A Simple Model of Ship Wakes by RAZA S. KHAN B.S., Denison University, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF COMPUTER SCIENCE We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1994 © Raza S. Khan, 1994 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) \ Department of Computer Science The University of British Columbia Vancouver, Canada Date 94/09/24 Abstract While ocean waves were among the first natural phenomenon to be modeled satisfactorily in computer graphics, waves from ships—ship wakes—have been largely ignored. The model presented in this thesis is suitable for animating wakes created by a ship moving along an arbitrary course. Instead of the dynamic solution of a free-surface problem that can be computationally expensive and unstable, the approach presented is kinematic and while ad hoc, is efficient and simple to implement. The model superimposes circular waves emanating from points along the ship's path to determine the wake profile. In doing so, characteristics of the ship's hull are ignored. The approach is similar to the mathematical treatment of Kelvin's method of stationary phase, where curves of constant phase are obtained by integrating point impulses over the ship's course. Accuracy in obtaining an exact profile for the surface, as determined by the stationary phase method for closed-form solutions, is sacrificed for the ability to specify an arbitrary path. The problem then becomes one of generating circles with a height field associated with them. This is done by adopting two different methods; one that uses a midpoint circle algorithm based on Bresenham's incremental circle generator and another that efficiently determines a profile for the circles. The path of the ship is represented by parametric piecewise-cubic curves and the water surface by a height field. An animation is obtained by generating the height field for successive positions of the ship along the curve. 11 Contents Abstract ii Table of Contents iii List of Figures v Dedication vi Acknowledgement vii 1 Introduction 1 1.1 Motivation 1 1.2 Objective 3 1.3 Previous Work in Modeling Waves 6 1.4 Overview 8 2 Mathematical Background 9 2.1 Determination of Region of Disturbance 10 2.1.1 Solution for disturbance created by a point impulse 10 2.2 Obtaining Curves of Constant Phase 13 2.2.1 Curves of constant phase for a circular course 13 2.2.2 Curves of constant phase for a straight course 15 2.3 Wake Geometry for a Straight Path 16 2.4 Estimating the Profile for a Circular Wave 19 iii 3 Graphical Model of Ship Wakes 21 3.1 Introduction 22 3.2 Ship Path Specification 22 3.2.1 The Bezier curve 23 3.2.2 Arc-length parameterization 24 3.3 Techniques Used to Generate Circular Waves 24 3.3.1 Calculating radius and amplitude of the circles 25 3.3.2 Midpoint circle scan-conversion algorithm 26 3.3.3 Computing a profile for circles 27 4 Results 29 4.1 Varying the Path 30 4.2 Changing the Number of Circles 34 4.3 Effect of Acceleration and Deceleration on Wake Produced 36 5 Conclusion 39 5.1 Future Work 40 A Algorithms 42 A.l Pseudocode for Arc-Length Parameterization 42 Bibliography 44 IV List of Figures 1.1 Ship wake characteristics 4 1.2 Geometry for Goss's representation of ship wakes 7 2.1 Notation used for the ship wave problem 11 2.2 Points influenced by a moving source 12 2.3 Curves of constant phase for a circular path 14 2.4 Curves of constant phase for a straight path 15 2.5 Wake geometry for a straight path 16 2.6 Wave groups for vessel on a straight path 17 2.7 Plot of ^ 18 X 2.8 Surface profile of Stokes's wave compared to a sinusoidal wave 20 3.1 Effect of arc-length parameterization on curve 25 3.2 Clipped region for circle profile calculation 28 4.1 Images obtained for a straight course before and after filtering 31 4.2 Images obtained for a curved path before and after filtering 32 4.3 Images obtained for a self-intersecting path before and after filtering. 33 4.4 Showing effect of number of circles on the wake 35 4.5 Wake produced by a vessel that is decelerating 37 4.6 Wake produced by a vessel that is accelerating 38 v To nana, nani, dada, dadi VI Acknowledgements I would like to thank my supervisor, Dr. Alain Fournier and my second reader, Dr. David Forsey for their guidance in the completion of this thesis. I am grateful to the following, without whom this work would not have reached fruition: Bill, for taking on the role of student reader without even being asked. My thanks to him for countless McD. runs, his counsel to exercise restraint in the use of Forms, and much useful advice over the course of our friendship. Chris R. for his glowing presence in the lab, his sympathetic ear to my many grievances, and for vacating Lucille on numerous occasions during the troublesome coding days (nights); Chris H. for preserving some sanity by taking me on movie runs; Rob, for his late night stories; Sameer, for being the permanent resident on the third floor no matter what the time! and Vishwa, for much encouragement. I would also like to express my acknowledgments to my brother, Ayaz, a constant source of inspiration for always being there and my parents for their endless support and love. I remain indebted to the department for providing me with financial support during the course of my studies here. vn Chapter 1 Introduction Once more upon the waters! yet once more! And the waves bound beneath me as a steed That knows his rider. -Lord Byron, 1788-1824 A wake is "the track left by a moving body (as a ship) in a fluid (as water)"1. This thesis presents a method for modeling ship wakes suitable for computer graphics. 1.1 Motivation While substantial research has been directed towards the theory of wake production by mathematicians and hydrodynamists alike [kelv87], [sher56], [stok57], [kost68], [mich69], [newm77], [crap84], it has not spurred similar interest among graphics researchers to model the phenomenon. This comes as more of a surprise when one realizes that waves, albeit ocean waves, were among the first natural phenomenon to be modeled satisfactorily 1 Webster's 7th dictionary. 1 Chapter 1. Introduction 2 in computer graphics [fole90]. Ocean waves themselves, however, were but a link in the chain of evolution that extends not so far back to the days of modeling static geometrical primitives such as lines, circles, and rectangles. Computer graphics has indeed evolved to capture some aspects of the complex motion of fluid flows [gate94], growth of botanical systems [prus93], explosions [reev83], and other phenomenon that are considered natural. It is not only in this context, though, that one sees the importance of modeling ship wakes. Another incentive comes from the interaction between two objects. Animating a boat and applying textures to enhance the image can produce convincing results, but the motion of a boat affects the surface underneath it and this phenomenon needs to be captured as well. It is with this purpose in mind that this model has been developed. An extension of the problem, modeling the interaction between two types of models, has not been much explored. However, it will gain attention as animation systems become further integrated. The few published examples of this approach include interaction of wind fields with plant models [shin92] and the effect of the environment on plants [prus94]. Goss models ship wakes to provide a motion cue for aircraft pilots in assessing the movement of ships [goss90]. Since the model was intended for flight simulation, a real­ time method was essential. Also, considering that the viewing distance from aircrafts flying overhead is large compared to wave amplitude, a 2-D model was sufficient. The present work is not limited by such considerations and therefore a more general 3-D model is desired. Animators interested in capturing the motion of vessels over water would benefit from a model that incorporates ship wakes. Such a model could accommodate one or several of the following criteria: shape and size of the ship, speed and path of motion, depth of Chapter 1. Introduction 3 water, and various factors not limited to wind and turbulence on the water surface. The model subsequently discussed allows specification of an arbitrary path in generating a wake and is simple to implement. 1.2 Objective The principle interest here is in modeling realistic shape and motion of the waves gener­ ated as a ship moves through water. Realistic does not imply the waves follow precisely the dynamics of ship waves. Considering that ship waves are influenced by several fac­ tors including but not limited to ship motion, wind, and wave motion [goss90], such a task would be quite difficult.