Happy Pi Day

March 14, 2019  - By

In honor of 3.14.2019, here is what GPS World’s Innovation column editor Richard Langley wrote about π in an article (“A Sideways Look at How the Global Positioning System Works“) nine years ago.

3.1415926…. π. Every nerd’s favorite number. It is the ratio of a circle’s circumference to its diameter in conventional or Euclidean space. We use it, for example, to convert angles measured in radians to degrees (π radians = 180 degrees). π is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats. But we sometimes use an easily remembered fraction, such as 22/7, to get an approximate value. In this case, 3.14. But, if we compute more digits with this fraction, we get 3.1428571…, clearly an incorrect result. A better way to remember π to eight digits is to count the number of letters in each word of the mnemonic “May I have a large container of coffee?”

In computations related to GPS, how many digits of π should be used? It depends. If you are developing your own algorithms and software for modeling GPS observations or determining precise orbits for the satellites, you’ll likely need π to 16 digits for double-precision floating-point calculations. But it would be a mistake to use π to this precision in computing the position of a satellite from the broadcast ephemeris. The GPS interface specification document, IS-GPS-200, specifies a 14-digit value for π (3.1415926535898) in the satellite coordinate computation. Use fewer or more digits, and the resulting satellite coordinates will not be as accurate.

Full article here.

Thank you, Dr. Langley.

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About the Author: Richard B. Langley

Richard B. Langley is a professor in the Department of Geodesy and Geomatics Engineering at the University of New Brunswick (UNB) in Fredericton, Canada, where he has been teaching and conducting research since 1981. He has a B.Sc. in applied physics from the University of Waterloo and a Ph.D. in experimental space science from York University, Toronto. He spent two years at MIT as a postdoctoral fellow, researching geodetic applications of lunar laser ranging and VLBI. For work in VLBI, he shared two NASA Group Achievement Awards. Professor Langley has worked extensively with the Global Positioning System. He has been active in the development of GPS error models since the early 1980s and is a co-author of the venerable “Guide to GPS Positioning” and a columnist and contributing editor of GPS World magazine. His research team is currently working on a number of GPS-related projects, including the study of atmospheric effects on wide-area augmentation systems, the adaptation of techniques for spaceborne GPS, and the development of GPS-based systems for machine control and deformation monitoring. Professor Langley is a collaborator in UNB’s Canadian High Arctic Ionospheric Network project and is the principal investigator for the GPS instrument on the Canadian CASSIOPE research satellite now in orbit. Professor Langley is a fellow of The Institute of Navigation (ION), the Royal Institute of Navigation, and the International Association of Geodesy. He shared the ION 2003 Burka Award with Don Kim and received the ION’s Johannes Kepler Award in 2007.