项目ive几何视角：从直观到代数的探索

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"《透视投影几何：现实与复数几何的探索》一书由Jürgen Richter-Gebert撰写，是对几何学这一数学领域中一种独特视角的深入探讨。本书关注的重点是投影几何，它将传统的视觉、逻辑和代数方法融合在一起，为理解空间对象的相互关系提供了多维度的思考途径。在传统的几何学中，我们熟知的对象包括点、线、圆和平面，这些实体之间的关系构成了几何的基础。作者强调，几何学既包含直观和可视化的元素，如测量和构造，也涉及严密的逻辑推理和公理系统的优雅。例如，欧几里得几何以其严谨的证明体系著称，而黎曼几何则在处理弯曲空间时更为抽象且富有挑战性。投影几何引入了代数的视角，通过向量、矩阵和方程的形式来表达几何概念。这使得几何对象及其关系可以被转化为算术操作，从而形成一个平行于传统几何的抽象世界。在这个世界里，不仅有算法上的精确计算，还有更深层次的范畴论和抽象代数思想的应用，比如通过线性变换和矩阵运算来理解和操作投影。这本书不仅适合那些对几何基础感兴趣的读者，也对于从事计算机图形学、三维建模和游戏开发等领域的专业人士具有重要意义。在现代3D图形技术中，投影几何原理被广泛应用，如透视投影、射影变换等，它们在渲染、模型构建以及虚拟现实等场景中扮演着核心角色。通过阅读《透视投影几何》，读者可以收获对几何学更深的理解，无论是从直观的形状到抽象的理论，都能从中找到桥梁，从而提高问题解决能力和创新思维。同时，书中提供的理论基础也为深入研究数学、计算机科学和工程实践提供了坚实的支撑。"

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

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 Quadrilateral 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 Perspectivity ................................. 167

10.1 Conic through Five Points ............................... 167

10.2 Conics and Cross-Ratios ................................. 170

10.3 Perspective 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 RP

....................................... 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

剩余592页未读，继续阅读

HenryRead

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