UNIVERSITY OF CINCINNATI Date: 25-Jun-2010 I, Justin Wesley Smith , hereby submit this original work as part of the requirements for the degree of: Master of Science in Computer Science It is entitled: Points and Lines in the Plane Student Signature: Justin Wesley Smith This work and its defense approved by: Committee Chair: George Purdy, PhD George Purdy, PhD Kenneth Berman, PhD Kenneth Berman, PhD Carla Purdy, C, PhD Carla Purdy, C, PhD 8/9/2010 908 Points and Lines in the Plane A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Master of Science In the Department of Computer Science of the College of Engineering By Justin W. Smith B.S. Arkansas State University December 2000 Committee Chair: George Purdy, Ph.D. Abstract This thesis will focus on two topics: (1) finding intersections determined by an arrangement of hyperplanes (e.g., lines in a plane), and (2) lower bounds on the number of various \types" of lines determined by a configuration of points. The first topic is algorthmic. Given an arrangement of n lines in R2, a O(n log n) algorithm is demonstrated for finding an ordinary intersection (i.e., an intersection of exactly two lines). This algorithm is then extended to finding an ordinary intersection among hyperplanes in Rd, under the hypothesis that no d hyperplanes pass through a line and not all pass through the same point. Algorithms are also given to find an ordinary intersection in an arrangement of pseudolines in time O(n2), and to find a monochromatic intersection in a bichromatic arrangement of pseudolines in time O(n2). The second topic is combinatorial. Let G and R be finite sets of points, colored green and red respectively, such that jGj = n, jRj = n − k, G \ R = ;, and G [ R are not all collinear. Lower bounds will be demonstrated for several types of lines (e.g., bichromatic and equichromatic) determined by few points in R2. Acknowledgements First, I would like to thank my advisor, Prof. George B. Purdy. He directed my research toward problems that were both interesting and solvable. I am grateful for his patience and persistence with me as his student. Many of the results of this thesis were a culmination of our joint efforts, and several of the results may be solely attributed to him. (I will attempt to mark those results as such.) His direction has led me to areas of research that I hope to study throughout my academic career. I also thank the faculty and staff at the University of Cincinati. They have created an excellent environment for learning and research. Most of all, I thank my wife Katrina. She sacrificed much so that I could leave a profitable career and return to graduate school. Her hard work, patience, and love, I will always admire. This thesis is dedicated to her. ii Contents 1 Introduction 1 2 Background 3 2.1 A Question Of J.J. Sylvester . 3 2.2 Duality and the Projective Plane . 4 2.2.1 Polar Dual . 8 2.2.2 Parabolic Dual . 8 2.3 Melchior's Inequality . 9 2.3.1 Euler's Polyhedral Formula . 10 2.3.2 Melchior's Proof of the Inequality . 11 2.4 How Many? . 12 2.5 Da Silva and Fukuda's Conjecture . 15 3 On Finding Ordinary Or Monochromatic Intersection Points 18 3.1 Introduction . 18 3.2 Ordinary Points in an Arrangement of Lines in R2 . 19 3.2.1 Existence of Ordinary Intersection Points . 19 3.2.2 Locating an Ordinary Intersection . 20 3.2.3 Algorithm to Find an Ordinary Point in Time O(n log n) 22 3.3 Ordinary Points in an Arrangement of Hyperplanes in Rd . 23 3.3.1 Duality . 23 3.3.2 Algorithm to Find an Ordinary Point in Time O(n log n) 24 iii 3.4 Arrangements of Pseudolines . 29 3.4.1 Ordinary Intersection Points . 29 3.4.2 A O(n2) Algorithm to Find an Ordinary Point . 30 3.4.3 Existence of Monochromatic Points in a Bichromatic Ar- rangement . 32 3.4.4 Algorithm to Find a Monochromatic Intersection in a Bichromatic Arrangement of Pseudolines . 33 4 Bichromatic and Equichromatic Lines 35 4.1 Introduction . 35 4.2 Equichromatic Lines . 37 4.2.1 Lower Bound in R2 ...................... 37 4.2.2 Proof of the Kleitman-Pinchasi Conjecture . 39 4.2.3 Equichromatic Lines With Few Points . 40 4.3 Lines in C2 .............................. 45 4.3.1 A Lower Bound For Bichromatic Lines . 45 4.3.2 Bichromatic Lines Through At Most Six Points . 46 4.3.3 Lower Bound On Total Number of Lines . 47 4.3.4 The Kleitman-Pinchasi Conjecture Revisited . 49 5 Conclusions and Future Work 51 5.1 Finding Ordinary or Monochromatic Intersection Points . 51 5.2 Bichromatic and Equichromatic Lines . 51 iv List of Figures 2.1 A perpendicular from point pk to line pipj is shorter than the perpendicular from pi to line pjpk.................. 4 2.2 The Kelly-Moser configuration has seven points that determine only three ordinary lines, and thus, t2(7) = 3. 13 2.3 McKee's configuration showing thirteen points determining only six ordinary lines. 14 2.4 Known values of t2(n) originally published in [1]. 14 2.5 A B¨or¨oczkyconfiguration of ten points determining five ordinary lines. 15 2.6 A counter example to Da Silva and Fukuda's conjecture. Origi- nally published in [2]. 16 3.1 A line passing through (non-ordinary) point Q will determine an intersection closer to L0 than X. .................. 21 3.2 If X is the lowest intersection determined by lines not parallel to L0, then Y must be ordinary. 23 3.3 If point R is not ordinary, then a third pseudoline L3 must cross either segment PQ or PS....................... 31 3.4 If point R is monochromatic, then a third pseudoline L3 with a different color must cross either segment PQ or PS. 33 v List of Tables 5.1 Best General Lower Bounds . 52 5.2 Best Equichromatic Lower Bounds . 52 5.3 Best Bichromatic Lower Bounds . 52 vi Chapter 1 Introduction Many (if not most) problems in discrete mathematics can be intuitively under- stood by a well motivated person. Some of these problems even have a semblance to the \brain teasers" one might see published in a newspaper. Although they seem intuitive, their solutions (if known) are often quite complicated. Among the most intriguing problems in discrete mathematics are those with a geometric aspect. When a problem is stated geometrically, one might feel that intuition provides a significant advantage in finding the solution. However tantalizing these may be to researchers, many of these \intuitive" problems are still open, i.e., no satisfactory solution is known. These attributes make such problems captivating for study. Several books are devoted to discussing open problems in combinatorial (or discrete) geometry (e.g., [3], [4] or [5]) with the hope that a reader will be inspired to find new insights or, perhaps, even find a solution. The present thesis presents several results from the fields of combinatorial and computational geometry. These fields each contain interesting questions about the nature of geometrical structures. As the name suggests, combinato- rial geometry studies discrete aspects of geometry, often attempting to answer questions about the enumeration (or counting) of geometric objects. Similarly, computational geometry is a search for the most efficient methods to perform 1 computation on geometric objects. These two fields often have a symbiotic relationship, i.e., results in one field affect the other. Combinatorial geometry, like several other fields of mathematics, has Paul Erd}osas a protagonist. It was a question published by him in 1943 ([6]) about points and the lines that instigated the study of such discrete structures. Sub- sequent to his question, other famous mathematicians (e.g., Th. Motzkin, G. A. Dirac, and L. M. Kelly) followed his lead into this subject. Erd}os'questions not only instigated, but also were a driving force behind the development of this field. A mention should also be made of Michael I. Shamos whose 1978 thesis ([7]) was the first to address computational questions about geometry. After his thesis, the field of computational geometry developed quickly. Within a few years, the first significant book on the subject was published, [8], for which Shamos is a co-author. I hope the reader would agree that only minimal background knowledge is necessary to understand and appreciate the results recorded here. For readers who are beginning their study of these topics, Chapter 2 will provide some of the necessary context from which to begin study. Chapters 3 and 4 contain the primary results for this thesis. Both of these chapters have been submitted for publication, and Chapter 4 has already ap- peared in [9]. Chapter 5 concludes with a brief discussion of possible future work to extend these results. Results were found, under the direction of my adviser, between Spring of 2008 and Summer of 2009. 2 Chapter 2 Background 2.1 A Question Of J.J. Sylvester In 1893, J.J. Sylvester posed the following problem [10], Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line. A simpler way might be to ask, \Does every set of noncollinear points necessarily determine an ordinary line (i.e. a line passing through exactly two of them)?".