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A Hypothetical Method of Attempting to Break the Current Sailing Record Around the World using Spherical Trigonometry

Ausarbeitung 2013 27 Seiten

Mathematik - Geometrie

Leseprobe

Table of Contents

Abstract

Table of Contents

A. Acknowledgment

B. Introduction

C. Spherical Trigonometry

D. Record Breaking Attempt

E. Conclusion

F. Works Cited

Abstract:

Ever since the creation of math, mathematicians have attempted to extend, or challenge the work of another mathematician with the intent to try and disprove their discoveries. The applications of math we now use to solve the problems of life, are due todiscoveries of these great minds. Mathematics is no longer a system to count objects, asthis examination will attempt to : Propose a Hypothetical Method of Attempting to Break the Current Sailing Record Around the World using Spherical Trigonometry.

The scope in which this examination will take into account is that of spherical trigonometry at its sole. Situations will be adjusted to make spherical trigonometry thetool to attempt to challenge the current record of sailing around the world. It will notinclude the vector components entirely. Needless to say, the majority of the trigonometryused in this examination will be explained just enough to be understandable for thecommon math enthusiast. The record breaking component, is only a form in which thissub-branch of spherical geometry can be applied in the real world.The result of this examination ended with a success. The method taken resulted in breaking the current record held by Loïck Peyron within an astonishing 45 days 13 hours

42 minutes and 53 seconds. But from this examination it was derived that, with respect tothe given points, that you could go around the world in 20 days 17 hours 5 minutes and 17 seconds when going at a speed of 40 knots. However, this result was attained by nottaking into consideration certain external factors that Loïck Peyron may have encounteredwhen he broke the record. Therefore, if the condition were just right, and a constant speedof 40 knots was kept consistent throughout, the results from this examination would bevalid.

A. Acknowledgment

Before we can attempt to hypothetically break a record, some distinctions have to beaddressed. Under no circumstances, is this examination defaming the current recordholder, Loïck Peyron, by providing an alternative route, or method. External factors suchas climate, the Earth not being perfectly spherical(radius of 6,371 Km will be used) andboundaries will not be taken into consideration, due to the uncertainties they mayimplicate in the examination; average speed of sailboat under regular circumstances willbe used. This examination, is only a hypothetical attempt; the routes presented during thisexamination have not been executed in the real world or will ever be by the author of thisexamination.

B. Introduction

Euclid is the father of geometry and, though he might not have been the one who discovered it, he is accredited for it because he develop the first comprehensive deductivesystem. Non-Euclidean geometries are based on Euclid’s postulates, but respectively usetheir own version of the parallel, fifth, postulate. Non-Euclidean Geometry is accreditedto four people: C.F. Gauss (1777-1855) for discovering the possibility of Non-Euclideangeometry, N. Lobachevsky (1792-1856) for the discovery of hyperbolic geometry, J.Bolyai (1802-1860) for adding onto Gauss’ and Lobachevsky’s work and B. Riemann(1826-1866) for his development in spherical, or elliptical, geometry. For thisexamination the sub-branch of Non-Euclidean geometry, spherical trigonometry, will be used to, Propose a Hypothetical Method of Attempting to Break the Current Sailing Record Around the World using Spherical Trigonometry.

In the world, especially on the surface of the Earth, lines are not always straight. Forexample if an airplane from California was heading to China, the route it would takewould not be a straight line; if so, the airplane would have to cut through the Earth.Spherical Trigonometry, is a sub-branch of Bernhard Riemann’s spherical geometry. Justlike in Euclidean geometry there is trigonometry, there is also for when the plane isspherical. Not just does this implicate that there is a difference in formulas, but also howproblems will be approached.

Before the attempt of hypothetically breaking a world record, it is important to analyze and study the record holder’s, Loïck Peyron, route, equipment, speed etc. Loïck Peyron, in his world-record-breaking run, used a Zoulou Extreme 40, which has anaverage speed of 40 knots(~46 mph, 74 km/h). The route taken by Loïck Peyron, beganand ended in an imaginary line between the Créac'h lighthouse on Ouessant (Ushant)Island, France, and the Lizard Lighthouse, UK located on the coordinates 48°27′34.23″N,5°7′45.4″W. It took him 45 days 13 hours 42 minutes and 53 seconds to break the record,hence this is the time to beat.

C. Spherical Trigonometry

It takes basic knowledge in geometry to know that the shortest distance between twopoints is a straight line; but on the surface of a sphere, there are no such things as straightlines. The shortest distance between two points on a sphere is the arc of a great circle passing through those points. “A great circle is defined to be the intersection with a sphere on a plane containing the center of the sphere, [such that of figure-1 and 2]. If the plane does not contain the center of the sphere, its intersection with the sphere is known as a small circle, [such that of figure-3 and 4]”(Dhillon).

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Figure-1

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Figure-2

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Figure-3

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Figure-4

The main concept that this investigation will deal with, is with the triangle; more specifically, spherical triangles. In Euclidean geometry when we connect three points on a plane using the shortest possible route, it will create a triangle. By analogy, in Non-Euclidean geometry when we want to connect three points on the surface of a sphere, “we would draw arcs of great circles and hence create a spherical triangle”(Dhillon).Because spherical triangles do not necessarily have to look like planar triangles, a triangle on the surface of a sphere is only a spherical triangle if all the following properties aretrue: the three sides are all arcs of great circles, any two sides are together longer than the third side, the sum of the three angles is greater than 180° (π radians), and if eachindividual spherical angle is less than 180°.

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Figure-5

As this examination will deal with the surface of the Earth, it is important to knowhow spherical trigonometry can be expressed. Two coordinates, latitude and longitude,are used to pin-point any location on the surface of the Earth. The longitude of a point, ismeasured angularly from east to west of the Greenwich Meridian(great circle that passesthrough both poles of the Earth) with respect to the equator. If the longitude is east of the Greenwich Meridian, then it is a positive angle. If the longitude is west of the Greenwich meridian, then it is a negative angle. The latitude of a point, is measured angularly from north to south of the equator with respect to the Greenwich Meridian. If the latitude is north of the equator, then the angle is positive. If the latitude is is south of the equator, then the angle will be negative.

Laws pertaining to Euclidean geometry, more specifically trigonometry, still exist in Non-Euclidean. Laws like the law of sine and cosine still, accurately, can be used to solve problems. There are some distinction the law of cosine, as it does not look entirely like the Non-Euclidean version, but the law of sine is extremely similar. Where print letters are the sides of the triangle and roman being the interior angles.

The law of Cosine is represented as such:

cos(a) cos(b) cos(c) cos( )sin(b) sin(c)

The law of Sine is represented as such:

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D. Record Breaking Attempt

To begin, we must first decide on the route that will be taken. The route taken by Loïck Peyron, began and ended in an imaginary line between the Créac'h lighthouse on Ouessant (Ushant) Island, France, and the Lizard Lighthouse, UK located on thecoordinates 48°27′34.23″N, 5°7′45.4″W. Latitude and Longitude’s standard notation issexagesimal (shown above), but for the purpose of this examination, they will be changed into decimal degrees, and then into radians.

The first step in doing so, is to change the seconds portion of the latitude or longitudeinto a decimal of minutes. The second step, is to set the minutes over sixty and it equal to“x”. Next, add the “x” value to the degrees. Lastly, multiply the degrees by π ,and divideby 180. For example, the coordinates of the Créac'h lighthouse on Ouessant (Ushant)Island, France, and the Lizard Lighthouse, UK located on the coordinates 48°27′34.23″N,5°7′45.4″W will be changed into decimal degrees, then into radians. Varying, wether thelatitude and longitude are going north, south, east or west put the correct + or - sign; asthis will be crucial in the subtraction.

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This process will be repeated with the rest of the latitudes and longitudes of the points remaining. Table-1 displays the given points of reference to guide the boat , with their respective latitude and longitude in both sexagesimal and decimal degree form (turned into radians).

The selection process for the points chosen was not a specific one. I merely looked ata globe with the destination and pin pointed points what seemed to me the mostreasonable, yet shortest to arrive at the destination. Figure - 7 represents the points on Table - 1 plotted.

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Figure-7

*Not to scale

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Details

Seiten
27
Jahr
2013
ISBN (eBook)
9783668748330
ISBN (Buch)
9783668748347
Dateigröße
641 KB
Sprache
Englisch
Katalognummer
v430763
Note
3.00
Schlagworte
hypothetical method attempting break current sailing record around world spherical trigonometry

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Titel: A Hypothetical Method of Attempting to Break the Current Sailing Record Around the World using Spherical Trigonometry