At the Heart of Every Great Pixar Film there is Math

As Pixar’s Senior Scientist, Dr. Tony DeRose is the genius that makes telling Pixar stories possible. While most animation might be considered an artistic discipline, computer-based animation relies on strong principles of mathematics. At a lecture DeRose gave at New York’s Museum of Math, he describes his job as “translating principles of arithmetic, geometry, and algebra into software that renders objects or powers physics engines.” Most everybody has seen a Pixar film or short and fallen for their near-flawless stories, but their advantage in computer animation that started in 1997 made Pixar stand out among their contemporaries.

Nearly every major element in a Pixar film have their own set of physics: hair, cloth, fluids, smoke, clouds, fire, etc. These components behave in specific and unique ways and to represent them realistically, Pixar scientists have to create physics engines for each element. But that’s just the beginning. Simulating water is easy,” said DeRose during his lecture. “What’s hard is, how do you make water more directable?” Every element in an animated film requires its own physics and rules that allow it to come to life and tell a story.

DeRose, who has a PhD in computer science with a specialty in computational physics and a decade of experience as a professor of computer science and engineering at the University of Washington, is concerned with not just making they physics of an animation realistic, but also be a part of the story being told. In the film Brave, the hair of the main character required its own physics engine so it could be beautiful, realistic, and expressive. Tim Carmody writes for The Verge:

“In the real world, hair keeps its bounciness and volume by constantly colliding with itself,” DeRose says. Merida’s hair is made of 100,000 individual elements. “If you know any combinatorics, you know that if you have n objects, you have n² possible collisions,” he says, or 10 billion. How can you render so many collisions quickly enough to be usable? You have to create a new spatial data structure that culls extraneous collisions without being too lossy. Instead of a quick-and-dirty compression algorithm like MP3 or JPEG, Pixar has to create the equivalent of the PNG or the FLAC for animating hair.

The detail of computer animation often requires more modeling calculations than what physicists might normally use. In order to make the production of a film feasible, DeRose spends a great deal of his time trying to create mathematical approximations for detailed calculations without sacrificing the fidelity of an animation. DeRose has worked extensively from converting the industry standard of polygons to smooth parabolas.

In most video game and film animation, models are built out of many planar surfaces called polygons, the concept being that if you increased the number of polygons by massive orders of magnitude, eventually the model would look organic and smooth. (This is similar to the idea of starting with a triangle, then adding another side to make a square; another to make a pentagon; then hexagon; heptagon; octagon; nonagon; decagon; on and on until the shape looks indistinguishable from a perfect circle.) But even the most detailed polygon models can be seen, so they were placed with parabolic curves that would be continuous at any level of detail. But they too were difficult to calculate, so mathematicians created subdivision surfaces, which quickly generate smooth shapes but finding and dividing the midpoints of lines.

This method of subdivision surfaces was first used in Pixar’s 1997 short film Geri’s Game. This radically changed how films were animated, and while the technique was originally just applied to parts of character models like the nose, ears, eyes, and expressions on the face, now nearly every model in animation uses advancements on the subdivision surfaces standard. DeRose and other computer scientists work vigorously to update and improve these methods, but Pixar no longer has the lead on the industry it used to. The open-source nature of the Internet has allowed many ambitious amateurs to come up with their own models of animation and allows anyone with a modem to have access to nearly the same software Pixar uses. Independent filmmakers are able to create models of the same fidelity, but the real quality of the film falls back on the nature of the story being told.


The Verge

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