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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 Section Section Section



2. 3. 4. 5.



Composition and Matrix Multiplication ..............................246 Frames and How to Represent Them .................................. 256 Properties of the Frame Matrix .............................................265 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.



Dependencies A student with a modest knowledge of Euclidean geometry—who knows how to multiply matrices and (for an optional section) has a grasp of differential calculus—can study this book. Here are the dependencies among the chapters: items in ellipses are matters presumed from other studies, but the amount is slight and can be provided by the instructor to make the geometry class more self-contained. The dotted line signifies some reference that might be made to axioms of 3-dimensional geometry. However, one can skip these matters and rely on the students’ intuitions.



Ch. 1. The Axiomatic Method in Geometry



Slight knowledge of translation, rotation, reflection



3rd semester calculus



Ch. 2. The Euclidean Heritage



Ch. 3. Non-Euclidean Geometry



Matrix multiplication



Ch. 6. Transformation Geometry II: Isometries and Matrices



Ch. 4. Transformation Geometry I: Isometries and Symmetries



Ch. 5. Vectors in Geometry



3D visualization



Ch. 7. Transformation Geometry III: Similarity, Inversion and Projections



Ch. 8. Graphs, Maps and Polyhedra



xi



Course Outlines Euclidean Geometry for Prospective Teachers • Chapter 1. Perhaps skip Axioms for Three‐Dimensional Geometry and Axioms for Areas and Volume, and go a little light on Interiors of Angles. The idea is to move quickly into Chapter 2 before students get bogged down. • Chapter 2. This is the heart of such a course, but with greater emphasis on the first three sections on Congruence, Perpendicularity, and Parallelism. • Chapter 3. If one wants a point of view on foundations. If not, proceed to Chapter 4. • Chapter 4. This is Transformation Geometry, which is an extended application of Euclidean geometry. Applications in art and design; composition of isometries—this paves the way for the abstract view found in Modern Algebra courses.



Applied Geometry For students of the applied sciences, especially computer science students wishing to take a mathematics course with applications. • Chapter 5. Students previously learned this in a calculus or linear algebra course, but not the geometric point of view or the applications. • Chapter 6. Using matrices to represent transformations. • Chapter 7. Image formation, thought to be an old hat aspect of projective geometry, has come roaring back for computer vision and computer graphics.



Topics in Geometry for Graduate Students For graduate students wanting advanced and diversified glimpses of geometry. • Chapter 3. From the point of view of foundations. xiii



xiv



Course Outlines



• Chapter 5. The computational point of view with modern applications. • Chapter 8. A look at problems of recent interest in two and three dimensions.



Axiomatic and Alternative Geometries For the philosophically inclined, or those headed to graduate school. • Chapter 1. To be done carefully. • Chapter 2. If the students are confident in elementary Euclidean, then the emphasis here should be on the rigour and the role of the axioms. • Chapter 3. Non-Euclidean geometry.



Glimpses Glimpses of Applications A Glimpse of Application: Mathematics and Building



Chapter 2



Section 1



A Glimpse of Application: Thales Estimates Distance A Glimpse of Application: Fermat and the Behavior of Light



Chapter 2 Chapter 2



Section 1 Section 2



A Glimpse of Application: Eratosthenes Measures the Earth



Chapter 2



Section 3



A Glimpse of Application: The Hidden Surface Problem A Glimpse of Application: Longitude



Chapter 5 Chapter 5



Section 2 Section 3



A Glimpse of Application: Stereographic Projection



Chapter 7



Section 2



A Glimpse of Application: Modeling Molecules



Chapter 8



Section 1



Glimpses of History A Glimpse of History: The Tragedy of Giordano Bruno A Glimpse of History: The Life of Georgy Voronoi



Chapter 1 Chapter 2



Section 1 Section 2



A Glimpse of History: Bolyai, Gauss, Lobachevsky



Chapter 3



Section 1



A Glimpse of History: Johann Lambert, A One-Man Band



Chapter 3



Section 2



A Glimpse of History: Josiah Gibbs, the Master of Vectors



Chapter 5



Section 1



A Glimpse of History: Cayley, Mister Matrix



Chapter 6



Section 2



A Glimpse of History: Albrecht Dürer, Mathematics and Art



Chapter 7



Section 3



A Glimpse of History: Johannes Kepler



Chapter 8



Section 4



xv



xvi



Glimpses



Main Applications Subject



Section



Area of Application



Deliveries



2.2



Operations research



Pattern recognition



2.2



Computer vision



Rigidity of frameworks Modern astronomy



2.3 3.1



Building Astronomy



Shortcomings of maps



3.2



Cartography



Symmetry San Ildefonso Pueblo designs



5.4 4.5



Art Art



Does this robot hit the pothole?



5.1



Robotics



Does this robot go through the doorway?



5.1



Robotics



Is this point in the polygon? Will this robot hit the desk?



5.1 5.1



Robotics Robotics



Will this X-ray hit the tumour?



5.1



Medicine



Stereographic vision Global positioning system



5.1 5.3



Computer vision Geolocation



Magnifying a curve



5.4



Computer graphics



Curve fitting with splines Computer graphics: waving the flag



5.4 6.1



Engineering Computer graphics



Robbie, the robot, turns



6.2



Robotics



Robbie’s location and orientation Computer vision



6.3 6.3



Robotics Computer vision



Robot collision avoidance Moving a robot arm A scaling up by pixels



6.3 (Example 6.11) 6.5 7.1



Circular motion goes straight



7.2 (Figure 7.12)



Image formation: perspective projection



7.3



Stereographic projection



7.3 (Figure 7.17)



Robotics Robotics Computer graphics Mechanical engineering Computer vision and graphics Cartography



Scheduling a symposium The Euler number and computer vision



8.1 8.2



Operations research Computer vision



Fullerene molecules



8.4



Chemistry



1 The Axiomatic Method in Geometry We human beings are at home with the physical world—our senses guide our movements and help us estimate sizes. But from earliest times, we have wanted to know things about our physical world that our senses and measuring instruments could not tell us—for example, what is the circumference of the earth? For this, we needed the kinds of geometry we will explore in this book: axiomatic geometry. This means that we start with assumptions people are willing to accept as true (perhaps “for the sake of argument”) and use logical arguments based on these agreed-upon principles, instead of our senses. The process is called deduction and the starting principles are called axioms. Geometry, that of Euclid in particular (which we start with), has paid dividends for over 2,000 years, but there are still frontiers to explore. For example, the development of robots has led to the desire to mimic whatever mysterious processes our human minds do, through geometry, to move about in the world safely and effectively. Prerequisites: high school mathematics, the notation of set theory (in just a few places)



Section 1. Axioms for Euclidean Geometry In this section, we describe the basic principles from which we drive everything else in our study of Euclidean geometry. These basic principles are called axioms. We then carry out rigorous proofs of the first few deductions from this axiom set. Our axiom set is a descendant of the five axioms provided by Euclid, but there are some differences. There are two main reasons why we do not use Euclid’s axioms as he originally gave them: 1. Euclid phrases his axioms in a way which is hard for the modern reader to appreciate. 2. It has been necessary to add axioms to Euclid’s set to be able to give rigorous proofs of many Euclidean theorems. DOI: 10.1201/9780429198328-1



1



2



Geometry and Its Applications



A number of individuals, and at least one committee, have taken turns in improving Euclid’s axiom set: notably, David Hilbert in 1899, G. D. Birkhoff in 1932, and the School Mathematics Study Group (SMSG) during the 1960s. Even though these axiom sets differ from one another—and from Euclid’s—they all lead to the well-known theorems in Euclid’s Elements. Consequently, we say that they are all axiom sets for Euclidean geometry. The axioms we list below for our use are a minor rewording of the SMSG axiom set. As we embark on our study of axiomatic Euclidean geometry, you will be asked to consider proofs of some statements you may have learned before. To enter the spirit of our study, you must put aside what you have learned before or find obvious. In earlier sections, we have relied on some geometry you have previously learned but in this section, we strive to construct proofs only from the axioms we are about to list, and any theorems we have previously proved from those axioms. Keep in mind that our objective in our axiomatic discussion of Euclidean geometry is not to learn facts about geometry, but to learn about the logical structure of geometry. Axioms About Points on Lines Axiom 1: The Point-Line Incidence Axiom. Given any two different points, there is exactly one line which contains them. We denote the line connecting A and B by AB . Our first theorem about lines uses proof by contradiction or indirect proof. It is based on the idea that the truths of Euclidean geometry do not contradict one another; if you reason correctly on true statements, then you can never deduce a statement that contradicts another which is known to be true. If you do find a contradiction, then one of the statements you have been reasoning from must be false. In our proof, we will make a supposition and show that it leads to a contradiction. This proves the supposition false. Theorem 1.1: Two lines cross in at most one point. Proof: Suppose lines L and M contain the two points A and B, then A and B would have two lines containing them, violating Axiom 1. This contradiction shows that our supposition that L and M contain two points must be false. ■ Our next axiom is just a mathematical way of saying what everyone who has ever used a ruler will find familiar: a line “comes with” a set of numerical markings we can use for calculations and proofs.



The Axiomatic Method in Geometry



3



Axiom 2: Ruler Axiom For any line, there exists a 1-1 correspondence f between the points of the line and the real numbers. This means: 1. Every point A on the line has a number f(A) associated with it. 2. Different points have different numbers associated with them. 3. Every number, positive or negative, has some point to which it is associated. The function f is called the ruler function for that line. The number f(A) is called the coordinate of A. This axiom allows us to use the properties of the real numbers to find out things about lines. For example, there are infinitely many real numbers, so we must have infinitely many corresponding points on a line. The ruler axiom also allows us to define the key geometric ideas of distance and betweenness. Definitions: Let A, B. and C be three points on a line and f be the ruler function for that line. 1. We say B is between A and C if either f(A) < f(B) < f(C) or f(C) < f(B) < f(A). We write A-B-C to indicate that B is between A and C. C-B-A is the same as A-B-C. 2. The segment from A to B, denoted AB, is defined to be the set consisting of A, B, and all points X where A-X-B. 3. The distance from A to B is defined to be |f(B) − f(A)| and we denote this distance by AB. Note that if A and B designate the same point, AB = 0. 4. If AB = CD, then the segments AB and CD are called congruent. Theorem 1.2: 1. AB = BA 2. If A-B-C, then AB + BC = AC 3. If A, B, C are three different points on a line, exactly one of them is between the other two. Proof: 1. AB = |f(B) − f (A)| = |− [ f (A) − f (B)]| = |f(A) − f(B)| = BA 2. There are two cases. First, suppose f(A) < f(B) < f (C). Then,



4



Geometry and Its Applications



AB + BC = = = = =



|f (B) f ( B) f (C ) |f (C ) AC



f (A)| + |f (C ) f (B)| f (A) + f (C ) f (B) f (A) f (A)|



We leave the second case as an exercise. 3. Let a, b, c be the coordinates of A, B, C according to the ruler function for the line they lie on. It is a well-known fact about numbers that, out of three different numbers, exactly one can lie between the other two. Consequently, by definition of what it means for one point to be between two others, the result follows. ■ The following theorem was assumed as an axiom by Euclid. If Euclid had included our Ruler Axiom among his, then, of course, he would not have needed to assume what we are about to prove. Theorem 1.3: Extendibility If A and B are any two points, then the segment AB can be extended by any positive distance on either side of segment AB (Figure 1.1).



Proof: Let e > 0 be the amount of extension wanted. Let’s say we want to extend past B to a point C, so that B is between A and C and BC = e. Let a and b be the real numbers f(A) and f(B) under the ruler function for the line AB . Case 1: a < b (Figure 1.1). Then, define c = b + e. By part 3 of the ruler axiom, there is a point C which corresponds to the number c. C is the point we want since: a. B is between A and C (since a < b < b + e). b. BC = |f(C) − f(B)| = |(b + e) − b| = |e| = e since e > 0.



FIGURE 1.1 The Ruler Function helps extend segment AB.



The Axiomatic Method in Geometry



Case 2: a > b. We leave this as an exercise. ⌘ A Glimpse of History. The Tragedy of Giordano Bruno.



FIGURE 1.2 Statue of Giordano Bruno. (Photograph by the author.)



5



6



Geometry and Its Applications



The statue shown here shows an Italian scholar of the 16th century, Giordano Bruno. It stands in the Campo de Florio in Rome, where tourists crowd the many restaurants at night. No doubt that they find it pleasing to dine in the shadow of history, having no idea that what is commemorated by the statue is gruesome. Bruno was burnt alive in this very spot because he professed many ideas that were deeply upsetting to the power structure of the time. One of Bruno’s heretical ideas comes right from Euclid—our Theorem 1.3. Bruno deduced from Euclid’s version that the universe must be infinite if you can always extend segments to be longer. He concluded further (and this is not in Euclid) that in an infinite universe, there would be many worlds like our own. This contradicted the prevailing idea that the earth was the centre of—and the very reason for—God’s creation of the universe. ■ Theorem 1.4: The Midpoint Theorem Every segment has a midpoint. That is, for any points A and B, there is a point M on segment AB so that AM = (1/2)AB. ■ Proof: We leave this as an exercise.



■



The Midpoint Theorem requires us to think of points as infinitely small. If they had any positive size, a line segment would be a bit like a necklace (Figure 1.3). If the number of points were even, there would not be one exactly in the middle by Theorem 1.4. FIGURE 1.3 A segment AB of 6 “fat” points would not have a midpoint.



Definition: If A and B are distinct points, the ray from A through B, denoted AB , is the set of all points C on line AB , such that A is not between B and C. We call A the endpoint of the ray.



The negative phrasing of this definition is sometimes awkward, so the following theorem is sometimes handy. Its proof is based on one of our previous results. Can you find it? Theorem 1.5: AB consists of segment AB together with all points X where A-B-X. ■ Definition: Let A, B, and C be points not on the same line. AB ∪ AC is called the angle BAC and denoted as ∠BAC (Figure 1.4). We may also denote this angle as ∠CAB.



The Axiomatic Method in Geometry



B



B A



A



7



B



C



A A



C ↔ AB



Segment AB



B



B



↔ Ray AB



A Angle ∠BAC



Line Triangle ABC



FIGURE 1.4 The cast of characters.



When we hear “angle”, we often think of it as the space between the rays that border it. Our definition of an angle does not capture that idea. We will need a separate definition of the interior of an angle, and base it on the next axiom.



Separation Axiom 3: Pasch’s Separation Axiom for a Line Given a line L in the plane, the points in the plane which are not on L form two sets, H1 and H2, called half-planes, so that: a. if A and B are points in the same half-plane, then AB lies wholly in that half-plane. b. if A and B are points not in the same half-plane, then AB intersects L. H1 and H2 are also called sides of L. L is called the boundary line of H1 and H2. Notice that the half-planes mentioned in Pasch’s axiom do not contain their boundary line. They are sometimes referred to as open half-planes for this reason. Definition: Let A, B, and C be three non-collinear points, as in Figure 1.5 (this means that there is no line which contains all three of them). Let HB be the half-plane determined by AC , which contains B. Let HC be the half-plane determined by AB , which contains C. The inside or interior of ∠BAC is defined to be HB ∩ HC. Pasch’s Axiom was only added to the axiom set for Euclidean geometry in the late 19th century when geometers became aware that, for many geometric figures, there was no way to rigorously define the inside or outside of the figure, much less prove theorems about them. For example, if you had



8



Geometry and Its Applications



FIGURE 1.5 An angle and its interior.



asked Euclid to prove that a line containing a point on the inside of an angle crosses at least one of the rays making the angle, he would have been unable to do so. He would undoubtedly have been unconcerned about this, thinking this theorem to be too obvious to bother with. Theorem 1.6: If a ray AB has endpoint A on line L, but B does not lie on L, then all points of the ray, except for A, lie on the same side of L as B. Proof: The proof is indirect. Assume there is a point C on the ray so that C and B are on opposite sides of L. By Pasch’s Axiom, BC crosses L at some AB and, by Theorem 1.1, AB crosses L in point. This must be A since BC just one point, namely A. Since A is not B or C, the fact that A is in BC means



B-A-C. But this means C is not in AB by the definition of a ray.



■



Triangles play a starring role in geometry, so it is time to define them. Let A, B, C be three non-collinear points. In that case, we define the triangle ABC to be AB ∪ BC ∪ AC . Can you see how to define the interior of a triangle? Suppose we have a triangle ABC and we extend side AC to D, thereby creating an exterior angle ∠BCD as in Figure 1.6. Pick any point M on BC which is not B or C, then extend AM past M to any point E. Will E be in the inside of the exterior angle ∠BCD? Our visual intuition says yes, but if we want the highest degree of rigour, we need a proof. Here it is, but with reasons for some steps left for you to supply. Theorem 1.7: If A, B, and C are not collinear and 1. A-C-D 2. B-M-C 3. A-M-E then, E is in the interior of ∠BCD.



The Axiomatic Method in Geometry



9



FIGURE 1.6 Illustrating Theorem 1.7.



Proof: According to our definition of the interior of an angle, we need to show two things: a. that E is on the same side as D of line BC ; and b. that E is on the same side as B of line CD . a. A and D are on opposite sides of BC (why?). A and E are also on opposite sides of BC (why?). Thus, we have shown both E and D to be on the opposite side of A of line BC . Therefore, E and D must be on the same side of BC , as we wished to prove. b. By hypothesis, B is not on AC , so B is not on CD , since AC and CD are the same line. Thus, by the previous theorem, CB lies wholly on the B side of CD (except for C). But B-M-C means M ε CB , and so M and B lie on the same side of CD Likewise, E and M lie on the same side of CD . Thus, E and B lie on the same side of CD as we wished to prove. ■



Axioms about Measuring Angles We have spoken of angles, but not about measuring them. To fill this gap, we come now to a group of axioms which do for angles what the ruler axiom does for lines. We might refer to them. as a group, as the protractor axioms. Axiom 4: The Angle Measurement Axiom. To every angle, there corresponds a real number between 0° and 180° called its measure or size. We denote the measure of ∠BAC by m∠BAC. Definition: If m∠BAC = m∠PQR, then we say ∠BAC and ∠PQR are congruent angles. Axiom 5: The Angle Construction Axiom.



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Geometry and Its Applications



B A



D



α A



X



β B



C



A’



B’



FIGURE 1.7 Supplementary angles α + β = 180°. b) ∠AXB and ∠A’XB’ are vertical angles; ∠BXA’ and ∠B’XA are vertical angles.



Let AB lie entirely on the boundary line L of some half-plane H. For every number r where 0° < r < 180°, there is exactly one ray AC where C ε H and m∠CAB = r. Axiom 6: The Angle Addition Axiom. If D is a point in the interior of ∠BAC, then m∠BAC = m∠BAD + m∠DAC. Definition: If A-B-C and D is any point not on line AC , then the angles ∠ABD and ∠DBC are called supplementary (Figure 1.7).



Axiom 7: The Supplementary Angles Axiom. If two angles are supplementary, then their measures add to 180°. Now, consider two lines crossing at X, making four angles as in the righthand side of Figure 1.7. Each angle has two neighbouring angles and one which is “across” from it. For example, ∠AXB is across ∠A’XB’. An angle and the one across from it are said to be vertical angles or form a vertical pair. A technical definition goes like this: if A and A’ are points on one line where A-X-A’ and B and B’ are on the other line with B-X-B’, then ∠AXB and ∠A’XB’ are vertical angles. Notice that an angle cannot be vertical by itself—it is only vertical in relation to another. Theorem 1.8: Vertical Angles Theorem Vertical angles are congruent. Proof: We use the notation of Figure 1.7. ∠AXB and ∠BXA’ are supplementary. Likewise, ∠BXA’ and ∠A’XB’ are supplementary. These facts, together with Axiom 7, give the equations



The Axiomatic Method in Geometry



11



m AXB + m BXA = 180° m A XB + m BXA = 180°



Subtracting one equation from the other, we deduce m∠AXB = m∠A’XB’.



Axioms about Congruence and Parallelism The next two axioms are the real workhorses of Euclidean geometry, so we just list them here, saving a more extensive discussion for the next chapter. Axiom 8: The Side-Angle-Side Congruence Axiom (SAS) If 1. one angle of a triangle, say A, is congruent to a certain angle of a second triangle, say A’; and if 2. one side forming the angle in the first triangle, say AB, is congruent to a side forming the congruent angle, say A B , in the second triangle; and if 3. the remaining side forming the angle in the first triangle, AC , is congruent to the remaining side forming the congruent angle in the second triangle, A C , then the triangles are congruent with the correspondence A A’, B B’, C C’.



Axiom 9: Euclid’s Parallel Axiom Given a point P of a line L, there is at most one line in the plane through P not meeting L. Axioms for Three-Dimensional Geometry The axioms we have given so far describe matters on a single plane. This is adequate for most of our work since plane geometry is our main objective. But for comprehensiveness, and for brief applications of three dimensions in this book, we now deal with the third dimension. First, all the previously mentioned axioms are still true with the understanding that they hold for all the planes in the three-dimensional space. For example, Pasch’s Separation Axiom needs to be understood as true for every plane which contains L. Likewise, the Parallel Axiom holds not just for “the plane”, but for any plane containing P and L. In addition to these reinterpretations, we need some extra axioms.



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Geometry and Its Applications



Axiom 10: Point - Plane Incidence Axiom Given any three points, there is at least one plane containing them. If the points are not collinear, there is exactly one plane passing through them. Axiom 11: Plane Intersection Axiom Two distinct planes either don’t intersect or intersect in a line. Axiom 12: Existence of the Second and Third Dimensions For every line, there is a point not on it. For every plane, there is a point not on it. Axiom 13: Line - Plane Incidence Axiom If two points of a line are in a plane, then the line lies entirely in that plane. Axiom 14: Pasch’s Separation Axiom for a Plane Given a plane N, the points which are not on N form two sets, S1 and S2, called half-spaces, with the properties: a. if A and B are points in the same half-space, then AB lies wholly in that half-space; b. if A and B are points not in the same half-space, then AB intersects N. S1 and S2 are also called sides of N. N is called the boundary plane of S1 and S2. Axioms about Area and Volume 15. To every polygonal region there corresponds a definite positive real number called its area. 16. If two triangles are congruent, then they have the same area. 17. Suppose that the region R is the union of two regions, R1 and R2, which intersect at most in a finite number of segments and individual points. The area of R is the sum of the areas of R1 and R2: area(R1 ∪ R2) = area(R1) + area(R2). 18. The area of a rectangle is the product of the length of the base and the length of the height. Although we shall not use them in this book, for completeness we finish with the following axioms concerning volume.



The Axiomatic Method in Geometry



19. The volume of a rectangular parallelepiped is equal to the product of the length of its altitude and the area of its base. 20. (Cavalieri’s Principle) Suppose two solids and a plane are given. Suppose also that every plane which is parallel to the given plane either does not intersect either solid or intersects both in planar cross-sections with the same area. In that case, the solids have the same volume.



Exercises Axioms About Points on Lines 1. Which of the three properties of f in the Ruler Axiom ensures that a line is infinite in extent? 2. Which of the three properties of f in the Ruler Axiom ensures that for any two different points A and B, AB ≠ 0? 3. Suppose we defined distance as AB = [ f(B) − f(A)]2. Which facts about distance, proven by Theorems 1.2, 1.3, and 1.4, are no longer true? Which are still true? 4. Suppose the absolute value was dropped from the definition of distance. Would parts 1 and 2 of Theorem 1.2 still be true? Explain. 5. Suppose we change the definition for A-B-C by changing the strict inequalities (“