Efficient Modeling and Rendering of Turbulent Water over Natural Terrain Nathan Holmberg ∗ and Burkhard C. W ¨ unsche † Graphics Group, Department of Computer Science University of Auckland, Auckland, New Zealand Abstract Water phenomena are some of the most visually spectacular effects found in nature. This paper presents an efficient hybrid method to model turbulent water such as fast flowing rivers and waterfalls with the intent that the model can be used as part of a larger environment or scene. The model presented uses hydrostatic theory to incorpo- rate a 2D height field and a particle system to model respectively the main volume and spray of turbulent water. The user is able to submit any environment formed from spheres and panels making the solution very flexible and adaptable. A smooth representation of the water surface is obtained by fit- ting a uniform B-Spline surface to the height field. Foam, spray and other turbulent effects are represented by particles which are rendered as spheres or billboards. Our results show that the model provides a nearly realistic simulation of turbulent water and for sim- ple scenes nearly interactive speeds are possible which compares favorably with alternative techniques. For non-interactive applica- tions ray tracing can be used to obtain higher quality results. CR Categories: I.3.5 [Computer Graphics]: Computational Ge- ometry and Object Modeling—Physically based modeling; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism— Animation; Keywords: physically-based modeling, turbulent water, water simulation 1 Introduction The complexity and power of water flow in nature is both impres- sive and beautiful. It is also a part of our everyday lives and so well known to the human perception that unrealistic motion is eas- ily discernible. It is no surprise therefore that much effort has been applied in computer graphics to try to capture water in a convincing way. This is far from trivial as most phenomena can be attributed to complex rapidly changing molecular interactions, and as such are beyond the modeling power of today’s computers. Instead com- puter graphics researchers try to produce models with underlying physical principles that create, as accurately as possible, large scale approximations of these local interactions. ∗ e-mail: nhol021@ec.auckland.ac.nz † e-mail: burkhard@cs.auckland.ac.nz Research on the simulation of the behavior of water has pro- gressed from models which were only able to represent certain ef- fects, such as the motion of deep water waves to the exclusion of all else, to more general models with a firmer basis in hydrodynam- ics capable of realistic motion that follows expected behavior in a range of situations. These more recent models allow animators to specify environments and start conditions and the model will do the rest. The purpose of this work is to build a general model with a focus on the natural movement of rivers, rapids and waterfalls. As such turbulence, spray, and the like should be accounted for while not being treated as special cases. While not yet being complete and fully realistic, our model has been able to produce results that show its potential. 2 Previous Work The earliest work on modeling water in computer graphics used mathematical models and explicit functions to simulate surface be- havior. Some, while obeying physical equations, were very inflex- ible outside their intended scope such as the modeling of deep sea waves by Schachter [Schachter 1980] while others tried to model surface phenomena in 2D velocity fields [Neyret and Praizelin 2001]. Later work introduced rudimentary particle systems. Initially particle interaction was ignored and particles simply bounced around the environment [Reeves 1983; Sims 1990]. This produced reasonable effects for waterfalls and other instances where particle movement had enough energy to break the molecular level bonds as needed. In particle systems the equations for interaction can be complex and computationally expensive as each particle may be af- fected by every other resulting in a computational complexity of O(n 2 ) which is unacceptably slow when dealing with hundreds of thousands of particles. However, without inter-particle forces vol- umes of water can not be modeled well and the usually identifiable effects of adhesion and cohesion are missing. Some of the work at modeling these effects includes Miller and Pearce’s [Miller and Pearce 1989] connected particle system which used forces to im- itate soft collisions based on the difference in particle’s positions. This works well for viscous fluids, as intended by the authors, but is unsuitable for the number of particles or scale necessary to model large bodies of water. Kass and Miller [Kass and Miller 1990] introduced column sys- tems which use simplified flow equations, based on hydrostatics, between columns and treat the water volume as a 2D height field. This implicitly allows for the modeling of surface phenomena such as waves and can efficiently model large bodies of water. O’Brien and Hodgins [O’Brien and Hodgins 1995] extended this model by incorporating the interaction with external objects and splashes. However, a column’s height is still represented by one height vari- able meaning that vertical isotropy is assumed and only very simple (flat) environments can be used. Mould and Yang [Mould and Yang 1997] furthered this work by dividing each column into a user de- fined number of cells, relaxing the assumption of vertical isotropy. Another improvement was made by allowing for more complex en-