Geometry and Its Applications (eBook)

Walter A. Meyer (Autor)

eBook Download: EPUB

2006 | 2. Auflage
560 Seiten
Elsevier Science (Verlag)
978-0-08-047803-6 (ISBN)

Infos

Meyer's Geometry and Its Applications, Second Edition, combines traditional geometry with current ideas to present a modern approach that is grounded in real-world applications. It balances the deductive approach with discovery learning, and introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text integrates applications and examples throughout and includes historical notes in many chapters.

The Second Edition of Geometry and Its Applications is a significant text for any college or university that focuses on geometry's usefulness in other disciplines. It is especially appropriate for engineering and science majors, as well as future mathematics teachers.

* Realistic applications integrated throughout the text, including (but not limited to):
- Symmetries of artistic patterns
- Physics
- Robotics
- Computer vision
- Computer graphics
- Stability of architectural structures
- Molecular biology
- Medicine
- Pattern recognition
* Historical notes included in many chapters
* Instructor's Manual with solutions available for all adopters of the text

Inhalt

Front cover 1
Title page 4
Copyright page 5
Table of Contents 6
Preface 10
Supplements for the Instructor 12
Introduction 14
Axiomatic Geometry 15
Rigidity and Architecture 15
Computer Graphics 16
Symmetries in Anthropology 17
Robotics 17
Molecular Shapes 20
Companion Website Information 21
1 The Axiomatic Method in Geometry 22
1.1 The Aims of Axiomatic Geometry 22
Section 1.1 Exercises 29
1.2 Proofs in Axiomatic Geometry 32
Section 1.2 Exercises 40
1.3 Axioms for Euclidean Geometry 41
Section 1.3 Exercises 55
2 The Euclidean Heritage 58
2.1 Congruence 58
Section 2.1 Exercises 69
2.2 Perpendicularity 72
Section 2.2 Exercises 84
2.3 Parallelism 89
Section 2.3 Exercises 101
2.4 Area and Similarity 108
Section 2.4 Exercises 120
3 Non-Euclidean Geometry 124
3.1 Hyperbolic and Other Non-Euclidean Geometries 124
Section 3.1 Exercises 134
3.2 Spherical Geometry: A Three-Dimensional View 137
Section 3.2 Exercises 148
3.3 Spherical Geometry: An Axiomatic View 150
Section 3.3 Exercises 162
4 Transformation Geometry I: Isometries and Symmetries 164
4.1 Isometries and Their Invariants 164
Section 4.1 Exercises 174
4.2 Composing Isometries 176
Section 4.2 Exercises 188
4.3 There Are Only Four Kinds of Isometries 190
Section 4.3 Exercises 199
4.4 Symmetries of Patterns 202
Section 4.4 Exercises 211
4.5 What Combinations of Symmetries Can Strip Patterns Have? 213
Section 4.5 Exercises 218
5 Vectors in Geometry 220
5.1 Parametric Equations of Lines 220
Section 5.1 Exercises 236
5.2 Scalar Products, Planes, and the Hidden Surface Problem 239
Section 5.2 Exercises 249
5.3 Norms, Spheres, and the Global Positioning System 252
Section 5.3 Exercises 260
5.4 Curve Fitting with Splines in Two Dimensions 263
Section 5.4 Exercises 273
6 Transformation Geometry II: Isometries and Matrices 278
6.1 Equations and Matrices for Familiar Transformations 279
Section 6.1 Exercises 287
6.2 Composition and Matrix Multiplication 290
Section 6.2 Exercises 296
6.3 Frames and How to Represent Them 299
Section 6.3 Exercises 307
6.4 Properties of the Frame Matrix 309
Section 6.4 Exercises 315
6.5 Forward Kinematics for a Simple Robot Arm 318
Section 6.5 Exercises 328
7 Transformation Geometry III: Similarity, Inversion, and Projection 332
7.1 Central Similarity and Other Similarity Transformations in the Plane 333
Section 7.1 Exercises 340
7.2 Inversion 343
Section 7.2 Exercises 352
7.3 Perspective Projection and Image Formation 355
Section 7.3 Exercises 366
7.4 Parallelism and Vanishing Points of a Perspective Projection 369
Section 7.4 Exercises 380
7.5 Parallel Projection 383
Section 7.5 Exercises 395
8 Graphs, Maps, and Polyhedra 400
8.1 Introduction to Graph Theory 400
Section 8.1 Exercises 419
8.2 Euler’s Formula and the Euler Number 425
Section 8.2 Exercises 435
8.3 Polyhedra, Combinatorial Structure, and Planar Maps 438
Section 8.3 Exercises 450
8.4 Special Kinds of Polyhedra: Regular Polyhedra and Fullerenes 453
Section 8.4 Exercises 463
8.5 List Coloring and the Five-Color Theorem 465
Section 8.5 Exercises 475
Bibliography 480
Chapters 1 and 2 480
Chapter 3 480
Chapter 4 481
Chapter 5 481
Chapter 6 481
Chapter 7 482
Chapter 8 482
Answers to Odd-Numbered Exercises 484
Chapter 1 484
Chapter 2 488
Chapter 3 496
Chapter 4 498
Chapter 5 505
Chapter 6 511
Chapter 7 520
Chapter 8 528
Index 544

Textauszug

Introduction

Introduction

Geometry is full of beautiful theorems, and its logical structure can be inspiring. As the poet Edna St. Vincent Millay wrote, "Euclid alone has looked on beauty bare." But beyond beauty and logic, geometry also contains important tools for applied mathematics. This should be no surprise, since the word geometry means "earth measurement" in Greek. As just one example, we will illustrate the appropriateness of this name by showing how geometry was applied by the ancient Greeks to measure the circumference of the earth without actually going around it. But the story of geometric applications is modern as well as ancient. The upsurge in science and technology in the last few decades has brought with it an outpouring ofnew questions for geometers. In this introduction we provide a sampler of the big ideas and important applications that will be discussed in this book.1

Individuals often have preferences, either for applications in contrast to theory or vice versa. This is unavoidable and understandable. But the premise of this book is that, whatever our preferences may be, it is good to be aware of how the two faces of geometry enrich each other. Applications can’t proceed without an underlying theory. And theoretical ideas, although they can stand alone, often surprise us with unexpected applications. Throughout the history of mathematics, theory and applications have carried out an intricate dance, sometimes dancing far apart, sometimes close. My hope is that this book gives a balanced picture of the dance at this time, in the early years of a new millennium.

Axiomatic Geometry

Of all the marvelous abilities we human beings possess, nothing is more impressive than our visual systems. We have no trouble telling circles apart from squares, estimating sizes, noticing when triangles appear congruent, and so on. Despite this, the earliest big idea in geometry was to achieve truth by proof and not by eye. Was that really necessary or useful? These ideas are explored in Chapters 1 and 2.

Creating a geometry based on proof required some basic truths — which are called axioms in geometry. Axioms are supposed to be uncontroversial and obviously true, but Euclid seemed nervous about his parallel axiom. Other geometers caught this whiff of uncertainty and, about 2000 years later, some were bold enough to deny the parallel axiom. In doing this they denied the evidence of their own eyes and the weight of 2000 years of tradition. In addition, they created a challenge for students of this so-called "non-Euclidean geometry," which asks them to accept axioms and theorems that seem to contradict our everyday visual experience. According to our visual experience, these non-Euclidean geometers are cranks and crackpots. But eventually they were promoted to visionaries when physicists discovered that the faraway behavior of light rays (physical examples of straight lines) is different from the close-to-home behavior our eyes observe. Astronomers are working to make use of this non-Euclidean behavior of light rays to search for "dark matter" and to foretell the fate of the universe. These revolutionary ideas are explored in Chapter 3.

Rigidity and Architecture

If you are reading this indoors, the building you are in undoubtedly has a skeleton of either wooden or steel beams, and your safety depends partly on the rigidity of this skeleton (see Figure I.1b). Neither a single rectangle (Figure I.1b), nor a grid of them, would be rigid if it had hinges where the beams meet. Therefore, when we build frameworks for buildings, we certainly don’t put hinges at the corners — in fact, we make these corners as strong as we can. But it is hard to make a corner perfectly rigid, so we will think of the corners as a little bit like hinges that need to be kept from flexing. If a rectangle is built of four sticks hinged together in such a way that any flexing produces a four-sided figure in a plane, and if we add a diagonal brace, then the rectangle can’t be flexed at all. Perhaps surprisingly, if we have a grid of many rectangles, it is not necessary to brace every rectangle. The braced grid in Figure I.1b turns out to be rigid even if every corner is hinged. In Section 2.3, we work out a procedure for determining when a set of braces makes a grid of rectangles rigid even though all corners are hinged.

Figure I.1 (a) A hinged rectangle flexing. (b) A braced grid that cannot flex even if totally hinged.

Computer Graphics

The impressionist painter Paul Signac (1863–1935) painted "The Dining Room" by putting lots of tiny dots on a canvas (Figure I.2a). If you stand back, the tiny dots blend together to make a picture. This painting technique, called pointillism, was a sensation at the time, and it foreshadows modern image technology. For example, if you take a close look at your TV screen, you’ll see that the picture is composed of tiny dots of light. Likewise, a computer screen creates a picture by "turning on" little patches of color called pixels. Think of them as forming an array of very tiny light bulbs, arranged in hundreds of rows and columns in the x-y plane so that each point with integer coordinates is the center of a pixel (Figure I.2b).

Figure I.2 Pixels in (a) art and (b) computer graphics.

Courtesy of Metropolitan Museum of Art.

When a graphics program shows a picture, how does it calculate which pixels to turn on and what colors they should be? Here is a simple version of the problem: If we are given two pixels (shaded in Figure I.2b) and want to connect them with a set of pixels to give the impression of a blue straight line, which "in between" pixels should be turned blue? Neither Signac nor you would have any problem with this, painting by eye, but how can the computer do it by calculations based on the coordinates of the centers of the start and end pixels? In this computer version of the problem, the desired answer is a list of pixel centers, each center specified by x and y coordinates. Chapter 1 gives you an idea of how to create such a list.

Symmetries in Anthropology

In studying vanished cultures, anthropologists often learn a great deal from the artistic patterns these cultures produced. Figure I.3 shows two patterns you might find on a cloth or circling around a clay pot. Each of these patterns has some kind of symmetry. But what kind? What do we mean by the word symmetry? Some would say the bottom pattern has more symmetry than the top pattern. How can symmetry be measured? The questions we pose here for patterns are similar to ones that arise, in three-dimensional form, in the study of crystallography. We study these questions in Chapter 4, with particular attention paid to the pottery of the San Ildefonso pueblo in the southwestern United States.

Figure I.3 Two strip patterns.

Robotics

Figure I.4 shows a robot about to drill a hole. Perhaps the hole is being drilled into someone’s skull in preparation for brain surgery, or maybe it’s part of the manufacture of an automobile. Whatever the purpose, the drill tip has to be in just the right place and pointing just the right way. The robot moves the drill about by changing its joint angles. If we specify the x,y,z coordinates of the drill tip, how do we calculate the values of θ1 θ2, and θ3 needed to bring the drill tip to the desired point? Can we also specify the direction of the drill? These are questions in robot kinematics. In Section 6.5 we study the basics of robot kinematics for a two-dimensional robot. This is the same kind of mathematics used for three-dimensional robot kinematics.

Figure I.4 A simplified version of a PUMA robot arm.

Molecular Shapes

As chemistry advances, it pays more attention to the geometric shapes of molecules. In 1985 a new molecular shape was discovered, called the buckyball,2 that reminded chemists of the pattern on a soccer ball. The molecule consisted of 60 carbon atoms distributed in a roughly spherical shape — like the corners of the pattern on the soccer ball. Coincidentally, this pattern had been discovered not only by soccer ball manufacturers but many centuries ago by mathematicians. Figure I.5 shows the pattern in a Renaissance drawing of a truncated icosahedron, by Leonardo da Vinci. To understand the structure of the buckyball, think of the corners of the truncated icosahedron as being occupied by carbon atoms and think of the connecting links as representing chemical bonds betweeen certain carbon atoms. Each carbon atom is connected to each of three other nearby carbon atoms with a chemical bond. This pattern contains 12 pentagons and 20 hexagons.

Figure I.5 The pattern of the buckyball molecule, drawn five centuries before the discovery of the molecule!

Courtesy of Jerry Blow.

Once chemists discovered this molecule, they looked for other molecules involving just carbons, where each carbon is connected to exactly...

Produkt-Details

• Erscheint lt. Verlag: 21.2.2006
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-08-047803-4 / 0080478034
• ISBN-13: 978-0-08-047803-6 / 9780080478036