Minimizing Map Distortion Using Oblique Projections Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Jiaqi Zhang Graduate Program in Geodetic Science The Ohio State University 2017 Thesis Committee Dr. Alan John Saalfeld, Advisor Dr. C.K. Shum 1 Copyrighted by Jiaqi Zhang 2017 2 Abstract We provide a precise mathematical definition of map distortion, and we introduce several map projections that have minimal distortion for certain specific regions of the Earth (namely, regions whose boundary consists of one or two parallels of latitude). The minimizing projections for these regions will all have normal aspect. We show that oblique aspect projections have the same distortion properties as their normal aspect counterparts on earth regions bounded by one lesser circle or by two lesser circles lying in parallel planes. We summarize the good consequences of having small distortion, and we outline a strategy for finding the best oblique map projection of least distortion for any region on the Earth. We then implement that strategy to find projections of least distortion for some unusually shaped countries, states and other regions. ii Acknowledgments It’s a great honor to have the opportunity to study and research at the Ohio State University and these years are the most precious time of my life. I would like to thank my advisor Dr. Alan John Saalfeld, the most important and kind person in my academic life here. Besides instructing me knowledge and the way to conduct research, he cultivates my capacity for independent thinking and attitude on scientific research. Also, my thanks go to Dr. Michael G Bevis, who inspires me a lot and supports our research. Moreover, I want to show my respect and appreciations to those excellent professors in my department like Dr. Jekeli and Dr. Schaffrin. Finally, I would like to thank my parents for supporting me financially and mentally. iii Vita 2010 - 2014 B.S., Geographic Information System and Cartography, Wuhan University, China 2014 – present M.S., Geodetic Science, School of Earth Sciences, The Ohio State University Fields of Study Major Field: Geodetic Science iv Table of Contents Abstract ............................................................................................................................... ii Acknowledgments.............................................................................................................. iii Vita ..................................................................................................................................... iv List of Tables .................................................................................................................... vii List of Figures .................................................................................................................. viii Chapter 1. Introduction ....................................................................................................... 1 1.1 Scale and Distortion .................................................................................................. 1 Chapter 2. Background and Knowledge ............................................................................. 4 2.1 Projections................................................................................................................. 4 2.2 Scales Calculation ..................................................................................................... 7 2.3 Normal Aspect Projections ..................................................................................... 10 2.3.1 Normal Aspect Azimuthal Projections ............................................................ 11 2.3.2 Normal Aspect Cylindrical Projections ........................................................... 11 2.3.3 Normal Aspect Conic Projection ..................................................................... 12 2.4 Oblique Projections ................................................................................................. 13 2.5 Comparison of Normal and Oblique Aspect Projections ........................................ 21 2.6 Normal and Oblique Aspect Projection’s Equivalent Properties of Distortion ...... 23 Chapter 3. Oblique Azimuthal Projection ......................................................................... 25 3.1 Methodology ........................................................................................................... 25 3.2 Experiments ............................................................................................................ 28 3.2.1 Data Preprocessing........................................................................................... 28 3.2.2 Main Programs ................................................................................................. 31 3.3 Results Analysis ...................................................................................................... 32 Chapter 4. Oblique Cylindrical Projection........................................................................ 35 4.1 Methodology ........................................................................................................... 35 v 4.2 Experiments ............................................................................................................ 37 4.2.1 Data Preprocessing........................................................................................... 37 4.2.2 Main Programs ................................................................................................. 38 4.3 Results Analysis ...................................................................................................... 40 Chapter 5. Oblique Conic Projection ................................................................................ 43 5.1 Methodology ........................................................................................................... 43 5.2 Experiments ............................................................................................................ 45 5.2.1 Data Preprocessing........................................................................................... 45 5.2.2 Main Programs ................................................................................................. 45 5.3 Results Analysis ...................................................................................................... 46 Chapter 6. Evaluation........................................................................................................ 48 6.1 Evaluation of Performance ..................................................................................... 48 6.2 Evaluation of Effectiveness .................................................................................... 50 Chapter 7. Conclusions ..................................................................................................... 56 7.1 Discussion on angle and area distortion.................................................................. 56 7.2 Practical significance of oblique projections .......................................................... 58 Bibliography ..................................................................................................................... 60 vi List of Tables Table 1. Classification of normal aspect tangent projections. .......................................... 10 Table 2. Distortion for the equidistant azimuthal projection of a circular area. ............... 14 Table 3. Distortion for cylindrical projections of band around a great circle. .................. 16 Table 4. Distortion for conic projections of band around a lesser circle. ......................... 18 Table 5. Distortion for conic projections of band around a lesser circle. ......................... 19 Table 6. Comparison of normal aspect projection VS. oblique aspect. ............................ 21 Table 7. Distortions under different equidistant oblique projections. .............................. 48 Table 8. US states fitting within a spherical cap. .............................................................. 58 Table 9. Countries fitting within a spherical cap. ............................................................. 59 vii List of Figures Figure 1. Geodesic curves on the datum surface and the projection surface. ..................... 3 Figure 2. Map projection scale determination by curve tangents. ...................................... 4 Figure 3. Maps of Ohio using different projections (cylindrical on left, conic on right). .. 6 Figure 4. Maps of South America using different projections. ........................................... 7 Figure 5. Normal aspect projections' scales. ....................................................................... 8 Figure 6. Equidistant normal aspect projections' scales. .................................................... 9 Figure 7. Distortion = ((x/R)/sin(x/R)-1)*100. ................................................................. 15 Figure 8. Distortion = ([2tan(x/2R)/sin(x/R)]-1)*100. ..................................................... 15 Figure 9. Distortion of a given width band region (Unit of x: KM). ................................ 17 Figure 10. Distortion of a given width band region (Unit of x: KM). ......................... 19 Figure 11. Distortion of a given width band region (Unit of x: KM). .............................. 20 Figure 12. Normal VS. Oblique aspect azimuthal