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Geometry and Its Applications PDF

489 Pages·2022·61.133 MB·English
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Geometry and Its Applications Geometry and Its Applications Third Edition Walter J. Meyer Third edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 Taylor & Francis Group, LLC [First edition published by Elsevier 1999] [Second edition published by Elsevier 2006] CRC Press is an imprint of Taylor & Francis Group, LLC The right of Walter Meyer to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs, and Patents Act 1988. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication — we apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we may rectify it in any future reprint. Except as permitted under US Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data A catalog record has been requested for this book ISBN: 978-0-367-18798-9 (hbk) ISBN: 978-0-367-68999-5 (pbk) ISBN: 978-0-429-19832-8 (ebk) DOI: 10.1201/9780429198328 Typeset in Palatino by MPS Limited, Dehradun Contents To the Instructor..................................................................................................vii To the Student........................................................................................................ix Dependencies..........................................................................................................xi Course Outlines..................................................................................................xiii List of Glimpses....................................................................................................xv 1. The Axiomatic Method in Geometry........................................................1 Section 1. Axioms for Euclidean Geometry...............................................1 2. The Euclidean Heritage..............................................................................17 Section 1. Congruence..................................................................................17 Section 2. Perpendicularity.........................................................................31 Section 3. Parallelism...................................................................................48 Section 4. Area and Similarity....................................................................65 3. Non-Euclidean Geometry...........................................................................79 Section 1. Hyperbolic and Other Non-Euclidean Geometries..............79 Section 2. Spherical Geometry – A Three‐Dimensional View..............93 Section 3. Spherical Geometry – An Axiomatic View.........................106 Section 4. The Relative Consistency of Hyperbolic Geometry...........115 4. Transformation Geometry I: Isometries and Symmetries................129 Section 1. Isometries and Their Invariants.............................................129 Section 2. Composing Isometries.............................................................140 Section 3. There Are Only Four Kinds of Isometries...........................152 Section 4. Symmetries of Patterns...........................................................162 Section 5. What Combinations of Symmetries Can Strip Patterns Have?..........................................................................171 5. Vectors in Geometry.................................................................................179 Section 1. Parametric Equations of Lines...............................................179 Section 2. Scalar Products, Planes, and the Hidden Surface Problem........................................................................197 Section 3. Norms, Spheres and the Global Positioning System.........210 Section 4. Curve Fitting With Splines.....................................................222 6. Transformation Geometry II: Isometries and Matrices....................235 Section 1. Equations and Matrices for Familiar Transformations........................................................................235 v vi Contents Section 2. Composition and Matrix Multiplication..............................246 Section 3. Frames and How to Represent Them..................................256 Section 4. Properties of the Frame Matrix.............................................265 Section 5. Forward Kinematics for a Simple Robot Arm....................273 7. Transformation Geometry III: Similarity, Inversion and Projections...................................................................................................287 Section 1. Central Similarity and Other Similarity Transformations in the Plane.................................................287 Section 2. Inversion....................................................................................296 Section 3. Perspective Projection and Image Formation.....................306 Section 4. Parallelism and Vanishing Points of a Perspective Projection..............................................................322 Section 5. Parallel Projection.....................................................................334 8. Graphs, Maps and Polyhedra.................................................................351 Section 1. Introduction to Graph Theory...............................................351 Section 2. Euler’s Formula and the Euler Number..............................373 Section 3. Polyhedra, Combinatorial Structure, and Planar Maps......................................................................385 Section 4. Special Kinds of Polyhedra: Regular Polyhedra and Fullerenes...................................................................................399 Bibliography........................................................................................................411 Answers to Even-Numbered Exercises..........................................................413 Index.....................................................................................................................463 To the Instructor This book is meant to provide a balanced and diversified view of geometry—modern as well as ancient, axiomatic as well as intuitive, applied as well as pure, and with some history. We cover Euclid in a more rigorous and foundational style than students have studied in high school. We also cover modern ideas, especially ones which show applied geometry. The book is completely accessible for upper-division mathematics majors. Although there may be new topics here, nothing is particularly advanced—for the most part, it all grows directly out of Euclid. It is a moderately large book so that instructors can design a variety of courses for different students. We have used it for the following: • Students who wish to become secondary school teachers and need a deeper look at Euclid apart from their high school course. • A course to provide students, especially those of computer science, with an applied view of modern geometry. • Graduate students who want to diversify and have a glimpse of newer geometry. • Students for whom axiomatics and alternative geometries are appropriate. Below are some pointers of how to put together such courses. A glance at our dependency chart (which follows) shows that not much is needed from earlier college mathematics courses (more detailed descriptions of prerequisites are given at the start of each chapter). What students need is the maturity to deal with proofs and a few careful calculations but calculation is less than in a calculus course). We also want students to witness applied geometry; nearly every section of the book offers an opportunity to introduce or elaborate on an application. Finally, we want students to see that geometry is part of human culture, so we included a number of historical vignettes. In writing this book, I am aware of the many people and organizations who have shaped my thoughts. I learned a good deal about applications of geometry at the Grumman Corporation (now Northrop-Grumman) while in charge of a robotics research program, when I had the opportunity to teach this material at Adelphi University, and during a year spent as a Visiting Professor at the US Military Academy at West Point. In particular, I thank my cadets and my students at Adelphi for finding errors and suggesting improvements in earlier drafts. Appreciation is due to NSF, the Sloan Foundation, and COMAP for involving me in programs dedicated to vii viii To the Instructor the improvement of geometry at both collegiate and secondary levels. Finally, I wish to thank numerous individuals with whom I have been in contact about geometry, in general, and this book in particular: Joseph Malkevitch, Donald Crowe, Robert Bumcrot, Andrew Gleason, Branko Grünbaum, Victor Klee, Greg Lupton, John Oprea, Brigitte Selvatius, Marie Vanisko, and Sol Garfunkel. To the Student The main novelty of this book is that it presents a wider view of geometry than the Euclidean geometry you will recall from high school. Here, you will find modern as well as ancient geometry, applied as well as pure geometry, all spiced with historical vignettes. There are a number of advantages from this presentation of geometry: 1. Most of the topics are useful. Many of them are being applied today, for example, in software we use everyday. An important objective of this book is to introduce applications of geometry, including the study of symmetry (useful in graphic design), chemistry, topics in computer science and robotics. About half the pages of the book concern an application or are part of the theory that supports an application. 2. There are a few long and complicated calculations where learning the steps is the main task but understanding the ideas is just as important. 3. There are parts of this book that help prepare you for advanced mathematics courses, especially abstract algebra. 4. There are jobs that use geometry—especially the vector geometry of Chapters 5, 6 and 7. 5. Euclid is well-represented for those who wish to become secondary school teachers. How to use this book Many students study a mathematics course by just examining the worked examples and hoping this will enable them to work the exercises. But our exercises are seldom close copies of worked examples. To prepare for the exercises, you should—with the help of the text and your professor—follow the storyline of the topic, understand the concepts, and study the proofs. Finally, it has been repeatedly shown that students do better if they have a well‐matched study partner. To understand some machines, people are trained to take them apart and put them together again blindfolded. We don’t recommend studying geometry blindfolded, but we do recommend studying the proofs in the text with that level of attention so you can easily reproduce them. ix x To the Student A bonus This book is one of the most student-centred courses of study you will encounter in college. In most courses, you have to accept a lot of what you are told because you don’t have the time, the energy, or the resources to verify it for yourself. Is water really composed of hydrogen and oxygen? Are the Great Lakes salty? Save yourself the trouble and ask your instructor. But you can check out the facts of geometry as we present them in this book. The method is called ’proof’, and you can learn it. There is a lot in this book and we hope you will take advantage of it.

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