Geometry and Its Applications

489 Pages·2022·61.133 MB·English

by  Walter J. Meyer

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
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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.