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 to discoveries of these great minds. Mathematics is no longer a system to count objects, as this 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 the tool to attempt to challenge the current record of sailing around the world. It will not include the vector components entirely. Needless to say, the majority of the trigonometry used in this examination will be explained just enough to be understandable for the common math enthusiast. The record breaking component, is only a form in which this sub-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 to the 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 not taking into consideration certain external factors that Loïck Peyron may have encountered when he broke the record. Therefore, if the condition were just right, and a constant speed of 40 knots was kept consistent throughout, the results from this examination would be valid.
Table of Contents
A. Acknowledgment
B. Introduction
C. Spherical Trigonometry
D. Record Breaking Attempt
E. Conclusion
F. Works Cited
Research Objective and Scope
The primary objective of this paper is to propose a hypothetical method for attempting to break the current world sailing record by applying principles of spherical trigonometry to navigation. The study explores how mathematical models can optimize sailing routes and calculates the time required to circumnavigate the globe at a constant speed, while acknowledging the limitations of a theoretical model compared to real-world maritime conditions.
- Application of spherical trigonometry to navigation
- Use of the Haversine formula to calculate great-circle distances
- Analysis of Loïck Peyron’s record-breaking route and speed parameters
- Mathematical modeling of a hypothetical circumnavigation route
- Comparison between theoretical time calculations and existing sailing records
Excerpt from the Book
C. Spherical Trigonometry
It takes basic knowledge in geometry to know that the shortest distance between two points is a straight line; but on the surface of a sphere, there are no such things as straight lines. 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).
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 are true: 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 each individual spherical angle is less than 180°.
Summary of Chapters
A. Acknowledgment: This section clarifies the hypothetical nature of the study and notes that the research is not intended to diminish the achievements of the current record holder.
B. Introduction: The introduction provides a brief historical context of non-Euclidean geometry and defines the mathematical framework for the proposed navigation method.
C. Spherical Trigonometry: This chapter defines the fundamental concepts of spherical geometry and explains the properties of spherical triangles and coordinate systems.
D. Record Breaking Attempt: This section details the practical application of the Haversine formula to calculate distances between various points on a theoretical global route.
E. Conclusion: The conclusion evaluates the findings, confirming the theoretical possibility of the route while noting that external real-world factors were excluded.
F. Works Cited: This section lists the academic and digital resources utilized to support the mathematical calculations and theoretical arguments.
Keywords
Spherical Trigonometry, Non-Euclidean Geometry, Haversine Formula, Great Circle, Sailing Record, Navigation, Mathematical Modeling, Latitude, Longitude, Circumnavigation, Spherical Triangles, Global Navigation, Distance Calculation
Frequently Asked Questions
What is the core focus of this research paper?
The paper explores the use of spherical trigonometry as a mathematical tool to plan a theoretically faster route for circumnavigating the Earth by sailboat.
Which specific mathematical tools are used for navigation?
The author primarily utilizes the Haversine formula, which calculates the distance between two points on a sphere based on their latitude and longitude.
What is the primary objective of the investigation?
The objective is to propose a hypothetical method to challenge the existing sailing world record by calculating a more efficient path using great-circle arcs.
What methodology does the author employ?
The author converts geographical coordinates into radians and applies spherical trigonometry laws, specifically the Haversine formula, to determine segment distances along a chosen route.
What content is covered in the main body?
The main body explains the principles of spherical triangles, establishes a reference table of coordinates, and performs step-by-step distance calculations for the route segments.
Which keywords best characterize this work?
Key terms include spherical trigonometry, Haversine formula, global navigation, and mathematical modeling in the context of maritime records.
How does the author treat external factors like climate?
The author explicitly excludes climate, ocean currents, and non-spherical Earth irregularities, treating the study as a controlled, purely mathematical, and hypothetical exercise.
Why was the record held by Loïck Peyron chosen as the benchmark?
The author uses Peyron’s record-breaking speed of 40 knots as the standard constant velocity to test whether the calculated route could theoretically outperform his total time.
- Arbeit zitieren
- Juan Casiano (Autor:in), 2013, A Hypothetical Method of Attempting to Break the Current Sailing Record Around the World using Spherical Trigonometry, München, GRIN Verlag, https://www.grin.com/document/430763
