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 irishtimes.com].

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.

The great French philosopher René Descartes had a brilliant idea that enabled him to give a name to every point in a plane. He fixed one point, called the origin, and drew two lines – the axes – through it and at right angles to each other. The distances of any point from these axes give a pair of numbers, (x, y), which we call the cartesian coordinates. They act to label every point, identifying it uniquely. The theorem of Pythagoras then gives the distance from the origin to the point: the square of this distance is the sum of the squares of x and y. Descartes’ idea had far-reaching consequences and served as a grand unification of algebra and geometry.

Riemann’s Geometry

Bernhard Riemann discovered a sweeping generalization of the Pythagorean theorem. He considered points in a multi-dimensional space and he allowed this space – or manifold – to be curved, not flat like the Euclidean plane. Riemann then wrote an expression for the distance between any two points in his manifold. This expression was analogous to the theorem of Pythagoras, but much broader in scope.

A simple way to visualize Riemann’s curved manifold is to think of the Earth’s surface. We learn in Euclidean geometry that a straight line is the shortest distance between two points. So, the shortest route from Dublin to Melbourne is a line passing close to the Earth’s centre. But traversing this path poses certain difficulties. In practice, we must travel upon, or slightly above, the Earth’s surface. The shortest route is then a great circle: its centre is at the Earth’s centre, but we travel via an arc along the surface. This path is called a geodesic.

The Flattened Earth

However, the Earth is not exactly spherical: the closest simple mathematical shape is a flattened sphere or oblate spheroid. To determine the shape precisely, the French Academy of Sciences organized a geodetic expedition in 1735 to make measurements of the meridian arc in polar and tropical regions. This confirmed that the Earth was slightly flattened like an orange, not elongated like a lemon [see previous post on the shape of the earth].

Excepting any two antipodal points, the geodesic, or minimal distance route, between any two locations on a sphere is uniquely determined. However, when we make allowances for the slight flattening of the Earth, something curious happens: the shortest route is no longer unique. For example, Singapore and Quito both lie on the Equator but, due to the bulging Earth, the equatorial trans-Pacific route between them is not the shortest route: there are two shortest routes, one to the north of Hawaii and one to the south over Australia.

The effect of non-sphericity is small but, with costly aviation fuel, it is important for long-haul flight planning. It also has consequences for astronomy, time measurement, mineral surveying, determining the positions of artificial satellites and GPS.

Sources

${\bullet}$ Karney, CFF. Drawing geodesics on Google Maps [  Website  ]

${\bullet}$ Previous post on geodesics on