Posts Tagged 'Navigation'

Geodesics on the Spheroidal Earth-II

Geodesy is the study of the shape and size of the Earth, and of variations in its gravitational field. The Earth was originally believed to be flat, but many clues, such as the manner in which ships appear and disappear at the horizon, and the changed perspective from an elevated vantage point, as well as astronomical phenomena, convinced savants of its spherical shape. In the third century BC, Eratosthenes accurately estimated the circumference of the Earth [TM137 or search for “thatsmaths” at].


Geodesic at bearing of 60 degrees from Singapore. Passes close to Quito, Ecuador. Note that it is not a closed curve: it does not return to Singapore.

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Geodesics on the Spheroidal Earth – I

Both Quito in Ecuador and Singapore are on the Equator. One can fly due eastward from Singapore and reach Quito in due course. However, this is not the shortest route. The equatorial trans-Pacific route from Singapore to Quito is not a geodesic on Earth! Why not?


A drastically flattened spheroid. Clearly, the equatorial route between the blue and red points is not the shortest path.

Continue reading ‘Geodesics on the Spheroidal Earth – I’

From Sailing on a Rhumb to Flying on a Geodesic

If you fly 14,500 km due westward from New York you will come to Beijing. The two cities are on the fortieth parallel of latitude. However, by flying a great circle route over the Arctic, you can reach Beijing in 11,000 km, saving 3,500 km and much time and aviation fuel.  [TM124 or search for “thatsmaths” at].


Great circle route from New York to Beijing (gnomonic projection).

On a gnomonic projection (as above) each point on the Earth’s surface is projected from the centre of the Earth onto a plane tangent to the globe. On this map, great circles appear as straight lines.

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Geometry in and out of this World

Hyperbolic geometry is the topic of the That’s Maths column in the Irish Times this week (TM031 or  click Irish Times and search for “thatsmaths”).

Living on a Sphere

The shortest distance between two points is a straight line. This is one of the basic principles of Euclidean geometry. But we live on a spherical Earth, and we cannot travel the straight line path from Dublin to New York: we have to stick to the surface of the globe, and the geometry we need is more complicated than the plane geometry of Euclid. Spherical geometry is central for the study of geophysics and astronomy, and vital for navigation.

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Where in the World?

Here’s a conundrum: You buy a watch in Anchorage, Alaska (61°N). It keeps excellent time. Then you move to Singapore, on the Equator. Does the watch go fast or slow? For the answer to this puzzle, read on. Continue reading ‘Where in the World?’

Shackleton’s spectacular boat-trip

A little mathematics goes a long, long way; in the adventure recounted below, elementary geometry brought an intrepid band of six men 800 sea miles across the treacherous Southern Ocean, and led to the saving of 28 lives. Continue reading ‘Shackleton’s spectacular boat-trip’

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