Spherical Geometry And Its Applications
| Main Author: | Marshall A. Whittlesey |
|---|---|
| Format: | Book xi, 335 hlm |
| Bahasa: | eng |
| Terbitan: |
CRC Press
, 2020
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| Subjects: | |
| Online Access: |
http://library.sttalhidros.web.id//index.php?p=show_detail&id=882 http://library.sttalhidros.web.id//lib/minigalnano/createthumb.php?filename=images/docs/9c424753bf51d753f9683accdb9d984f.jpg.jpg&width=200 |
Daftar Isi:
- It has been at least fifty years since spherical geometry and spherical trigonometry have been a regular part of the high school or undergraduate curriculum. It is an unusual mathematics program that has a course in them today, except as a topic in a survey of geometry. This work is an attempt to bring a comprehensive coverage of spherical geometry and its applications to a modern audience that is unfamiliar with it. The study of geometry on the sphere dates back at least two thousand years. Educated people have long understood that the earth is round, and use of spherical geometry has been valuable in navigation, surveying, and other work involving understanding the nature of the earth. It is also apparent to the nighttime observer that the stars and planets in the sky appear to lie on a sphere about the earth. Their positions can be used to keep track of time and seasons. A traveler can also use them to keep track of his/her position on the surface of the earth. A basic fact in the field of navigation is that the angular elevation of the North Star (Polaris) above the horizon is the same as the observer's latitude. But if that star is not visible, other stars can be used to determine position with methods that are more complex. More recentiy. spherical geometry has found use in the field of plate tectonics. In the twentieth century it was shown that the surface of the earth is covered by a series of fairly rigid plates which move relative to each other. These motions cause garthguakes and volcanoes. Spherical geometry helps us understand how these plate movements work. Lastly, spherical geometry has found application in the field of crystallography. Crystals are objects in which atoms or molecules are carefully arranged im a regular pattern. Typically, these patterns include both lines and planes of atoms and molecules. Measurement of the angles among these lines and planes on a crystal yields clues as to the internal molecular structure of the crystal. Internal structure can also be studied by studying the pattern of scattering of an x-ray fired into the crystal. Spherical geometry can be helpful in analyzing these patterns. Hopefully, these applications will persuade the reader that spherical geomretry is worth mastering. My reasons for writing a new book on the subject are several. It is intended to be a comprehensive coverage of spherical geometry and its applications in a mathematically rigorous manner. I think that Sphcrical geometry is perhaps a bcttcr route by which to introduce a student to an axiomatic system of non-Euclidean ideas, since a number of facts about gcometry on the two-dimensional sphere differ notably from those with which the student is familiar in plane geometry. The applications of spherical gcometry are also guite accessible, making it easier to persuade the student that geometic on a curved surface has some value. All mathematicians should know something about spherical geometry, but for the high school teacher, broad knowledge of the applications of mathematics in the natural sciences is particularly important. Spherical geometry was a significant part of the mathematics curriculum until the 1950s. Many standard books in trigonometry in this period include topics in both plane and spherical trigonometry. Its use in navigation was probably important enough for it to be regarded as worth teaching to a general audience. But high school mathematics became increasingly geared toward calculus after the 1950s, and spherical geometry is difficult to fit in if One wishes to learn calculus in high school. When I began working on this book, there appeared to be no recent book in English covering spherical geometry exclusively. In the meantime, Glen Van Brummelen published his book (VB2012). His book covers much more of the history of the subject than mine, but avoids a number of technical points in order to focus on the overall beauty of the subject. I attempt to fill in all technical details but spend less time on the history of the subject. For classroom use, the instructor might find either of these approaches better, depending on the emphasis of the course. This book is intended as a course in spherical geometry for mathematics majors. Prereguisites include knowledge of plane and solid Euclidean geometry. trigonometry, and coordinate geometry. The properties of all the trigonometric functions and their inverses are essential, including identities such as the double-angle and half-angle formulas, sum-to-product formulas, and the laws of sines and cosines. The student should also be familiar with methods of logic and formal proof similar to that obtained either in a proof-based high school geometry class or a course in transition to proof-based mathematies. It helps if the student is familiar with modular arithmetic, as it is necessary to add angles modulo 27. These prereguisites are enough to understand most of the book. But I do use other more advanced methods when convenient. I make brief use of Taylor series to compare the spherical Pythagorean theorem to the planar Pythagorean thcorem. 1 make some use of calculus and the logarithm function in discussion of the Mercator projection. In Chapter 7, it helps to be familiar with the notions of mappings and related ideas such as domain, range, and what it means for a mapping to be injective or surjective. 1 assume some familiarity with basic Incar algebra in the sections on crystallography and the stercographic projection. In Chapter 8, it helps if the student is familiar with the complex numbers and vectors in three dimensions, including the dot and cross product and their properties. My approach here is to motivate the basic properties on the sphere as a three-dimensional object informally with pictures in Chapter 2. Here the idea of the great circle is introduced as the interseetion of a sphere with a plane through its center. Distance on the sphere is mcasured via the central angle of an arc. Angles between grcat circles are measured via the dihedral angles of the planes containing the circles. I also discuss how to cakculate the surface area of a sphere with clementary methods. In Chapter 3, I create an axiomatic system for spherical gcomectry similar to that sometimes used in plane gconietry: first we have axioms involving incidence, then involving distance, and then involving measure of angles. These axioms avoid reference to the sphere as a subject of three-dimensional space. I feel this exercise is good for a class where the instructor wishes to teach proof in a geormnetric setting. I have tried to set up an axiomatic system for spherical geometry with as few axioms as possible. The usual problem arises: one can spend a lot of time proving propositions that are not very interesting! The instructor should feel free to skim over sections 9 and 10 in a first course and simply take for granted propositions there which seem clear to the student. For example, I have avoided studying the proofs of Proposition 10.8 and Proposition 10.9 in an introductory class. In section 11, I prove the basic results about spherical triangles. In section 12, I prove the theorems about congruence of triangles. Here the differences between plane and spherical geometry begin to emerge: some of the theorems are the same, but some are dramatically different. In section 13, I delve into inegualities in spherical triangles. Again, some theorems are like those for plane triangles and some are not. A highlight of this section is what I call the spherical exterior angle theorem: that the measure of an exterior angle of a spherical triangle is less than the sum of the measures of the opposite interior angles but greater than their difference. While these inegualities are not really new I am not aware that any other work states this useful theorem in this form. Lastly in section 14, I discuss the areas of spherical triangles and polygons. Chapter 4 considers the main formulas of spherical triangle trigonometry and introduces their basic applications. Tools from three-dimensional geometry are avoided. Chapter 5 shows how to use spherical trigonometry in the field of spherical astronomy. The student learns about coordinate systems for the sky and how to change coordinates. I then discuss the problem of determining the time when the sun and stars rise and set. This problem is more complex than most people would guess. It is not too difficult to obtain answers that are within ten minutes or s0 of the actual time, but answers that are within a minute of the actual time reguire considerably more effort and can be omitted from a conrse. Chapter 6 apphes spherical trigonometry to polyhedra in three dimcnsions. I feature the formula for the angle between faces in a Platonic solid. I also brieflv show how to use that formula to determine the number of fowdimen-sional regular polytopes. In Chapter 7. I discuss the mappings of a sphere to itself and projections toa planc. Here is where I discuss the application in platc tectonics. In Chapter 8 Ishow how to prove thc main formulas for spherical triangles With guaternion and vector methods. I think this chapter helps feature th, Guaternions as a tool in thrco-dimensional geometry that most math major Are not exposcd to have tried to write a book by which one can understand spherical geom. etry from various levels of background knowledge. But I also make clear in the text that more sophisticated tools are sometimes appropriate, especially Where I sliow how to use thhe guaternions to streamline proofs and broaden our perspective on the subject. In the hope of appealing to the largest audience possible, I have taken the step of presenting several proofs of some theorems. Most modern approaches to spherical geometry include considerable usage of vectors, and while this has some advantages, intuitive motivation for certain theorems is often lost. As an example, vector proofs of the spherical law of sines often start with a guadruple vector product identity that must be checked. For a theorem as intuitively simple as the law of sines, this seems like overkill. As with many synthetic proofs in plane and spatial geometry. some synthetic proofs have logical gaps, or extra cases, which the reader is invited to fill in. Different approaches to a subject often offer different insights. The text features many exercises inspired by propositions from an ancient text on spherical geometry called the Sphaerica of Menelaus of Alexandria. This text dates to about 100 AD and originally appeared in Greek. That text has been lost but it was translated and reworked into Arabic in various versions. These are the oldest versions of the text that we now have. My student Rani Hermiz translated a manuscript of Sphaerica into English. (See (He2015).) That translation appears online at the web site of the library of California State University San Marcos. All my references to propositions in Sphaerica use the numbering system used in (Kr1936| and (He2015). A critical translation is now available in English: see (RP2017). I hope readers will find the content of the ancient propositions intriguing. It should be clear that these ancient mathematicians were extremely capable people who could do much with limited mathematical technology. Many of the propositions seem to appear in no modern work. There are a number of online resources in spherical geometry. John Sullivan of the University of Illinois has a Java applet for demonstrating certain features in spherical geometry such as parallel transport. James King of the University of Washington has a web page with many links to spherical geometry pages. Tevian Dray of Oregon State University has a spherical drawing program Sphcrical Eascl. The web site for the U.S. Naval Observatory and www.timcanddato.com provide resources for determining rise and set for varjous celestial ohjects including the sun and the moon. The Department of Mathematics and Statistics at Saint Louis University has an online feature of mathematics and the mt of M.C. Escher which includes a page on spherical geometry. The reader should also be aware of the Lenart Sphere as a useful teaching and learning tool. There are surely other resources I am not aware of. A number of people have been helpful in the creation of this book. I am grateful for valuable commentary on the manuscript from Kenneth Rosen (AT&T Labs) and Glen Van Brummelen (@uest University). The people at CRC Press, including Robert Ross, Michele Dimont, Suzanne Lassandro, and Shashi Kumar, have been terrific to work with. I am most grateful to Rani Hermiz for his translation of Sphaerica. I owe a debt of gratitude to Thomas Banchoff (Brown University) for mentioning Sphaerica to me in the first place and for many conversations about spherical geometry over the years. 1 would also like to thank Professors Stephen Nelson (Tulane University), V. Frederick Rickey (US Military Academy), Francis Ford (Providence College), and David Robbins (Trinity College). Lam also grateful to students from my classes wlio made corrections or suggestions: Megan Amely, Edgar Ayala, Marilena Beckstraud, Kenny Courser, Abigail Dunlea, Rani Herimiz, Jesus Hernandez, Robert Kendrick, Yuan Lin (Annie) Lee, Sean Malter, Jonathan Singh, Gregory Guayante, Jill Richard, and Xiaodan Xu. I would like to thank my department and my deans Victor Rocha and Katherine Kantardjieff for approving professional leave time to work on this book.