Maps and mathematical cartography

SUMMARY: Scales and projections are used to display geographic features on maps. 

The word “map” is the name given to any representation of the Earth’s features—natural and artificial—usually on a plane using a given scale and map projection. In scientific and mathematics applications, the term “map” is more broadly interpreted. The purpose of a map is to register and transmit information about those features and the spatial relations between them. 

A common characteristic of all maps is that they are reduced and conventional representations of reality, which makes them significantly different from an aerial photograph. While an aerial photograph depicts all the physical objects that a sensor could detect and register, a map is a selection of natural and artificial objectsvisible and invisiblechosen to fit the cartographer’s purpose and the limits imposed by the available space. These objects are represented on maps in a conventional way by means of symbols; this is not the case with photographs, in which they are depicted by the visual image they present when viewed from above by the sensor. The symbols on a map are designed to categorize features by type and to optimize the document’s legibility. Very often, their size is not proportional to the size of the objects they represent. For example, roads are symbolized by lines of variable thickness and pattern, often much larger than the corresponding width of the actual roads, since representing them to exact scale would often make them too thin, even invisible. In other cases, such as with cities, features are symbolized by punctual symbols whose color and shape depend on the classification scheme chosensuch as administrative status or population.

Maps are usually classified in three main categoriesgeneral reference maps, thematic maps, and charts. A general reference map depicts generic geographic information of various types considered useful to a large spectrum of users. This information may include topography, political and administrative borders, and land cover. The best example of a general reference map is the topographic map. A thematic map, on the other hand, represents the geographic distribution of a specific theme or group of themes such as geological features, population, or air temperature. A chart is a special type of map designed to support navigation, either maritime with nautical charts or aerial with aerial charts. 

History

Maps were first made by the ancient civilizations of Europe and the Middle East several centuries before the Common Era. One of the oldest known is a Babylonian clay map of the world c. 600 B.C.E., now kept in the British Museum. Though it is documented in the testimony of Ptolemy of Alexandria (c. 90–169 C.E.) and others that maps were drawn in Greece as early as the seventh century B.C.E., none are known to have survived. However, several medieval manuscript maps have survived that represent the ecumene (the known inhabited part of the world around the Mediterranean basin). Few had any practical purpose, and most were symbolic representations inspired by religion and myth rather than by reality. In his Geography, published for the first time in the second century C.E., Ptolemy describes three map projections in detail and presents a list of more than 8000 places in the ecumene, defined by their latitudes and longitudes. 

This list permitted others to redraw the maps that may have accompanied the original text once the work was translated into Latin and disseminated throughout Europe during the fifteenth century. The publication of several editions of Geography did much to bring about the rebirth of scientific cartography. By this time, nautical charts had already been used to navigate in the Mediterranean for at least two centuries. And while terrestrial cartography quickly adopted the geographic coordinates and map projections proposed by Ptolemy, nautical charts remained based on the magnetic directions and estimated distances observed by pilots at sea. Still, these representations were of astonishing accuracy and detail compared with the traditional maps of the time. 

It is now known that the first nautical charts, commonly known as “portolan charts,” were constructed in the first half of the thirteenth century, probably in Genoa, after the introduction of the magnetic compass and the adoption of the decimal system in Europe. This basic model continued to be used in nautical cartography for a long time, though much improved by the introduction of astronomical navigation during the fifteenth century. The resulting modality, based on observed latitudes and magnetic directions, became known as the “latitude chart”or “plane chart”and played a fundamental role in the discoveries and maritime expansion periods. In 1569, an important world map specifically conceived for supporting maritime navigation was constructed by the Flemish cartographer Gerard Kremer (1512–1594), better known by the Latinized name of “Gerardus Mercator.” Contrary to traditional portolan charts, this map was based on the latitudes and longitudes of places and represented all rhumb lines (lines of constant course) as straight segments making true angles with the meridians. 

Though Mercator did not explain how the planisphere was made, a geometric method was most likely used. The mathematics of the projection is not trivial and its formalization had to wait until after calculus was developed, more than one century later. As for its full adoption as a navigational tool, that did not occur until the middle of the eighteenth century, when the marine chronometer was invented and longitudes could finally be determined at sea. 

Mathematical Cartography

Maps may depict only a small part of the whole surface of the Earth. The word “scale” means the quotient between a length measured on a map and the corresponding distance measured on the Earth’s surface. Because it is not possible to represent the spherical surface of the Earth in a plane without distorting the relative position of the places (and thus, the shape of all objects), the scale of a map is not constant, always varying from place to place and, in the generality of cases, also with the direction. In large-scale maps, like the plant of a city or the topographic map of a small region, these distortions can be ignored and the scale considered constant for most practical purposes. That is not the case when a large area of the Earth’s surface is represented, like in a planisphere or a map of a whole continent. Here shapes may be strongly deformed and the scale varies significantly from place to place. Measurements made on those maps with the purpose of evaluating distances between places, using their graphical or numerical scale, are only approximations, as the scale strictly applies only to certain parts of the maps— like the central meridian or parallel—and their use in the other regions may lead to very large errors. 

“Map projection” refers to any systematic way of representing the surface of the Earth on a plane. The process consists of two independent steps. First, one has to replace the irregular topographic surface, with all its mountains and valleys, with a simpler geometrical model, usually a sphere or an ellipsoid where a system of geographic coordinates (latitude and longitude) is established. Second, one has to project that model onto a plane surface. This step may be accomplished by some geometric construction or by a mathematical function that transforms each pair of geographic coordinates latitude (j) and longitude (l) into a pair of Cartesian coordinates x and y, defined on the plane. Depending on the purpose of the map, there are many different map projections to choose from. Knowing that none of them conserves the relative position of all places on the surface of the Earth, the choice is usually driven by the type of geometric property one wants to preserve. For example, equivalent or equal-area projections conserve the relative areas of all objects and are typically used in political maps. Conformal projections conserve the angles around any point on the map (the scale does not vary with direction), as well as the shape of small objects, and are utilized in nautical charts and topographic maps. Equidistant projections conserve the scale of certain lines and are used whenever one wants to preserve distances measured along those lines. This is the case of the azimuthal equidistant projection, where distances measured from the center of the projection along all great circles are conserved. This property is useful, for example, for quickly determining the distance of any place in the world measured from a chosen location. 

However, it is not possible for a map projection to have all these properties at the same time, and the conservation of some properties is usually accompanied by significant distortions of the others. A significant example is the Mercator projection (which is conformal), where all rhumb lines are represented by straight segments making true angles with the meridians. However, the scale increases with latitude in this projection, strongly affecting the proportion of the areas. The branch of cartography dealing with map projections is known as “mathematical cartography.” Though some map projections have been well known since remote antiquity, when they were often used for representing the sky, a more formal approach became possible only after the development of calculus. The most important contributions in the formalization of mathematical cartography were those of Johann Heinrich Lambert (1728–1777), Joseph-Louis Lagrange (1736–1813), Carl Friedrich Gauss (1777–1855) and Nicolas Auguste Tissot (1824–1897). 

Computers and geographic information systems have made it possible for previously unforeseen numbers of users to produce good-quality maps tailored to their specific needs and at a reasonable cost. They also allow scientists and mathematicians to map increasingly complex systems and concepts, such as the universe and the World Wide Web. They can also often render in three dimensions and beyond. In mathematics, maps can be used to alternatively express functions or connect mathematical objects. In conceiving those systems, as well as in acquiring the geographic data necessary to construct the representations within, mathematics continues to play a fundamental role. 

Bibliography

Brown, Lloyd A. The Story of Maps. Dover Publications, 1977.

Bugayevskiy, Lev, and John Snyder. Map Projections. A Reference Manual. Taylor & Francis, 1995.

Ehrenberg, Ralph E. Mapping the World: An Illustrated History of Cartography. National Geographic, 2005.

Snyder, John. Flattening the Earth. University of Chicago Press, 1993.

Zuravicky, Orli. Map Math: Learning about Latitude and Longitude Using Coordinate Systems. Rosen Publishing, 2005.