Technische Universität München Fakultät für Mathematik Lehrstuhl für Geometrie und Visualisierung Non-standard Analysis in Projective Geometry Michael Strobel Vollständiger Abdruck der von der Fakultät für Mathematik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Prof. Dr. Michael Marc Wolf Prüfer der Dissertation: 1. Prof. Dr. Dr. Jürgen Richter-Gebert 2. Prof. Jacques Fleuriot, Ph.D. University of Edinburgh Die Dissertation wurde am 29.05.2018 bei der Technischen Universität München einge- reicht und durch die Fakultät für Mathematik am 19.09.2018 angenommen. Contents 1. Geometry and Infinity1 2. Site Map3 3. Singularities of Geometric Constructions6 3.1. Modelling Dynamic Geometry.........................7 3.2. The Analytic Case............................... 13 3.3. Singularities and Derivatives.......................... 15 3.4. The Radical of all Evil............................. 21 3.5. Methods for Non-differentiable Functions.................. 25 3.5.1. Classical Power Series......................... 25 3.5.2. Law of Large Numbers......................... 26 3.5.3. Cauchy’s Integral Formula....................... 27 3.5.4. Algebraic Curves: Blow Up...................... 27 4. Non-standard Analysis 30 4.1. Constructing the Hyperreal and Hypercomplex Numbers.......... 31 4.2. Mathematical Logic and the Transfer Principle............... 35 ∗ 4.3. Basic Operations on K ............................ 41 4.4. Derivatives and Newton Quotients...................... 50 4.5. Some Useful Series Expansions........................ 51 5. Non-standard Projective Geometry 55 5.1. Incidence of Points and Lines......................... 65 5.2. Non-standard Projective Transformations.................. 75 5.3. Non-standard Cross-Ratios.......................... 82 5.4. Non-standard Conics.............................. 84 6. Removal of Singularities & Numerics 88 6.1. Non-standard Analysis and Removal of Singularities............ 88 6.2. Stability of a Solution............................. 97 3 6.3. The Relation to Perturbation Theory..................... 101 6.4. Fast Numerical Methods............................ 103 6.4.1. Derivatives Revisited.......................... 103 6.4.2. There is no such Thing as a Free Non-standard Lunch....... 104 6.4.3. Implementation of a Non–Archimedean Field............ 105 6.4.4. Levi-Civita Field............................ 107 6.4.5. Implementation of the Levi-Civita Field............... 114 6.4.6. Application to Singularities in Geometric Constructions...... 119 6.5. Experimental Results & Examples...................... 121 6.5.1. Von-Staudt Construction....................... 121 6.5.2. Disjoint Circle Intersection...................... 127 6.6. Limitation of the Levi-Civita Field...................... 132 6.7. A priori Avoidance of Singularities...................... 138 6.7.1. Modeling the Avoidance of Singularities............... 138 6.7.2. From Plücker’s µ to Efficient Implementation............ 139 6.7.3. Limits of a Priori Resolution..................... 141 7. Open Problems & Future Work 143 7.1. Randomized Proving using Infinitesimal Deflection............. 143 7.2. Möbius Transformations............................ 144 7.3. New Approaches to Tracing.......................... 147 7.3.1. Infinitesimal Tracing.......................... 148 7.3.2. Tracing and Automatic Differentiation................ 149 Appendix A. CindyScript Implementation of the Levi-Civita Field 150 Appendix B. Computer Algebra System Code Tangential Circle Intersection 160 Appendix C. Law of Large Numbers 162 Appendix D. Projective Midpoint via Plücker’s µ 163 Bibliography 164 Acknowledgments “Alle guten Dinge haben etwas Lässiges und liegen wie Kühe auf der Wiese.” Friedrich Nietzsche, [100] First of all I want to thank J. Richter-Gebert for the freedom of choice in the subject and the means of my thesis and for being a lighthouse in the sea of science. And also I want to thank Jacques Fleuriot for examining this thesis and the numerous helpful remarks. Furthermore, I want to thank my family Silvia, Willi, Johannes and Andi for constantly supporting me. This thesis is also your merit! My special thanks go to Steffi for her support and the infamous coffee breaks. I want to thank my friends who accompanied me over all the years: Wolle, Daniel, Peppie, Stocke, Maze, Mario, Hannah, Sonja, Michi, Don, Jo, Anja, Karin, Jakob and Ella. Many thanks to Katharina and her family Dominik, Lydia and Norbert. The university and Munich would not have been the same without my friends and colleagues who always brought joyful distraction and discussions: Christoph, Peter, Benni, Patrick, Flo, Felix, Mario, Angela, Karin, Max, Benedict, Georg, Steffi, Horst, Rebecca, Eva, Kathi, Johanna, Anna, Matthias, André, Leo, Laura, Tim, Jutta, Diane, Carsten, Uli (and his Italian espresso machine), Florian and the rest of the chair of Geometry and Visualization. The coffee breaks will always be one of my favourite memories of my doctorate. I also want to thank my office roommate Aaron for happily discussing my ideas and always being so enthusiastic. Also I want to thank Thomas, the Servicebüro Studium and the Fachschaft for making the university a better place to be. Also I want to thank the group of Emo Welzl at the ETH Zürich for the nice and fruitful stay in Switzerland. Additionally, I want to thank the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics” for funding my position. Last but not least: I want to thank the CindyJS team. Jürgen and Uli (the benevolent dictators for life), Martin, Stefan and Aaron. It has not been always easy, but always a lot of fun! 1. Geometry and Infinity “Wahrlich es ist nicht das Wissen, sondern das Lernen, nicht das Besitzen, sondern das Erwerben, nicht das Da-Seyn, sondern das Hinkommen, was den grössten Genuss gewährt. Wenn ich eine Sache ganz ins Klare gebracht und erschöpft habe, so wende ich mich davon weg, um wieder ins Dunkle zu gehen; so sonderbar ist der nimmersatte Mensch, hat er ein Gebäude vollendet so ist es nicht um nun ruhig darin zu wohnen, sondern um ein anderes anzufangen.” Carl Friedrich Gauß, [49] p. 94 In this thesis we will integrate concepts of non-standard analysis into projective geometry. One major application of the developed theory will be the automatic removal of singularities in geometric constructions. Non-standard analysis allows the precise handling of infinitely small (infinitesimal) and infinitely large (unlimited) numbers which admits statements like: “there is an > 0 such + + that < r ∀r ∈ R ” or “there is an H such that H > r ∀r ∈ R ”. This obviously violates the Archimedean axiom. Therefore such fields that contain infinitesimal and unlimited numbers are also called “non-Archimedean”. To embed this theory into the “standard” ∗ theory, one may use field extensions of R and C. We will call them R (hyperreal numbers) ∗ and C (hypercomplex numbers), and one may seamlessly integrate these numbers into well-established (we will refer to it by the adjective “standard”) analysis in the sense of Weierstrass. Although the construction of these fields is highly non-trivial, its mere application is very intuitive. One strength of projective geometry is the natural and comprehensive integration of infinity. For example: consider the intersection of two (disjoint) lines. Euclidean geometry has to distinguish two cases: the lines intersect or they do not intersect (if they are parallel). In projective geometry there is always a point of intersection, it just may be infinitely far way. In practice, projective geometry is usually carried out over R or C, but its not restricted to these canonical fields. One may consider arbitrary fields for geometric operations and this will enable us to combine the mathematical branches of projective geometry and non-standard analysis. 1 1. Geometry and Infinity Using infinitesimal elements in geometry is not a new concept, au contraire it is one of the oldest concepts. Already Leibniz and Newton used them for geometric reasoning. Dynamic (projective) geometry allows us to describe the movement of geometric constructions and implement them on a computer. In dynamic constructions, singularities naturally arise. We will look into methods for resolving singularities and situations in which standard theory does not offer concise and consistent solutions. This will naturally lead to the integration of non-standard analysis into projective geometry. Non-standard analysis in projective geometry is quite an exciting topic on its own and we will explore basic properties that are sometimes surprising. Using this extension of projective geometry, we will be able to develop the theory of proper desingularization. After expanding this theory we will also develop algorithms and a real time capable implementation to integrate the theory into a dynamic geometry system. 2 2. Site Map Abstruse Goose (CC BY–NC 3.0, http://abstrusegoose.com/440) 3 2. Site Map If you are looking for a more streamlined version of this thesis, we already published two preprints that cover most of the novelties: [128, 129]. This thesis is essentially divided into five parts: Singularities of Geometric Construc- tions, an introduction to Non-standard Analysis, Non-standard Projective Geometry, Removal of Singularities & Numerics and Open Problems & Future Work. We will assume that the reader is familiar with projective geometry, if this is not the case we refer to the books