Behind the scenes of most movie special effects are computers crunching intense mathematical equations. And some of the most complex types of animation equations describe fluid motion: anything from lava flow to an explosion to the rise and disappearance of smoke rings. But many times, the equations available to animators aren't good enough to accurately represent and control fluids, says Mathieu Desbrun, professor of computer science at the California Institute of Technology, in Pasadena. In order to make fluid animation look good enough, he says, some animators opt to draw it by hand--a time-consuming process.

But Desbrun's research could make fluids flow more nicely on screen. He and his team are developing an entirely new approach to fluid motion, based on new mathematics called discrete differential geometry, that use equations designed specifically to be solved by computers rather than people. Ultimately, he says, they have the potential to cut the cost and time of making a piece of animation. "Now that we're using computers, it's a whole new ball game," he says.

Before computers, Desbrun explains, mathematicians and physicists developed equations for the motion of objects such as solids and fluids, and many of them were solvable by hand. Over the past few decades, it became clear that computers could be used to solve many of the more difficult equations, so computer scientists and mathematicians took the known set of equations and tried to modify them for the task. They reworked the equations explaining the physical rules, effectively breaking them into hundreds of numerous chunks so that the digital brain of a computer, which is good at working on a lot of these chunks at one time, could solve them.

While much successful work has been done using this approach, says Desbrun, these equations still only approximate motion, and they tend to produce unnaturally flowing liquids. For instance, in the case of a whirlpool, over time the traditional approach introduces errors into the motion, producing artificial viscosity: the visual result is a swirling whirlpool that slows down for no obvious reason. An animator must step in to modify the frames to make sure that the liquid keeps moving the right way.

Desbrun's approach is to write new equations that are based on physical properties that aren't expressed in the traditional equations. For instance, traditional equations include information about the velocity of a liquid, and this is used to approximate, or provide an inexact description of, a liquid's motion if it starts to swirl around. But Desbrun's equations bypass simple velocity and instead describe the swirling motion exactly, and in a way that computers can easily crunch. "Instead of just approximating them, we can capture the dynamics faithfully," he says. "And we show it pays off visually."