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Univerza Na Primorskem Fakulteta Za Matematiko, Naravoslovje in Informacijske Tehnologije UNIVERZA NA PRIMORSKEM FAKULTETA ZA MATEMATIKO, NARAVOSLOVJE IN INFORMACIJSKE TEHNOLOGIJE Master's thesis (Magistrsko delo) Properties and applications of the moment curve (Lastnosti in aplikacije momenta krivulje) Ime in priimek: Jasmina Begovi´c Studijskiˇ program: Matematiˇcneznanosti 2. stopnja Mentor: Izr. prof. dr. Gy¨orgyKiss Koper, januar 2020 Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020 II Kljuˇcnadokumentacijska informacija Ime in PRIIMEK: Jasmina BEGOVIC´ Naslov magistrskega dela: Lastnosti in aplikacije momenta krivulje Kraj: Koper Leto: 2020 Steviloˇ listov: 58 Steviloˇ slik: 12 Steviloˇ tabel: 2 Steviloˇ prilog: 3 Steviloˇ strani prilog: 17 Steviloˇ referenc: 52 Mentor: Izr. prof. dr. Gy¨orgyKiss UDK: 514.14(043.2) Kljuˇcnebesede: klasiˇcnamoment krivulja, zvit kubik, cikliˇcnipolitop, k-lok, MDS kodi Math. Subj. Class. (2010): 51N15, 52B05, 51E22 Izvleˇcek: Glavna tema magistrske naloge je lastnosti in nekatere aplikacije klasiˇcnemomentne krivulje. Magistrsko nalogo zaˇcnemos konstrukcijo krivulje kot preseka doloˇcenihob- jektov in pokaˇzemonjeno pomembnost in uporabo. Nato konstruiramo cikliˇcnipolitop kot konveksno ogrinjaˇcotoˇckna momentni krivulji in upoˇstevamo nekatere od na- jpomembnejˇsihlastnosti (skupaj z dokazi veˇcineod njih). Z uporabo zvite kubiˇcne krivulje klasificiramo vse racionalne kubiˇcnekrivulje z ustreznimi projekcijami. Kas- neje predstavimo pomemben razred kod, uporabljenih za odstopanje napak (MDS kode) in njihovo povezavo z loki in momentno krivuljo. Na koncu omenimo znano MDS pred- postavko in njeno geometrijsko razliˇcico. Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020 III Key words documentation Name and SURNAME: Jasmina BEGOVIC´ Title of Master's thesis: Properties and applications of the moment curve Place: Koper Year: 2020 Number of pages: 58 Number of figures: 12 Number of tables: 2 Number of appendices: 3 Number of appendix pages: 17 Number of references: 52 Mentor: Assoc. Prof. Gy¨orgyKiss, PhD UDK: 514.14(043.2) Keywords: classical moment curve, twisted cubic, cyclic polytope, k-arc, MDS codes Math. Subj. Class. (2010): 51N15, 52B05, 51E22 Abstract: The main topic of the Master thesis is properties and some applications of the classical moment curve. In the Master thesis we start by constructing the curve as intersection of certain objects and by showing its importance and use. Further, we construct the cyclic polytope as a convex hull of points on the moment curve and consider some of their most important properties (along with the proof for most of them). Using the twisted cubic, we classify rational cubic curves by appropriate projections. Finally, we introduce an important class of codes used for fault tolerance, the MDS codes, and their link to arcs and the moment curve. Lastly, we mention famous MDS conjecture and its geometric version. Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020 IV List of Contents 1 Introduction 1 1.1 Affine and Projective spaces . 2 1.2 Curves . 8 1.3 The Moment Curve . 10 1.3.1 A Geometric Construction . 12 1.3.2 Properties and Applications . 13 2 The Cyclic Polytope 16 2.1 Polytopes . 16 2.2 Euler's Polyhedral Formula . 21 2.3 An Application of Euler's Polyhedral Formula in R3 . 24 2.4 The Cyclic Polytope . 25 2.5 An Observation About the Cyclic Polytope . 26 2.6 Properties of the Cyclic Polytope . 28 3 The Twisted Cubic 33 3.1 Cubic Curves . 33 3.2 The Line at Infinity l1 ........................... 37 3.3 Tangents and Intersections . 39 3.4 Classes of Cubic Curves . 40 3.4.1 Cubic Curves With µ =1 ..................... 40 3.4.2 Cubic Curves With µ =2 ..................... 40 3.4.3 Cubic Curves With µ =3 ..................... 41 4 Channel Coding 44 4.1 MDS Codes . 44 4.2 MDS Conjecture . 49 5 Conclusion 52 6 Povzetek naloge v slovenskem jeziku 53 Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020 V 7 Bibliography 55 Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020 VI List of Tables 1 Cubic curves as projections of the twisted cubic . 43 2 Pairs (n; q) for which it is proved that every normal rational curve in PG(n; q) is a complete (q + 1)-arc. 51 Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020 VII List of Figures 1 Geometries. 6 2 Twisted cubic with xy and xz surfaces in the interval [−1; 1]. (Credit: Claudio Rocchini/Wikimedia Commons) . 11 3 The Platonic Solids. (Credit: NASA Blueshift) . 16 4 Schlegel diagrams. (Credit: Don Steward/Mathematics teaching) . 22 2 5 The cyclic polytope C2(5) in R ....................... 26 6 Curves with an inflection point. 34 7 Curves with singularity at the origin. 36 8 Construction of the line at infinity. 38 9 Affine and projective planes. (Credit: Marcelino Ram´ırez-Ib´a~nez) . 10 Types of double point on (x2 + y2 − x)2 = b2(x2 + y2). 11 Conics and the line at infinity l1...................... 12 The Fano Plane. Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020VIII List of Appendices A Analytical Approach to Projective Space B Lines, Tangents and Conics C Coding Theory and Channel Coding Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020 IX List of Abbreviations i.e. that is iff if and only if e.g. for example w.r.t. with respect to A; B; : : : points a; b;::: vectors V the vector space K an arbitrary field N set of natural numbers R set of real numbers C set of complex numbers Rd d-dimensional space over reals P2 the projective plane Pd projective d-space A2 the affine plane Ad affine d-space l1 the line at infinity dim(A) dimension of affine subspace A conv(S) convex hull of set S Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020 X Acknowledgement I would like to thank my mentor Professor Gy¨orgyKiss for helping me with the writing of this thesis, for his time and patience, especially for all the reviews of my thesis. I also express my profound gratitude to my parents and to my friend Merita Sokoli for providing me with unfailing support and continuous encouragement. This accomplishment would not have been possible without all of you. Thank you! Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020 1 1 Introduction The earliest known work on curves and conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas. In addition to mathematics, the parabolic curve is present in physics, astronomy, wireless communications, industry, solar energy, engineering, and even opti- cal illusions. From a parabolic antenna or parabolic microphone, building of suspension bridges, to automobile headlight reflectors to the design of ballistic missiles. A well known instance of the parabola in physics is the trajectory of a particle or body in motion under the influence of gravity, which is always an approximation of a parabola. On the other hand, parametric polynomial cubic curve segments are widely used in computer-aided design and computer-aided geometric design applications because their flexibility makes them suitable for use in the interactive design of curves and surfaces. Further, for animations on computers, special cubic polynomial called "cubic tweening functions" are being used to ease (or smooth) the animation while in solid modeling and graphics applications curve interpolation (cubic splines) are widely used. For instance, the curves used in Postscript (including Postscript fonts), and in most drawing and graphics programs (like Adobe Illustrator, Powerpoint, etc.), are cubic B´eziercurves. The outlines of the characters you are reading right now might be strings of cubic B´eziercurves and the seemingly circular dot above the "i" is actually four cubic curves. Similarly, whenever you read a PDF document, or almost any printed document, you are (often) looking at strings of cubic curves. Graphics packages like OpenGL and Direct3D use cubic curves heavily so the curves you see in games and other 3D programs are also cubics. In thermodynamics, the most commonly used equations of state in industry (for research and development in particular in oil and gas industry) are called cubic, even if they do not look to be at first glance.1 Cubic equations are used widely in Elliptic Curve Cryptography and very popular Bitcoin crypto-currency and many other crypto-currencies based off Bitcoin. List of the applications can continue, but what connects these notions and what will be the topic of this thesis, is that both parabola and the twisted cubic (a cubic curve) are special cases of the moment curve. 1The first of them was introduced by Van der Waals in 1873. Begovi´cJ. Properties and applications of the moment curve. Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, leto 2020 2 1.1 Affine and Projective spaces Metrical geometry is a part of descriptive geometry, and descriptive geom- etry is all geometry.2 The ordinary geometry which deals with circles, angles, parallel lines, similar triangles and so on, is called Euclidean geometry because it was first collected into a systematic account by the Greek geometer Euclid. The plane geometry of the first six books of Euclid's Elements may be described as the geometry of lines and circles: its tools are the straight-edge (or unmarked ruler) and the compasses.
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