# MATHEMATICS, MAP MAKING AND NAVIGATION

YOU ARE HERE BUT WHERE?

Map Making and Navigation have long depended on Mathematics, early results go back to the Greek geometers. But this is an area that is not so well known in the general mathematics community and most mathematics students do not see these applications at all. The AMS volume Portraits of the Earth: A Mathematician looks at Maps is based on a course intended for mathematics majors

Nathaniel Bowditch-1, Nathaniel Bowditch-2, was an early American mathematician. Updated versions of his The American Practical Navigator are well-known to boaters and those who go to sea. . Bowditch was not only an accomplished navigator he contributed greatly to the art and science of navigation. Galileo worked on problems relating to navigation.
Position on the earth is normally given in terms of Latitude and Longitude It was long known that position, i.e., latitude and longitude, could be determined by taking "sights" on stars, the sun, the moon or some of the planets but it was also necessary to know the "time". The book Longitude" chronicles the story of the famous English clockmaker (also made into a TV program for Public Television).
One minute of latitude is one nautical mile but one minute of longitude is not (the meridians get closer together as you approach one of the poles). The Mercator projection not only provides a way to construct a "flat" map, it adjusts for this discrepancy. Mercator needed to use something like an integral before calculus had been "invented". He not only made the meridians (of longitude) and parallels (of latitude) appear as straightlines intersecting at right angles he accounted for the change noted above. The solution was to spread out the parallels, the amount of spreading is related to a function known as the Gudermann. A very important feature of this projection is that shortest distance paths (geodesics) are straight lines.
In ordinary day to day activities we don't use "latitude and longitude", we need other coordinate systems such as UTM (Universal Transverse Mercator), online conversion. Fortunately GPS systems can do even better. The Mathematics of GPS is found here.For a more elementary discussion of mathematics and GPS see Mission Mathematics
REFERENCES
• Bowditch,N., American Practical Navigator Pub. No. 9 Defense Mapping Agency
• G.D.Dunlap and H.H. Shufeldt, Dutton's Navigation and Piloting, Naval Institute Press
• Hannah, J., Map Projections: Their Development and Use in New Zealand (Seminar Notes) Professional Development Committee, New Zealand Institute of Surveyors, Auckland NZ, August 1984 [This booklet contains a very clear and concise mathematical derivation of the conformality conditions (the Cauchy-Riemann equations). It also contains an excellent short discussion of the types of possible distortions in a map projection, and pratical formulae for calculation, especially for the New Zealand mapping grid.]
• Pijls, Wim, Some properties related to Mercator Projection", The American Mathematical Monthly 108 (2001) 537-543
• Sachs, J.M, "A Curious Mixture of Maps, Dates, and Names", Mathematics Magazine 60 (1987) 151-158
• Tuchinsky, Ph.M., Mercator's World Map and the Calculus, UMAP Modules-tools for Teaching, Birhauser 1981

More to come April 2003