The swirling maelstrom in "Pirates of the Caribbean: At World's End" or the beguiling rats turning out gourmet food in "Ratatouille" were a product of computer-generated imagery (CGI) grounded in mathematics, opening a whole new world of enchantment in cinema.
Mathematics provides a critical translation from the physical world to computer simulations.
The use of complex calculations in cinematic special effects has been described by three University of California, Los Angeles (UCLA) mathematicians:Aleka McAdams, Stanley Osher and Joseph Teran, who have made significant contributions to research in the area.
Mathematics provides the language for expressing physical phenomena and their interactions, often in the form of partial differential equations.
The equations are usually too complex to be solved exactly, so mathematicians have developed numerical methods and algorithms that can be implemented on computers to obtain approximate solutions.
The kinds of approximations needed to, for example, simulate a firestorm, were in the past computationally intractable.
With faster computing equipment and more-efficient architectures, such simulations are feasible today and they drive many of the most spectacular feats in the visual effects industry.
The area of computational fluid dynamics (CFD) provides many of the tools used in simulations of phenomena such as smoke, fire and water.
Before the use of CFD, computer-generated special effects such as explosions were driven by force fields applied to passive unconnected particles, a method that produced rather unrealistic results.
Today, a combination of improved hardware and faster algorithms for CFD models have made such special effects much more realistic.
Mathematics also plays a key role in computer-generated animations of all kinds of solids, from animated characters to cityscapes.
Virtually every computer-generated solid has an explicit mathematical representation as a meshed surface or volume. Flesh simulations can endow computer-generated characters with realistically bulging muscles and rippling fat.
Hair simulation provides a realistic way to depict the highly complex phenomenon of thousands of hairs interacting and colliding, says a UCLA release.
The effects industry is emerging as an exciting new frontier for mathematicians, one that uniquely combines mathematical insights with the art of moviemaking.
The findings are slated for publication in the May issue of the Notices of the American Mathematical Society.