The Cone Tessellation Model for Three-Dimensional Networks Gözde Sarışın Submitted to the Institute of Graduate Studies and Research In Partial Fulfilment of the Requirements for the Degree of Master of Science in Computer Engineering Eastern Mediterranean University May 2011 Gazimağusa, North Cyprus Approval of the Institute of Graduate Studies and Research Prof. Dr. Elvan Yılmaz Director I certify that this thesis satisfies the requirements as a thesis for the degree of Master of Science in Computer Engineering. Assoc. Prof. Dr. Muhammad Salamah Chair, Department of Computer Engineering We certify that we have read this thesis and that in our opinion it is fully adequate in scope and quality as a thesis for the degree of Master of Science in Computer Engineering. Assoc. Prof. Dr. Muhammed Salamah Supervisor Examining Committee 1. Prof. Dr. Rza Bashirov ___________________________ 2. Assoc. Prof. Dr. Muhammed Salamah _ 3. Asst. Prof. Dr. Gürcü Öz _ ABSTRACT Wireless terrestrial networks are usually designed in two dimensional plane, but in real life they form three dimensional space. For navy, air force and under water applications, full connectivity and coverage are necessary to achieve good surveillance. Coverage and connectivity issues become more important to achieve full connectivity with less number of nodes. In three dimensional networks, node placement strategy is one of the most important design problems. In this thesis, main aim is to deploy the minimum number of nodes in an effective way to get communication among nodes over a multi-hop path. Sensing range and transmission range values are main factors for our calculations. The volumetric quotient, which is the ratio of the transmission range (represented as the volume of the shape) to the sensing range (represented as the volume of the shape’s circumsphere) of each node, is used as the main measure of the placement strategy. Researchers use polyhedrons to model three dimensional networks. As the volumetric quotient increases, we need less number of nodes for full coverage. The contribution of this research is finding a cone model which gives a higher volumetric quotient than polyhedrons. The inspiration comes from satellite foot- print. The cone model was tessellated using 3Ds Max which is a modelling, animation and rendering software. Cones are deployed in special manner that has no empty space between nodes. So we can achieve full connectivity with cone model. We compared our results with previous researches and we achieve better results in terms of volumetric quotient and number of nodes needed compared to other models. Our result achieved the minimum of transmission ranges for both axes. Best results found in terms of maximum of minimum transmission ranges for u and v axes. This research shows us we can achieve full coverage with using a shape rather than polyhedrons. Keywords: Modelling, three dimensional networks, tessellation, Wireless Sensor Networks, satellite foot-print ÖZ Yeryüzündeki kablosuz ağlar genellikle 2 boyutlu olarak dizayn edilir ama gerçek hayatta 3 boyutlu bir alan oluştururlar. Askeri, hava ve su altı uygulamalarında iyi bir gözlem yapabilmek için tam kapsama ve bağlantıya ihtiyaç vardır. Az sayıda sensör kullanarak tam bağlantı sağlamada kapsama ve bağlantı önemli bir rol oynamaktadır. 3 boyutlu ağlarda en önemli dizayn problemlerinden biri sensörü yerleştirme stratejisidir. Bu tezdeki asıl amaç çoklu-sıçrama yolundaki bağlantıyı sağlamak için en az sayıda sensörü en uygun şekilde yerleştirmektir. Algılama ve transfer alanı bizim hesaplamalarımızın ana faktörleridir. Volumetrik katsayı her bir sensörün transfer alanının algılama alanına bölümüyle bulunur, ki bu da yerleştirme stratejisinin ana hesaplamasıdır. Araştırmacılar 3 boyutlu ağları modellemek için çok yüzlü olan 3 boyutlu şekilleri kullanmışlardır. Volumetrik katsayı arttıkça tam kapsamayı elde etmek için daha az bağlantı noktasına ihtiyacımız olur. Bu araştırmanın katkısı, koni modellinin çok yüzlü 3 boyutlu şekillerden daha yüksek volumetrik katsayıya sahip olmasıdır. Fikir uydu ayak izinden ortaya çıkmıştır. Koni modeli, bir modelleme, animasyon ve şekil çevirme programı olan 3Ds Max ile 3 boyutlu düzleme yerleştirilmiştir. Koniler özel bir strateji ile yerleştirilmiş olup bağlantı noktaları arasında boşluk yoktur. Böylelikle tam bağlantıyı koni modeliyle elde ettik. Sonuçlarımızı önceki araştırmalarla karşılaştırdık ve volumetrik katsayı, gerekli olan bağlantı noktası sayısı, tüm eksenler için minimum transfer alanı ve u ve w eksenlerinde minimum transfer alanında en iyi sonuçları elde ettik. Bu araştırma bize tam bağlantıyı elde etmek için çok yüzlü 3 boyutlu şekiller yerine başka şekiller kullanılabileceğini gösterdi. Anahtar kelimeler: modelleme, 3 boyutlu ağlar, kablosuz sensör ağları, uydu ayak izi, Kelvin varsayımı, Kepler varsayımı ACKNOWLEDGMENTS I would like to thank my supervisor, Assoc. Prof. Dr. Muhammed Salamah, for his continuous support and guidance through my research. This thesis would not have been completed without his encouragement and valuable ideas. I am very grateful to Mustafa Erbilen. We discussed mathematical calculations in a long period of time, and I gained lots of information from his ideas. I would like to thank my mum for her financial and psychological support. At last, but not trivial, I would like to thank my precious friend Erdem, for his love, effort and patience to me. This thesis dedicated to my mum and to him. To My Mum and Erdem Ulutaş TABLE OF CONTENTS ABSTRACT..................................................................................................................iii ÖZ..................................................................................................................................iv ACKNOWLEDGMENTS........................................................................................... vii DEDICATION............................................................................................................viii LIST OF TABLES........................................................................................................xi LIST OF FIGURES.....................................................................................................xii LIST OF SYMBOLS/ABBREVIATIONS.................................................................xv 1 INTRODUCTION......................................................................................................1 1.1 Introduction to Wireless Networks.....................................................................1 1.2 Three Dimensional Wireless Networks..............................................................4 1.3 Thesis Contributions...........................................................................................5 1.4 Thesis Outline.....................................................................................................6 2 RELATED WORK.....................................................................................................7 2.1 Filling a space with polyhedron.........................................................................7 2.2 Kelvin’s Conjecture...........................................................................................8 2.3 Kepler’s Conjecture.........................................................................................10 2.4 Voronoi Tessellation .......................................................................................12 2.5 Satellite Foot-print...........................................................................................18 2.6 Assumptions and Goals of Related Works ......................................................19 2.7 Volumetric Quotient ........................................................................................20 2.7.1 Volumetric Quotients for Polyhedrons .....................................................23 2.8 Placement Strategies for Polyhedrons.............................................................27 2.8.1 Placement Strategy for Cube....................................................................27 2.8.2 Placement Strategy for Hexagonal Prism.................................................27 2.8.3 Placement Strategy for Rhombic Dodecahedron.....................................28 2.8.4 Placement Strategy for Truncated Octahedron........................................28 2.9 Simulation.........................................................................................................29 2.10 Related Works About Coverage in History....................................................32 3 THE PROPOSED CONE MODEL.........................................................................34 3.1 General Information About the Proposed Cone Model ...................................34 3.2 Optimal Height for Cone Model ......................................................................35 3.3 Idea of Satellite Foot Print ...............................................................................36 3.4 Placement Strategy for Cone Model ................................................................37 3.5 Tessellation for Cone Model in Three Dimension............................................38 3.6 Advantage of Cone Model................................................................................39 4 PERFORMANCE ANALYSIS................................................................................40 4.1 Comparison of Volumetric Quotients and Number of Nodes.........................40 4.2 Comparison of Minimum