# Math in the Movies

## Mathematicians to Thank for Great Graphics

May 1, 2007

100 powerful supercomputers perform geometrical, algebraic and calculus-based calculations to animate Pixar's characters. The laws of physics that inform the dynamics of fabric movement are most used in the computations.

BACKGROUND: Pixar Animation Studios is undergoing a digital revolution thanks to advances in areas such as computer technology, computational physics, and approximation theory. Tony Derose provided a behind-the-scenes look at the role that geometry plays in the revolution using examples drawn from Pixar's feature films, such as Toy Story I and II. Upcoming movie characters will be animated using a new advancement in geometry recently developed at Pixar.

ABOUT ANIMATION: The term animation refers generally to graphical displays in which a sequence of images with gradual differences results in the same effect as a photographed movie. Computer generated animations are getting more and more common, replacing hand drawn images and other special techniques. There are several ways to generate dynamic changes in computer graphics. Geometry animation is the most complex, and requires changing the geometric elements of a scene dynamically. This is also what most people generally refer to when using the term "animation," evidenced by motion pictures like "Toy Story" and "A Bug's Life."

HOW PIXAR DOES IT: Perhaps the most difficult aspect of animation is making people and clothing look real. Pixar's software is based on complex studies of how cloth moves when draped on a character, based on the laws of physics. For instance, drape a bedsheet between two points, and the center will hang downward, adjusting itself until it comes to rest in a state of pure tension. The animators begin with drawings of the characters, which they use to build computer puppets, later adding digital "strings" that correspond to various geometric points on the puppet. These strings serve as animation controls, ensuring that as each string is "pulled," the puppet's movements reflect what would occur in real life. Color and lighting effects are added last before the puppet is "animated." Pixar uses 100 powerful supercomputers that run 24 hours a day, seven days a week. It still takes the computers five to six hours to render a single frame lasting 1/24th of a second. For every second of film, it takes the computer six days.

WHAT IS GEOMETRY? Geometry is the field of mathematical knowledge dealing with spatial relationships. The earliest written records -- dating from Egypt and Mesopotamia about 3100 BC -- demonstrate that ancient peoples had already begun to devise mathematical rules and techniques useful for surveying land areas, constructing buildings, and measuring storage containers. Beginning about the 6th century BC, the Greeks gathered and extended this practical knowledge and from it generalized the abstract subject now known as geometry, from the combination of the Greek words geo ("Earth") and metron ("measure") for the measurement of the Earth.

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