mathematical cartography from the late professor, Dr. Branko Borcic, at the Geodetic University of Zagreb. Dr. Sybren de Jong at the University of British Columbia revitalized the interest for the subject. The greatest influence on the author's knowledge however was made by a Russian geodesist, V.V. ~avraiskii,whom the author has unfortunately never met, but whose book, unsurpassed in mathematical cartography, has influenced the author more than he is probably aware of. The author is grateful to his supervisors, Dr. Thomas Poiker and Dr. Robert Russell; the former for his constant encouragement in the research and the latter for his guidance through the complicated world of numerical analysis and for his careful proofreading of the thesis. The importance of Dr. Waldo Tobler for the present form of the thesis must also be acknowledged. He made many good suggestions and supplied the author with a list of helpful articles in the field of mathematical cartography. The colleagues at the British Columbia Institute of Technology, Dave Martens and Ken Gysler helped the author; the former in computer programming and the latter in proofreading the text. vii 8 The illustrations for various map projections were made by Wayne Luscombe at the World Bank in Washington, D.C. Mrs. Bernice Tong typed the manuscript overloaded with mathematical formulation in a remarkable manner. The author is grateful to her. viii TABLE OF CONTENTS Page Abstract iii Dedication v Acknowledgement vi List of Tables viii ~istof Figures ix 0. INTRODUCTION 1. Grimms' Introduction 2. Introduction to Cartographic Problems 3. Objectives of Research 4. Practical Optimizations for Canadian Maps I. GENERAL THEORY OF CARTOGRAPHIC MAPPINGS Introduction to Theory of Surfaces Cartographic Mappings Theory of Distort ions Indicatrix of Tissot Fundamental Differential Equations Classification of Mappings Conformal Mappings Equiareal Projections 11. OPTIMAL MAP PROJECTIONS 1. Ideal and Best Map Projections 2. Local Qua1 it at ive Measures 3. Qualitative Measure for Domains 4. Optimization of Conical projections 5. Optimization of Modified projections 6. Optimization and the Method of Least Squares 7. Criterion of Chebyshev , Page 111. THE CHEBYSHEV MAP PROJECTIONS 1. Introduction 2. '.Method of Ritz 3. The Method of Finite Differences 4. Symmetric Chebyshev Projections by Least Squares 5. Non-Symmetric Chebyshev Projections IV. OPTIMAL CARTOGRAPHIC PROJECTIONS FOR CANADA 1. Introduction and Historical Background 2. Optimal Conic Projections for Canada 3. Optimal Cylindric Projections 4. Optimal Azimuthal Projections 5. Optimal Modified Equiareal Projections 6. Optimization Results of Conical ~rojections 7. Optimization Results of Modified Equiareal projections 8. Chebyshev' s Projections for Canada 9. Conclusion and Recommendations 10. Epilogue, Written by Lewis Carroll APPENDIX 1 GRAPHICAL PRESENTATION OF TYPICAL CONICAL PROJECTIONS APPENDIX 2 GRAPHICAL REPRESENTATION OF TYPICAL EQUIAREAL MODIFIED PROJECTIONS 224 APPENDIX 3 DERIVATION OF FORMULAE FOR OPTIMIZED MODIFIED PROJECTIONS BIBLIOGRAPHY I LIST OF TABLES Table Page IV- 6- 1 Optimized parameters of conic projections 21 1 IV-6- 2 Optimized parameters of cylindric and azimuthal projections IV- 7- 1 Optimized parameters of modified equiareal projections IV-8- 1 Coefficients of harmonic polynomials LIST OF FIGURES Figure Page 1-2-1 Coordinate systems in cartographic mappings 26 1-2-2 Differentially small surface element of the sphere and its projection in the plane 1-4-1 Unit circle and the indicatrix of Tissot 11-4-1 Graticule and metagraticule 111-3-1 A section of the grid IV- 1-1 Distribution of points which approximate Canadian territory 0. INTRODUCTION Song For Five Dollars Five learned scholars were each paid a dollar to see if they could find out something new but they met with some resistance for according to the distance they noticed things got smaller or they grew. A passion in them burned they left no worm unturned they flattened all the bumps to fill in holes and from Leicester to East Anglia made continents rectangular and evenly distributed the poles. 2. INTRODUCTION TO CARTOGRAPHIC PROBLEMS The mathematical aspect of cartographic mapping is a process which establishes a unique connection between points of the earth's sphere and their images on a plane. It was proven in differential geometry (Eisenhart, 1960), (Goetz, 1970), (Taschner, 1977) that an isometric mapping of the sphere onto a plane with all corresponding distances on both surfaces remaining identical can never be achieved since the two surfaces do not possess the same Gaussian curvature. In other words, it is impossible to derive transformation formulae which will not alter distances in the mapping process. Cartographic transformations will always cause a certain deformation of the original surface. These deformations are reflected in changes of distances, angles and areas. The main task of mathematical cartography is to determine projection formulae to transform a mapped territory onto a plane with a minimum deformation of the original sphere. It is possible to derive transformation equations which have no deformations in either angles or areas (Richardus and Adler, l972), (Frankich, 1977). These projections are called conformal and equiareal, respectively. The transformation processes, however, always change distances and therefore the deformation of distances