Spherical Geometry

Published: 2021-07-07 00:11:01
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Category: Mathematics

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The curiosity to learn more about the universe and the way it works has done wonders for the human race. The travelers and researchers over the time have been working to find the shortest paths between their starting points and destinations. The perspective of the flat earth was debunked almost a millennia back. Geocentric earth concept lasted up until the end of seventeenth century. However, Friedrich, Gauss, and Riemann, working independently, were able to decipher the codes of shortest paths on a spherical surface a couple of centuries back from today. A century later, their theories were reinforced by Einstein’s Theory of General Relativity. The accomplishment from these scientists resulted in the creation of an academic discipline called Geodesics.Although the current understanding of the shape of the earth is best to describe the geodesics, yet the flat earth concept is useful too to estimate the complex concepts in a convenient function. However, by observing a flat-earth map one can wrongly perceive the distance between two points since flat surface does not take into account the earth’s warp. For instance, two places that appear at the same latitudinal coordinate may not really be parallel to the equator line. This is due to the fact that a flat map gives a perspective without considering curvature. This is why, airplanes appear to be diverting from the shortest path but, in reality, they are taking the shortest path. Some places that are horizontally on the same plane may appear to be drifted far apart at a flat-earth map.In order for an observer to find the shortest path on the globe, an elastic string is a decent choice for estimation. A string can be placed between the points under observation at its maximum stretch. The path traversed by the string would be the shortest route. Apart from the shortest route, this approach goes for other routes too that are bound to traverse the entire earth before ending up at the destination. However, the string should not be subject to any weights, hurdles or biases since it will not give the correct measurement for the straightest course. If the biases are added and then the string is released again, it will reform its original orientation which establishes the fact that it is the best path as compared to the ones around it. Videos from renowned mathematician Polthier visualize this effect. Furthermore, one of the videos also argues that a rider would not have to realign his bicycle every now and then while riding on a straight geodesic path.These concepts of differential geometry were further enhanced by Albert Einstein’s general relativity theory which proves the bending of light from a distant star due to sun’s gravity. This discovery reveals the fact that light follows geodesics too. In fact, every object influences other objects in its domain with the warping effect.Up until now, this document has discussed the shortest path at a regular surface. However, in reality, one rarely experiences the regular surfaces. Usually, one needs to find out the route at an arbitrary surface. For instance, a vehicle traveling from one part of a city to another would most likely confront turns, roundabouts, bridges and underpasses at different routes. To find the shortest path, one cannot simply use the string analogy. Thus, the path would be divided into a large number of tiny but straight paths on which string analogy can be applied. Measuring the path of each smaller routes and combining them would find the estimated total distance of each route. This is how online maps and route finder applications work.Works CitedSTROGATZ, STEVEN. “Think Globally”. Opinionator.Blogs.Nytimes.Com, 2010, https://opinionator.blogs.nytimes.com/2010/03/21/think-globally/?hp. Accessed 22 Apr 2018.

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