Map of the World

An Introduction to Mathematical Geodesy

Martin Vermeer, Antti Rasila

• Publisher: CRC Press
• Year: 2019
• Print ISBN:
  9780367217730
• eBook ISBN:
  9780429560972
• Topic: Technology & Engineering
• Subtopic: Complex Analysis
• Index: Technology & Engineering

Carl Friedrich Gauss, the "foremost of mathematicians," was a land surveyor. Measuring and calculating geodetic networks on the curved Earth was the inspiration for some of his greatest mathematical discoveries. This is just one example of how mathematics and geodesy, the science and art of measuring and mapping our world, have evolved together throughout history.

This text is for students and professionals in geodesy, land surveying, and geospatial science who need to understand the mathematics of describing the Earth and capturing her in maps and geospatial data: the discipline known as mathematical geodesy. Map of the World: An Introduction to Mathematical Geodesy aims to provide an accessible introduction to this area, presenting and developing the mathematics relating to maps, mapping, and the production of geospatial data. Described are the theory and its fundamental concepts, its application for processing, analyzing, transforming, and projecting geospatial data, and how these are used in producing charts and atlases. Also touched upon are the multitude of cross-overs into other sciences sharing in the adventure of discovering what our world really looks like.

FEATURES

• Written in a fluid and accessible style, replete with exercises; adaptable for courses on different levels
• Suitable for students and professionals in the mapping sciences, but also for lovers of maps and map making

Chapter 1

A brief history of mapping

The ancient Greek knew that the Earth is a sphere. They travelled long distances and wrote detailed travel accounts. Their understanding of the size of the Earth was however fuzzy. Only much later, at the dawn of the Renaissance, narratives started to give way to measurements. Thus, a better understanding started to form of the Earth's true dimensions. At the same time arose a need, a drive to augment the tales from far-away lands with mathematical models based on geography, as well as collect quantitative data on the Earth and the world more generally.

The early Renaissance was the era when, for the first time, geometrically correct maps of the whole world were produced. Drafting these was complicated by the target being on a curved surface, and thus a direct depiction onto the map plane as a scaled image is impossible. The art of book printing, developed in the 15^th century, made the dissemination of documents a lot easier than before. The voyages of discovery and the subsequent military and commercial seafaring — often the same thing — made the drafting of sufficiently precise maps a profitable commercial activity.

Of the map makers of the era, the most famous are Gerardus Mercator, Willem Janszoon Blaeuw, Abraham Ortelius (Ortels) and Jodocus Hondius (Joost de Hondt). It was a time of blossoming of science, erudition and craftsmanship. Also geodetic measurement instruments were developed, and in the Netherlands, Willebrord Snellius¹ , a student of Frisius² (Haasbroek, 1968), executed an extensive triangulation to determine the true size of the Earth. This was a time of change, but also — or perhaps just because of the changes, which were experienced as traumatic — an age of fanatical religiosity and dark superstition.

At the same time, also the true size of the Earth started to become evident. A little later, the Earth started to be studied as one planet among others. Studies were undertaken on how the Earth and other celestial bodies move in their orbits, like parts of a celestial clockwork. Also this research was, back then, a pre-condition to, and an enabler of, seafaring. This was the beginning of an era of ever more precise measurement and mapping, which continues to this day.

1.1 Map and scale

Ideally, a map of the world would be isometric. An isometric map displays the terrain on a reduced scale. Distances measured on the map correspond, directly through the scale, to distances in the terrain. The idea of an isometric map works in mapping a small area (i.e., on a large scale³), but, due to the curvature of the Earth, it is impossible to draft such a plane map for a larger area. Because globes are impractical, we will in this book concentrate on plane maps.

The impossibility of an isometric map can be visualized by thinking of a circle drawn on a sphere, and its radius. We assume here the centre to be also on the surface of the sphere. In the plane, the ratio between the radius of a circle and the length of its circumference is 2π. Because of the assumed isometricity of the map, the same should also be true on the surface of the sphere, when we take the distance between two points to mean the length of the shortest path between them on the surface of the sphere. On the surface of the sphere, however, the ratio is always a different one: the magnitude of the distortion depends on the radius of the circle. Therefore, a map drawn on the plane will always depict part of the surface of the sphere in some map projection that distorts sizes, though from the concept of map scale that is used for maps, one could infer otherwise. The magnitude of the error depends on the extent of the area depicted in the map. In some map projections, like the Mercator projection, the poles are located at an infinite distance from any other point.

1.2 Gerardus Mercator and Mercator's map of the world

Gerard Kremer (1512 – 1594) was born in Rupelmonde near Antwerp. His parents were German merchants, who had fled religious war and persecution of protestants in Flanders. Like many scholars in those days, he latinized his name, and it is better known as Gerardus Mercator, see Figure 1.1.

Figure 1.1: Gerardus Mercator, engraving Nicolas III de Larmessin 1682.

Mercator studied at the University of Leuven, as a student of the famous humanist Macropedius⁴. He was, however, disappointed with the teachings of theology and philosophy, which offered explanations that, in his opinion, did not satisfactorily describe the world and contradicted each other. Mercator traveled widely, returning however back to Leuven, this time as a student of the leading mathematician and astronomer of the time, Gemma Frisius. Frisius had other famous students, like the astronomer Johannes Stadius⁵ and advisor to Queen Elisabeth I of England, John Dee⁶ . (A crater on the Moon is named after Frisius, and another one after Mercator.)

Mercator became familiar with map making while working as a student of Frisius, who had made many significant invent...

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Contents
• Preface
• Acknowledgments
• Acronyms
• 1 A brief history of mapping
• 2 Popular conformal map projections
• 3 The complex plane and conformal mappings
• 4 Complex analysis
• 5 Conformal mappings
• 6 Transversal Mercator projections
• 7 Spherical trigonometry
• 8 The geometry of the ellipsoid of revolution
• 9 Three-dimensional co-ordinates and transformations
• 10 Co-ordinate reference systems
• 11 Co-ordinates of heaven and Earth
• 12 The orbital motion of satellites
• 13 The surface theory of Gauss
• 14 Riemann surfaces and charts
• 15 Map projections in light of surface theory
• A The stereographic projection maps circles into circles
• B Isometric latitude on the ellipsoid
• C Useful relations between the principal radii of curvature
• D Christoffel's symbols from the metric
• E The Riemann tensor from the Christoffel symbols
• F Conformal mappings in spaces of any dimension
• References
• Index