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Visualization of Curve and Surface Data Using Rational Cubic Ball Functions CORE Metadata, citation and similar papers at core.ac.uk Provided by Repository@USM VISUALIZATION OF CURVE AND SURFACE DATA USING RATIONAL CUBIC BALL FUNCTIONS WAN NURHADANI BINTI WAN JAAFAR UNIVERSITI SAINS MALAYSIA 2018 VISUALIZATION OF CURVE AND SURFACE DATA USING RATIONAL CUBIC BALL FUNCTIONS by WAN NURHADANI BINTI WAN JAAFAR Thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy February 2018 ACKNOWLEDGEMENT In the name of Allah S.W.T, the Most Gracious and the Most Merciful. First and foremost, I am most grateful to Allah, the One who gives me the strengths and the knowledge to complete this unforgettable journey. For this opportunity, I would like to thank those who helped me and encouraged me to complete this PhD journey. Firstly, I would like to express my deepest gratitude and appreciation to my main supervisor, Professor Dr. Abd. Rahni Mt Piah for his valuable guidance and advice throughout this study. Secondly, I would like to thank my field supervisor, Dr. Muhammad Abbas from University of Sargodha, Pakistan, who has constantly and consistently help me. Most importantly, I would like to thank University Malaysia Perlis and Ministry of Higher Education for the financial support during my PhD. Last but not least, special thanks to my beloved parents Che Jamilah Sulong, Wan Jaafar Wan Mohamad and all my friends, especially, Siti Jasmida, Hartini, Noor Wahida, Ezzah Liana, Shakila and Nur Fatihah. Not to forget, all my lab friends and the staffs at School of Mathematical Sciences, Universiti Sains Malaysia for their undivided attention, motivation, support and advice. ii TABLE OF CONTENTS ACKNOWLEDGEMENT .......................................................................................... ii TABLE OF CONTENTS ........................................................................................... iii LIST OF TABLES .................................................................................................... vii LIST OF FIGURES ................................................................................................. viii LIST OF ABBREVIATIONS ..................................................................................... x LIST OF SYMBOLS ................................................................................................. xi ABSTRAK…… ........................................................................................................ xii ABSTRACT………………………………………………………………………..xiii CHAPTER 1 - BACKGROUND OF STUDY AND LITERATURE REVIEW 1.1 Data Visualization .................................................................................................. 1 1.2 Shape Characteristics of Data and Applications .................................................... 3 1.2.1 Shape Characteristics ................................................................................... 3 1.2.2 Applications ................................................................................................. 7 1.3 Ball Functions and Advantages .............................................................................. 8 1.4 Continuity Conditions .......................................................................................... 14 1.5 Literature Review ................................................................................................. 15 1.5.1 Positivity Preserving Curve and Surface ................................................... 16 1.5.2 Monotonicity Preserving Curve and Surface ............................................. 21 1.5.3 Convexity Preserving Curve and Surface .................................................. 25 1.5.4 Constrained Preserving Curve and Surface ............................................... 28 1.6 Motivation of Study ............................................................................................. 29 1.7 Objectives of Study .............................................................................................. 31 1.8 Outline of the Thesis ............................................................................................ 32 iii CHAPTER 2 - RATIONAL CUBIC BALL WITH SHAPE CONTROL 2.1 Introduction .......................................................................................................... 35 2.2 C1 Rational Cubic Ball Interpolant ..................................................................... 36 2.3 Shape Control Analysis ........................................................................................ 37 2.4 Conclusion ........................................................................................................... 49 CHAPTER 3 - SHAPE PRESERVING RATIONAL CUBIC BALL CURVE INTERPOLATION WITH FOUR SHAPE PARAMETERS 3.1 Introduction .......................................................................................................... 51 3.2 Rational Cubic Ball Function with Four Shape Parameters................................. 53 3.2.1 Shape Preserving C1 Curve Visualization.................................................. 54 3.3 Positivity Preserving Rational Cubic Ball Interpolation ...................................... 55 3.4 Monotonicity Preserving Rational Cubic Ball Interpolation ............................... 59 3.5 Constrained Rational Cubic Ball Interpolation .................................................... 63 3.6 Shape Preserving Rational Cubic Bézier to Rational Cubic Ball using ............... 67 Conversion ........................................................................................................... 67 3.6.1 Definition Rational Cubic Bézier Function and Rational Cubic Ball ........ 68 Function ..................................................................................................... 68 3.7 Conversion Matrices between Rational Cubic Bézier Curve and Rational ......... 70 Cubic Ball Curve .................................................................................................. 70 3.7.1 Coefficients Matrix of Bézier Curve.......................................................... 70 3.7.2 Coefficients Matrix of Ball Curve ............................................................. 71 3.8 Relationship of Weights and Control Points between Rational Cubic Bézier ..... 72 Curve and Rational Cubic Ball Curve .................................................................. 72 iv 3.9 Rational Cubic Ball Interpolation using Conversion Control Points and ............ 74 Weights ................................................................................................................ 74 3.10 Positivity Rational Cubic Ball with Conversion Interpolation .......................... 75 3.11 Monotonicity Rational Cubic Ball with Conversion Interpolation .................... 78 3.12 Convexity Rational Cubic Ball with Conversion Interpolation ......................... 82 3.13 Constrained Rational Cubic Ball with Conversion Interpolation ...................... 87 3.14 Conclusion ......................................................................................................... 90 CHAPTER 4 - SHAPE PRESERVING RATIONAL CUBIC BALL CURVE INTERPOLATION WITH THREE SHAPE PARAMETERS 4.1 Introduction ......................................................................................................... 93 4.2 Rational Cubic Ball Function with Three Shape Parameters ............................... 94 4.3 Shape Preserving C1 Curve Visualization ............................................................ 95 4.4 Positivity Rational Cubic Ball Interpolation ........................................................ 96 4.5 Monotonicity Rational Cubic Ball Interpolation.................................................. 99 4.6 Convexity Rational Cubic Ball Interpolation ..................................................... 102 4.7 Constrained Rational Cubic Ball Interpolation .................................................. 105 4.9 Conclusion ......................................................................................................... 108 CHAPTER 5 - RATIONAL BI-CUBIC BALL SURFACE INTERPOLATION 5.1 Introduction ........................................................................................................ 110 5.2 Partially Blended Rational Bi-Cubic Ball Function (Coons Patches) ................ 111 5.3 Visualization of Positive Shaped Data by Partially Blended Rational .............. 115 Bi-Cubic Ball ..................................................................................................... 115 v 5.4 Visualization of Monotonic Shaped Data by Partially Blended Rational .......... 121 Bi-Cubic Ball ..................................................................................................... 121 5.5 Visualization of Constrained Data by Partially Blended Rational Bi-Cubic ..... 129 Ball ..................................................................................................................... 129 5.6 Rational Bi-Cubic Ball Function (Tensor Product Patches) .............................. 138 5.7 Visualization of Positive Shaped Data by Rational Bi-Cubic Ball .................... 142 5.8 Visualization of Monotonic Shaped Data by Rational Bi-Cubic Ball ............... 147 5.9 Visualization of Constrained Data by Rational Bi-Cubic Ball .......................... 162 5.10 Conclusion ....................................................................................................... 174 CHAPTER 6 - CONCLUSION AND FURTHER WORKS 6.1 Conclusion ........................................................................................................
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