April 14, 2010
Whether they're used to create a maelstrom in Pirates of the Caribbean: At World's End or rats cooking a gourmet meal in Ratatouille, computer-generated effects have opened a world of enchantment in cinema. These kinds of special effects are grounded in, and depend upon, mathematics.
Aleka McAdams, Stanley Osher, and Joseph Teran (all from the University of California, Los Angeles) expound on this claim in "Crashing Waves, Awesome Explosions, Turbulent Smoke, and Beyond: Applied Mathematics and Scientific Computing in the Visual Effects Industry" (pdf). Their paper was published in the May 2010 Notices of AMS.
Mathematics plays a key role in computer-generated animations, from animated characters to cityscapes. Nearly every computer-generated solid has an explicit mathematical representation as a meshed surface or volume. Flesh simulations, the authors noted, now endow computer-generated characters with realistically bulging muscles and rippling fat. And hair simulation provides a realistic way to depict the highly complex phenomenon of thousands of hairs interacting and colliding.
Mathematics, the researchers noted, offers the language for expressing physical phenomena and their interactions, often in the form of partial differential equations. Because such equations are usually too complex to be solved in their entirety, mathematicians have turned to numerical methods and algorithms, which can be implemented on computers to obtain approximate solutions to simulate, for example, firestorms.
Moreover, computational fluid dynamics (CFD) provides many of the tools used in simulations of phenomena such as smoke, fire, and water.
Better hardware and faster algorithms for CFD models have made such special effects much more realistic. CFD has also been used to simulate water-based phenomena. Mathematician-computer scientist Ronald Fedkiw (Stanford University), in fact, received an Academy Award for Technical Achievement for that kind of simulation. Examples of his work on computer-generated fluids can be seen in the seas in The Pirates of the Caribbean films and the dragon's flaming breath in Harry Potter and the Goblet of Fire.
Source: Notices of the AMS (May, 2010)
The three stages of animating Bill Nighy's character (Pirates of the Caribbean) image via Wikipedia.