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Perspectives on Projective • Jürgen Richter-Gebert

Perspectives on

A Guided Tour Through Real and

123 Jürgen Richter-Gebert TU München Zentrum Mathematik (M10) LS Geometrie Boltzmannstr. 3 85748 Garching Germany [email protected]

ISBN 978-3-642-17285-4 e-ISBN 978-3-642-17286-1 DOI 10.1007/978-3-642-17286-1 Springer Heidelberg Dordrecht London New York

Library of Congress Control : 2011921702

Mathematics Subject Classification (2010): 51A05, 51A25, 51M05, 51M10

c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of ,reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

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Springer is part of Springer Science+Business Media (www.springer.com) About This Book

Let no one ignorant of geometry enter here!

Entrance to Plato’s academy

Once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, “and what is the use of a book,” thought Alice, “without pictures or conversations?”

Lewis Carroll, Alice’s Adventures in Wonderland

Geometry is the mathematical discipline that deals with the interrelations of objects in the , in , or even in higher . Practicing geometry comes in very different flavors. More than any other mathematical discipline, the field of geometry ranges from the very concrete and visual to the very abstract and fundamental. In the one extreme, geometry deals with very concrete objects such as points, lines, , and planes and studies the interrelations between them. On the other side, geometry is a benchmark for logical rigor, the elegance of systems, and logical chains of proof. There is a third way of thinking about geometry that stands alongside the visual and the logic-based approaches: the algebraic treatment. Here algebraic structures such as vectors, matrices, and equations are used to form a kind of world, in which each geometric object and relation has an algebraic manifestation. In this parallel world, too, the considerations may be very concrete and algorithmic or very abstract and functorial. This book explains

v vi About This Book how to treat the fundamental objects of geometry using appropriate algebraic methods. Many of the techniques presented in this book have their roots in the work of the great geometers of the nineteenth century like Pl¨ucker, Grassmann, M¨obius, Klein, and Poincar´e (to mention only a few). The algebraic representations are, however, more by far than a way to express geometric objects by . Very often, finding the right algebraic structure unveils the “true” nature of a geometric concept. It may open new perspectives on and deep insights into matters that seemed elementary at first sight and help to generalize, connect, interpret, visualize, and under- stand. This is what this book is about. Its ultimate aim is to present the beauty that lies in the rich interplay of geometric structures and their al- gebraic counterparts. A warning should be issued right at the beginning. It is relatively easy to transform geometric objects into algebraic ones. For in- stance, points in the plane may be easily represented by their xy-coordinates. However, these “naive” approaches to representing geometric objects are very often not the ones that lead to far-reaching conclusions. Often it is useful to introduce more sophisticated algebraic methods that may seem more abstract at first sight but are ultimately more powerful and elegant. Guided by these more abstract and elegant structures, it may even happen that one is willing to modify the original concept of the geometric first-class citizens (say points or lines) and for instance add some new type of (more abstract) objects. When we talk about , one of the most fundamen- tal concepts of this book, exactly this will happen. We will first see that in the plane very elementary operations such as computing the through two points and computing the intersection of two lines can be very elegantly expressed if lines as well as points are represented by three-dimensional coor- dinates (where nonzero scalar multiples are identified). Taking a closer look at the relation of planar points and their three-dimensional representing vec- tors, we will observe that certain vectors do not represent points in the real Euclidean plane. This motivates the search for a geometric interpretation of these nonexistent points. It turns out that they may be interpreted as “points that are infinitely far away.” We will then extend the usual two-dimensional plane by these new points at infinity and obtain a richer and more elegant geometric system: the system of projective geometry. In a certain sense this way of thinking is quite similar to the work of chemists at the time when the periodic table of the elements was about to be discovered. Based on the elements known so far, they looked for ways to explain their behavior. At some they spotted a structure and certain principles into which all the known chemical elements could be fit- ted (the periodic table of the elements). However, some places in the periodic table did not correspond to known elements. It soon became more reasonable to claim the existence of these undiscovered elements than to give up the inner beauty and explanatory power of the periodic table. Later on, all ele- ments whose existence had been conjectured were indeed discovered. The role About This Book vii of “discovering elements” in mathematics is played by the “interpretation of concepts.” We will meet such situations quite often in this book.

The spirit of this book. In a sense, this book is much more about the “how” than about the “what” of geometry. The reader will recognize that very often we will study very simple objects and their relations. Elementary objects such as points, lines, circles, conics, , and distances are the real first-class citizens in our approach. Also, the operations we study will be quite elementary: intersecting two lines, intersecting a line and a conic, calculating tangents, etc. Most of these operations may in principle be per- formed with some advanced high-school mathematics. Regardless of that, our emphasis will be on structures that at the same time allow us to express the fundamental objects as well as the operations on them in a most elegant way. So the algebraic representation of an object never stands alone; it is always related to the operations that should be performed with the object. As mentioned before, these advanced representations often lead to new in- sights and broaden our understanding of the seemingly well-known objects. In this respect our philosophy here is very close to ’s famous book “Elementary Mathematics From an Advanced Standpoint.” While reading this book, the reader will find that the definitions and con- cepts are more important than the theorems. Very often the same (sometimes elementary) theorems are re-proved with different approaches. A topic that will show up over and over again is the question of how elegantly and gen- eralizably these proofs can be performed with the various methods. I hope that the reader will find these multiple perspectives on related topics a good way to gain a deeper understanding of what is going on.

A little history. As mentioned before, many of the techniques in this book go back to what could be called the golden age of geometry, the hundred years between 1790 and 1890. In this period, starting with Gaspard Monge many fundamental geometric concepts were discovered that went far beyond Eu- clid’s Elements (which until then had dominated geometric thinking). Many of these new concepts were intimately related to the underlying algebraic structures. In that period, algebra and geometry underwent a kind of coevo- lution, inspiring and enhancing each other. Projective geometry turned out to be one of the most fundamental structures that at the same time had the most elegant algebraic representation. The concepts of linear and multilinear algebra were developed in close connection to their geometric significance. The development culminated in the revolutionary discovery of what now is called “”: a geometric structure that violates the fifth postulate of and is still logically on an equal footing with his geom- etry. At its time, this discovery was so revolutionary that C.F. Gauss, who was one of the main protagonists in this discovery and at the same time one of the world’s leading mathematicians, kept it a secret and never published anything on that topic. (We will dedicate several chapters to this topic.) The viii About This Book key to an elegant treatment of hyperbolic geometry again lies in projective approaches. Nowadays, hyperbolic geometry is a well-established, amazingly rich mathematical subject with flourishing connections to many other fields, such as topology, theory, number theory, combinatorics, numerics, and many more. Unfortunately, in the nineteenth century the field of geometry grew per- haps a bit too fast. Many books with many pictures, many theorems, and many proofs of varying mathematical quality were published. Some of the proofs heavily relied on pictorial reasoning. At some time around the turn of the century, a point was reached where it was difficult to say which of these results were to be trusted and which were not. As a kind of antithetical development, this time was the beginning of a school of new and until then unmatched mathematical rigor. was one of the leading figures in the process of rewriting all geometry from scratch in order to place it on a reliable and safe foundation. His book “Grundlagen der Geometrie” (Foun- dations of Geometry) [58]startswithanaxiomsystem that even fixed gaps in Euclid’s and postulates to develop a watertight building of geom- etry. Hilbert’s famous saying that one must be able to say “tables, chairs, beer mugs” each time in place of “points, lines, planes” refers to the demand that an axiom system must be completely formal and not at all depend on the imagination. Following this strict approach, he and several other mathe- maticians triggered a development in which geometry was treated as a purely formal science. The hardliners of this program claimed that pictures, and in particular pictorial reasoning, had to be abandoned from geometry books.1 This development was a kind of catharsis for geometry, and many impor- tant and subtle points were revealed in this time (from 1900 to approximately 1970). However, this formal approach also had its disadvantages. There is a famous half-joking quotation from Johann Wolfgang von Goethe about math- ematical abstractions:

Mathematicians are a kind of Frenchmen; whatever you say to them they translate into their own language, and forthwith it is something entirely different.

Something like this happened to geometry in the time of rigorous abstraction. Abstraction opened mathematicans’ eyes to many far-reaching concepts, such as alternative axiom systems, , and combinatorial gener- alizations. At the same time, it changed the concept of what was considered a first-class mathematical citizen. Germs, schemes, , and configura- tion spaces became more important than (the old-fashioned) points, lines, and planes. As a sideeffect of this process many important concepts were almost for- gotten. Large parts of the still valuable “old geometry” were no longer taught at the universities. The following personal anecdote shall exemplify this. It was around 1993 when I gave a talk at KTH (Kungliska Tekniska H¨ogskolan)

1 It is a kind of historical irony that Hilbert, jointly with Cohn-Vossen, wrote a beautiful and highly visual book entitled “Geometry and the Imagination”[59]. About This Book ix in Sweden, where I mentioned a certain (and I think really cool) way to con- struct the foci of an simply by drawing four specific (complex) tangents and intersecting them (see Figure 19.6). After the talk, a much older colleague came to me and said, “Oh, I am so glad. I thought that today nobody re- membered this construction and that I might be one of the last ones who knew it.” In fact, I learned this construction from a book by Blaschke from the 1940s [6], and I hardly know a modern textbook in which it is taught. Perhaps this was one of the points at which I decided to write this book.

Geometry and computers. Since the 1970s, the role of geometric rea- soning has again undergone a structural change. The reason for this is that computers, and in particular computer graphics, have come to play a more and more important role. This has had a twofold effect. On the one hand, in order to obtain good visualizations (also in nonmathematical fields such as CAD, animated movies, games) it is essential to have a good and far-reaching modeling of the objects that are to be visualized, be it the newest automobile design, the dinosaurs in Jurassic Park, or chemical molecules. For such visu- alizations, even on a very elementary level the elegant treatment of primitives such as points, lines, and circles becomes a key issue. On the other hand, the computer became a tool that allowed mathematicians to visualize abstract concepts and to do precise research on a level that is still quite visual. In particular, computers have made it possible to interact directly with mathe- matical (and in particular with geometric) structures. All these developments brought a more concrete and more algorithmic treatment of geometry to the mathematical world’s attention once more. In fact, it turned out that many concepts related to nineteenth-century geometry were highly appropriate for dealing with geometric structures in a computational way. I myself began my research career at a time (around 1985) at which com- putational methods were seriously entering the everyday work of mathemati- cians. From then until now I have gone through a chain of thoughts, concepts and problems that definitely shaped the selection of topics in this book. For me, an amazing experience was that this chain went from quite abstract con- cepts in combinatorial geometry to increasingly elementary (or let us rather say fundamental) concepts and questions. Following these experiences it be- came more and more clear that the key to an elegant treatment of geometric structures lies in a good algebraic representation and goes straight to the heart of ninetheenth century geometry. Since many of these topics I was work- ing on form a kind of “knowledge base” for this book, I will briefly mention this chain. I started working on the structural and computational treatment of so-called realizability questions on combinatorial geometry (we will meet this topic briefly in Section 27.2). In this it turned out that theoretic and projective methods (see Chapter 6 and Chapter 7) are funda- mental. In fact (and this was part of my own doctoral thesis), these methods could be used to implement that were able to generate “readable algebraic proofs” for many geometric theorems (see Chapter 15) x About This Book and by this can form the basis of a kind of geometric expert system. After implementing this prover, I had the desire to have a nice interactive input device for geometric configurations that could be used to feed the prover. What started as a small and seemingly simple project turned out to be a task that is still occupying quite a substantial fraction of my research time. The original demands for this input interface were comparatively simple. The user should be able to use the mouse to construct geometric configurations containing points, lines, circles, conics, etc. After the construction is finished it should be possible to grab basis elements with the mouse, move them, and watch the dependent elements change according to the rules of the construc- tion. If the configuration encodes an incidence theorem, it should be possible to ask the prover for a proof of it. My experience in combinatorial geometry and made it immediately clear that such a system, if it was to be elegant, must be based on projective methods, since they have the nice feature of eliminating many special cases. What started at this time (a first prototypical project was undertaken together with Henry Crapo in 1992) for me turned out to be an ongoing search for elegant structures to represent the fundamental objects in geometry. In a sense, this book tells roughly half of this story. In 1996 I started the development of a less prototypical system for dynamic geometry (Cinderella), jointly with Ulrich Kortenkamp [112, 113]. In this system we tried to represent the geometric objects in a way that al- lows for a smooth implementation of geometric primitive operations. One can read the present book as a guide to the representation of these objects and operations. One fundamental breakthrough in the Cinderella project was the discovery that in order to achieve a continuous dynamic behavior in the ge- ometric elements it is necessary to embed the whole situation in an ambient complex space and in a sense navigate on Riemann surfaces (see [72, 73, 74]). This is the other half of the story, on which we will only very briefly touch in the very last section of the very last chapter. To tell it in full would require another book.

Applications, beauty, pictures, and formulas: This book is intended to serve two purposes. On the one hand, it should be very “hands-on” and purposely focuses on elementary objects such as points, lines, circles, conics, and their interrelations. The reader will find many concrete and directly ap- plicable formulas and recipes for performing operations, measurements, and transformations on them. On the other hand, the book is intended to com- municate some of the inner beauty of the subject. For me it is one of the most beautiful mathematical topics, with many amazing twists, surprises, and sub- tleties and still of fundamental importance for many practical applications. Although this book presents many such explicit algebraic and algorith- mic methods for performing primitive operations, the observant reader may recognize that in this book there are comparatively few long algebraic deriva- tions and calculations. This is intimately related to the approach of working on a conceptual level. We will try to derive conceptual setups that make About This Book xi explicit calculations superfluous whenever possible. In doing so we are close to the philosophy of one of the most important persons in nineteenth century geometry, Julius Pl¨ucker. Felix Klein, who was his student, wrote about him:

In der Pl¨uckerschen Geometrie wird die bloße Kombination von Gleichungen in geometrische Auffassungubersetzt ¨ und r¨uckw¨arts durch letztere die analytische Op- eration geleitet. Rechnung wird nach M¨oglichkeit vermieden, dabei eine bis zur Virtuosit¨at gesteigerte Beweglichkeit der inneren Anschauung, der geometrischen Ausdeutung vorliegender analytischer Gleichungen ausgebildet und in reichem Maße verwendet.

Or in the translation by M. Ackermann:

In Pl¨ucker’s geometry the bare combination of equations is translated into geomet- rical terms, and the analytic operations are led back through the geometric. Com- putation is avoided as far as possible; but by doing this, a mobility, heightened to the point of virtuosity, of inner intuition of the geometric interpretation of given analytic equations, is cultivated and extensively applied.

Many of the formulas and derivations that are given here are not only used to do a formal derivation that takes one from a statement A to a statement B. More importantly, formulas very often have a structural component. Many of them have interesting symmetry properties, a certain rhythm, so to speak. It is perhaps advisable that the reader pause at some point and meditate a bit on this inner structure and symmetry of some of the formulas. In the book you will also find many pictures, diagrams, and illustrations (so hopefully Alice will find it useful after all). They are intended to illustrate and not to replace the proofs and concepts that are presented. As with the formulas, while reading the book it is highly recommended that one spend a substantial amount of time looking closely at some of the pictures. A picture is worth a thousand words, and not everything that one might see and observe in the pictures is also in the text. So I recommend that the reader take some time for meditation on the pictures, their hidden symmetry structures, their spatial interpretations, their dynamic behavior.

Why this book? One might wonder why one should take the effort to write a 570-page book about projective geometry that contains so much “old geometry.” There are several reasons, and I will try to explain a few of them. My experience over the past few years: As already mentioned, much of my own work has been closely related to the representation of geometric objects on the computer. In the area of automated theorem-proving as in the area of dynamic geometry, the classical approaches turned out to be extremely useful. Homogeneous coordinates, invariant theoretic methods, Grassmann- Pl¨ucker relations, Cayley-Klein and many other topics that are central in this book were the key to understanding and implementing versatile and flexible tools. This book presents a selection of those topics that I found most helpful either from a structural point of view (how things are related) or from a pragmatic point of view (what is needed for implementations). xii About This Book

Furthermore, many aspects have been added to the purely classical viewpoint that will hopefully reveal some new interrelations between the topics. Backing up knowledge: I had to learn many of these concepts from the old original literature. Mathematical language changes over time, and sometimes it takes quite a bit of decoding to understand what some concept in some original paper really means. Although much of the old mathematics may still be valuable from a modern point of view, it might become increasingly inaccessible. In particular, if (as in the case of classical projective geometry) some concepts are no longer regularly taught at universities, they enter a self- reinforcing loop of fading from commonly available knowledge. Fortunately, the advent of computer visualization has made classical projective geometry an important topic, again. However, many of the deeper concepts are still accessible only to the experts. A few months ago I had a discussion with my colleague Tim Hoffmann on this topic, and in the discussion we found a nice metaphor for what is going on. Writing about classical topics in a modern language is like copying films from videotape to DVDs. The old media still exist; however, it becomes increasingly unlikely that they are used. It needs a refreshing copy procedure that puts the data/knowledge in a format accessible by modern readers (i.e. DVD players). So part of this book project is a kind of backup process. Still I can truly recommend to everyone to read at least once Felix Klein’s Vorlesungenuber ¨ nicht-euklidische Geometrie [68]orPl¨ucker’s System der analytischen Geometrie [100]. The audience: This book is intentionally written in a style that should be accessible to students who have basically finished their elementary course. It should be accessible to mathematicians as well as computer scientists and physicists. Most of the topics in this book are presented in a relatively self-contained way, allowing even geometry novices to profit from reading it.

A guided tour: Here is a brief summary of the topics you will meet in the following chapters. Except for Chapter 1 (which is a bit special, as you will see), this book is divided into three parts. The first part is entitled “Projec- tive Geometry” and deals with the very fundamental objects and concepts. Projective spaces are introduced, first on an axiomatic level (Chapter 2) and then in direct relation to spaces related to real geometry (“real” in the sense of the real numbers R). Homogeneous coordinates are introduced as the main tool for dealing with projective geometry on an algebraic level (Chapter 3). Their transformations are also studied. In particular, it is shown how vari- ous transformations can all be handled by a unified framework. Chapter 4 deals with first simple invariants under these transformations. Cross-ratios are prominently introduced. They will form the foundation of many inves- tigations in the later chapters. Chapter 5 is perhaps the theoretically most complicated chapter of the first part. There we show that projective transfor- mations can also be characterized by certain invariant properties (for instance About This Book xiii ). This chapter could be skipped on first reading. Chapters 6 and 7 demonstrate the importance of determinants in this context. We outline how one could alternatively build up the framework of projective geometry by tak- ing determinants instead of points as first-class citizens. The second part is entitled “Working and Playing with Geometry.” In this part a selection of topics is presented that can be handily treated by means of projective concepts. In a way, this part is also largely about the “flexibility of thinking” in Pl¨ucker’s sense. Here we try to demonstrate the conceptual power of projective geometry and homogeneous coordinates. Chapter 8 in- troduces more elaborate invariants. Chapters 9 to 11 deal intensively with conics. These chapters are of fundamental importance for the rest of the book and should not be skipped. Chapter 12 explains how the concepts gen- eralize to higher dimensions. Chapters 13 and 14 are in a sense special again. They introduce a beautiful method of dealing with projective geometry on a diagrammatic level. In this language, each formula can be expressed by a graphical diagram. Algebraic derivations translate to graph manipulations. These two chapters can be skipped at first reading; however, skipping them means missing a wealth of beautiful concepts. Chapter 15 finally tries to present all previously mentioned concepts in a combined way and highlights several interesting geometric incidence theorems and invariant-theoretic prov- ing methods. The third part is entitled “Measurements.” It deals with a fundamental problem that remains after the first two parts. Over the real numbers, projec- tive geometry and homogeneous coordinates are a powerful system. However, they have one great disadvantage. The only concepts that can be dealt with are those that are stable under projective transformations. This implies that such elementary geometric operations as measuring a distance and measuring an have no direct analogue in real projective geometry. Also, such fun- damental objects as circles are not objects of real projective geometry. This problem has a beautiful solution. Performing projective geometry over the complex numbers allows for the utilization of the geometric properties of this number field. Since multiplication by complex numbers of unit length corre- sponds to a and rotations implicitly encode distances, this implies that using complex numbers allows one to express measurements in projective geometry. We will see that, for instance, circles can be expressed as special conics that pass through two special complex points I and J. Adding these two points to projective geometry will essentially allow us to perform Eu- clidean operations. The entire third part is about the utilization of complex numbers for performing measurements. Chapter 16 provides a brief introduc- tion to the geometry of complex numbers. Chapter 17 introduces the complex , a first structure in which cocircularity can be expressed in a purely projective framework. Chapter 18 merges the structure of the real and the complex projective line to arrive at a system that combines the advantages of both spaces. Chapter 19 gives many concrete ex- amples of how this general philosophy applies to various Euclidean concepts. xiv About This Book

Chapters 20 to 26 deal with a bold generalization of this approach. It is shown how measurements can be based on projective calculations with respect to a conic. Here all three branches (projective invariants, conics, and complex numbers) are combined to form the very general framework of Cayley-Klein geometries. Chapter 20 introduces the basic concepts, while Chapter 21 in- troduces the general framework for measurements. Chapters 22 and 23 deal with various special geometric properties and theorems in these spaces. The historically very important topic of hyperbolic geometry is a special Cayley- . We dedicate Chapters 24 to 26 to it as the representation of hyperbolic elementary geometry. Hyperbolic geometry turns out to be a so- called nondegenerate Cayley-Klein geometry. This gives it various symmetry properties not shared by general Cayley-Klein geometries. Finally, in Chapter 27 we briefly mention a few topics that demonstrate how projective geometry influences other parts of mathematics, among them algebraic geometry, combinatorics, quantum information theory, and dynamic geometry.

Acknowledgements

It has taken what seems to me like an eternity to complete this book, and there were many people involved in reading through the drafts in its various stages. Some of them commented, some of them corrected, some of them protested (at the right places), some of them encouraged. I am sure that I will forget to mention many of them by name here. So first of all a great ‘thank you’ to everyone who gave me any kind of feedback on the manuscript during the last six years. There are some people who were very active in the final stages of this manuscript. They corrected numerous typos, improved my written English and went through some index battles in the formulas. Among them were Michael Schmid, Thorsten Orendt, Johann Hartl, Susanne Apel, Hermann Vogel, Tim Hoffmann, Peter Lebmeir, Vanessa Krummeck and Martin von Gagern. My special thanks go to Oswald Giering, who went through large parts of the manuscript and made very valuable mathematical comments. Three people deserve a very great Thank You regarding the final phase of writing the book. David Kramer and Stephan Lembach took it on them- selves to go through the entire (pre-)final version of the manuscript and tried to correct all the spalling2, punctuation, formula layout, unidiomatic use of terms and so on. The third person is Jutta Niebauer, who was incredibly patient while entering all these piles of corrections into my original TEXfiles. Drawings are essential to this book and most of the drawings have been cre- ated with suitable software. A great ‘thank you’ goes to Ulrich Kortenkamp, my coauthor of the Cinderella project. Writing the software and the book has

2 I wrote this acknowledgement after they finished their work. About This Book xv been a tightly interwoven process and I am quite convinced that without our mathematical discussions on the software several sections of this book would never have been written. In Section 26.5 you will see beautiful pictures of hyperbolic ornaments. They have been produced with the software project morenaments by Martin von Gagern (using hand-drawn sketches of myself as input). Also this project was essential for shaping some of the mathematics presented in this book. I cannot count the mathematical discussions I have had with colleagues and students on various topics in this book. Many students who attended my classes on Projective Geometry helped to clarify several mathematical and stylistic issues, and many of them definitely helped to clarify the exposition. I am especially grateful to those of them who encouraged me to leave the book in its present, rather explanatory style. They convinced me that even today students are willing to read fat books and that it is worse to approach the same topic from very different directions. Discussions with colleagues were also essential; here I would like to mention a few of the main players who have consciously or unconsciously contributed to the book in its present form (i.e. I learned a lot from them): G¨unter Ziegler, Jim Blinn, J¨urgen Bokowski, Bernd Sturmfels, Henry Crapo, Walter Whiteley, Tim Hoffmann, Ulrich Ko- rtenkamp, AleXX Below, and Martin von Gagern. A special thank you goes to Anders Bj¨orner, who in the earliest stages of this project encouraged me to write this book. Perhaps here I should also mention two other mathemati- cians from whom I learned a lot although I will here on earth unfortunately never have the chance to meet them: Julius Pl¨ucker and Felix Klein. Publishing this book with Springer Verlag means a lot to me. I am very happy to be able to collaborate with Martin Peters and Ruth Allewelt. They were always friendly and remained patient, even though writing this book took much longer than promised. My greatest, warmest, and foremost thanks go to the one person who has influenced me most throughout my entire life. Without the love, encourage- ment, faith in me, patience, and deep understanding of my wife, Ingrid, all else would mean nothing at all. Thank you! A special thank you also goes to my daughter Angie. I am surprised that she is still willing to give stylistic advice on so many fine points. Her fresh and modern look on things helped to improve the layout and graphical appearance in many points. A final thanks goes to Jimmy, our new “family member.” He helped me to stay grounded in the very final stages of the manuscript.

J¨urgen Richter-Gebert Garching, October 2010 • Contents

1 Pappos’s Theorem: Nine Proofs and Three Variations .... 3 1.1 Pappos’s Theorem and Projective Geometry ...... 4 1.2 Euclidean Versions of Pappos’s Theorem ...... 6 1.3 Projective Proofs of Pappos’s Theorem ...... 13 1.4 Conics ...... 19 1.5 More Conics ...... 22 1.6 Complex Numbers and Circles ...... 24 1.7 Finally...... 29

Part I Projective Geometry

2 Projective Planes ...... 35 2.1 Drawings and Perspectives ...... 36 2.2 The Axioms ...... 40 2.3 The Smallest Projective Plane ...... 43

3 Homogeneous Coordinates ...... 47 3.1 A Spatial Point of View ...... 47 3.2 The with Homogeneous Coordinates ...... 49 3.3 Joins and Meets ...... 52 3.4 Parallelism ...... 55 3.5 Duality ...... 56 3.6 Projective Transformations ...... 58 3.7 Finite Projective Planes ...... 64

4 Lines and Cross-Ratios ...... 67 4.1 Coordinates on a Line ...... 68 4.2 The Real Projective Line ...... 69 4.3 Cross-Ratios (a First Encounter) ...... 72 4.4 Elementary Properties of the Cross-Ratio ...... 74

xvii xviii Contents

5 Calculating with Points on Lines...... 79 5.1 Harmonic Points ...... 80 5.2 Projective Scales ...... 82 5.3 From Geometry to Real Numbers ...... 83 5.4 The Fundamental Theorem ...... 86 5.5 A Note on Other Fields ...... 88 5.6 Von Staudt’s Original Constructions ...... 89 5.7 Pappos’s Theorem ...... 91

6 Determinants ...... 93 6.1 A “Determinantal” Point of View...... 94 6.2 A Few Useful Formulas ...... 95 6.3 Pl¨ucker’s μ ...... 96 6.4 Invariant Properties ...... 99 6.5 Grassmann-Pl¨ucker relations ...... 102

7 More on Bracket Algebra ...... 109 7.1 From Points to Determinants ...... 109 7.2 ...and Back ...... 112 7.3 A Glimpse of Invariant Theory ...... 115 7.4 Projectively Invariant Functions ...... 120 7.5 The Bracket Algebra ...... 121

Part II Working and Playing with Geometry

8 Sets and Liftings ...... 129 8.1 Points on a Line ...... 129 8.2 Quadrilateral Sets ...... 131 8.3 Symmetry and Generalizations of Quadrilateral Sets ...... 134 8.4 Quadrilateral Sets and von Staudt ...... 136 8.5 Slope Conditions ...... 137 8.6 Involutions and Quadrilateral Sets ...... 139

9 Conics and Their Duals ...... 145 9.1 The Equation of a Conic ...... 145 9.2 Polars and Tangents ...... 149 9.3 Dual Quadratic Forms ...... 154 9.4 How Conics Transform ...... 156 9.5 Degenerate Conics ...... 157 9.6 Primal-Dual Pairs ...... 159

10 Conics and ...... 167 10.1 Conic through Five Points ...... 167 10.2 Conics and Cross-Ratios ...... 170 10.3 Generation of Conics ...... 172 10.4 Transformations and Conics ...... 175 Contents xix

10.5 Hesse’s “Ubertragungsprinzip”¨ ...... 179 10.6 Pascal’s and Brianchon’s Theorems ...... 184 10.7 Harmonic points on a conic ...... 185

11 Calculating with Conics ...... 189 11.1 Splitting a Degenerate Conic ...... 190 11.2 The Necessity of “If” Operations ...... 193 11.3 Intersecting a Conic and a Line ...... 194 11.4 Intersecting Two Conics ...... 196 11.5 The Role of Complex Numbers ...... 199 11.6 One Tangent and Four Points ...... 202

12 Projective d-space ...... 209 12.1 Elements at Infinity...... 210 12.2 Homogeneous Coordinates and Transformations ...... 211 12.3 Points and Planes in 3-Space ...... 213 12.4 Lines in 3-Space ...... 216 12.5 Joins and Meets: A Universal System ...... 219 12.6 ... And How to Use It ...... 222

13 Diagram Techniques ...... 227 13.1 From Points, Lines, and Matrices to Tensors ...... 228 13.2 A Few Fine Points ...... 231 13.3 Tensor Diagrams ...... 232 13.4 How Transformations Work ...... 234 13.5 The δ-tensor ...... 236 13.6 ε-Tensors ...... 237 13.7 The ε-δ Rule ...... 239 13.8 Transforming ε-Tensors ...... 241 13.9 Invariants of Line and Point Configurations ...... 245

14 Working with diagrams ...... 247 14.1 The Simplest Property: A Trace Condition ...... 248 14.2 Pascal’s Theorem ...... 250 14.3 Closed ε-Cycles ...... 252 14.4 Conics, Quadratic Forms, and Tangents ...... 256 14.5 Diagrams in RP3 ...... 259 14.6 The ε-δ-rule in Rank 4 ...... 262 14.7 Co- and Contravariant Lines in Rank 4 ...... 263 14.8 Tensors versus Pl¨ucker Coordinates ...... 265

15 Configurations, Theorems, and Bracket Expressions ...... 269 15.1 Desargues’s Theorem ...... 270 15.2 Binomial Proofs ...... 272 15.3 Chains and Cycles of Cross-Ratios ...... 277 15.4 Ceva and Menelaus ...... 279 xx Contents

15.5 Gluing Ceva and Menelaus Configurations ...... 285 15.6 Furthermore ...... 291

Part III Measurements

16 Complex Numbers: A Primer ...... 297 16.1 Historical Background...... 298 16.2 The Fundamental Theorem ...... 301 16.3 Geometry of Complex Numbers ...... 302 16.4 Euler’s Formula ...... 304 16.5 Complex Conjugation ...... 307

17 The Complex Projective Line ...... 311 17.1 CP1 ...... 311 17.2 Testing Geometric Properties ...... 312 17.3 Projective Transformations ...... 315 17.4 Inversions and M¨obius Reflections ...... 320 17.5 Grassmann-Pl¨ucker relations ...... 322 17.6 Intersection Angles ...... 324 17.7 Stereographic ...... 326

18 ...... 329 18.1 The points I and J ...... 330 18.2 Cocircularity ...... 331 18.3 The Robustness of the Cross-Ratio ...... 333 18.4 Transformations ...... 334 18.5 Translating Theorems ...... 338 18.6 More Geometric Properties ...... 339 18.7 Laguerre’s Formula ...... 342 18.8 Distances ...... 345

19 Euclidean Structures from a Projective Perspective ...... 349 19.1 Mirror Images...... 350 19.2 Angle Bisectors ...... 351 19.3 Center of a ...... 354 19.4 Constructing the Foci of a Conic ...... 356 19.5 Constructing a Conic by Foci ...... 360 19.6 Theorems ...... 362 19.7 Hybrid Thinking ...... 368

20 Cayley-Klein Geometries ...... 375 20.1 I and J Revisited ...... 376 20.2 Measurements in Cayley-Klein Geometries ...... 377 20.3 Nondegenerate Measurements along a Line ...... 379 20.4 Degenerate Measurements along a Line ...... 386 Contents xxi

20.5 A Planar Cayley-Klein Geometry ...... 389 20.6 A Census of Cayley-Klein Geometries ...... 393 20.7 Coarser and Finer Classifications ...... 398

21 Measurements and Transformations ...... 399 21.1 Measurements vs. Oriented Measurements ...... 400 21.2 Transformations ...... 401 21.3 Getting Rid of X and Y ...... 407 21.4 Comparing Measurements ...... 408 21.5 Reflections and Pole/Polar Pairs ...... 413 21.6 From Reflections to Rotations ...... 419

22 Cayley-Klein Geometries at Work ...... 423 22.1 ...... 424 22.2 Constructive versus Implicit Representations ...... 427 22.3 Commonalities and Differences ...... 429 22.4 and Angle Bisectors ...... 431 22.5 Trigonometry ...... 437

23 Circles and Cycles ...... 443 23.1 Circles via Distances ...... 444 23.2 Relation to the Fundamental Conic ...... 446 23.3 Centers at Infinity ...... 448 23.4 Organizing Principles ...... 450 23.5 Cycles in Galilean Geometry ...... 459

24 Non-Euclidean Geometry: A Historical Interlude ...... 465 24.1 The Inner Geometry of a Space ...... 466 24.2 Euclid’s Postulates ...... 468 24.3 Gauss, Bolyai, and Lobachevsky ...... 470 24.4 Beltrami and Klein ...... 474 24.5 The Beltrami-Klein Model ...... 476 24.6 Poincar´e ...... 479

25 Hyperbolic Geometry...... 483 25.1 The Staging Ground ...... 483 25.2 Hyperbolic Transformations...... 485 25.3 Angles and Boundaries ...... 487 25.4 The Poincar´eDisk...... 489 1 25.5 CP Transformations and the Poincar´eDisk ...... 496 25.6 Angles and Distances in the Poincar´eDisk...... 501

26 Selected Topics in Hyperbolic Geometry ...... 505 26.1 Circles and Cycles in the Poincar´eDisk ...... 505 26.2 Area and Angle Defect ...... 509 26.3 Thales and ...... 514 xxii Contents

26.4 Constructing Regular n-Gons ...... 517 26.5 Symmetry Groups ...... 519

27 What We Did Not Touch ...... 525 27.1 Algebraic Projective Geometry ...... 525 27.2 Projective Geometry and Discrete Mathematics ...... 531 27.3 Projective Geometry and Quantum Theory ...... 538 27.4 Dynamic Projective Geometry ...... 546

References ...... 557

Index ...... 563