In DreamWorks Animation's Shrek franchise, Princess Fiona almost exclusively wears her hair pulled back. The character's preference for braids has more to do with physics than fashion. Letting a cartoon character's hair down requires calculating a string of complex equations to create a realistic effect, so computer animators often just opt for short hair and updos rather than long, loose tresses. Likewise, most animated characters turn up on the big screen with straight locks because rendering them in three dimensions is a simpler mathematical task.
Animators' tool kits may be about to expand, and a convincing rendition of curls might one day abound in features by DreamWorks and Pixar. A team of researchers recently untangled the physics of a single strand of curly hair, publishing the results in Physical Review Letters. “This is the first time someone described the full 3-D configuration of a single naturally curved hair,” says co-author Pedro Reis, an assistant professor of mechanical engineering and of civil and environmental engineering at the Massachusetts Institute of Technology. “I would attribute that to the fact that the geometry of a curly hair is highly nonlinear— a word we often use for something complicated.”
Reis and his colleagues did not set out to model curly hair. They wanted to study curvature of long, thin structures. Think of submarine cables, oil and gas pipes, and even the tiny tails on bacteria. The team first laid hollow, tubular molds out straight or wrapped them around cylindrical objects ranging in diameter from 3.2 centimeters to one meter. Then they injected the molds with a rubberlike material, which dried to produce flexible rods with different curvatures. They suspended the rods to study how gravity affected their shape. With curls hanging one beside another, they realized the rods bore a striking similarity to the single strands that combine to form coifs ranging from rail-straight to the kinks of Afro-textured hair.
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The researchers carried out some 11,000 computer simulations, using the results to create a phase diagram depicting different geometric shapes a hanging strand will assume as a function of four properties: curvature, weight, length and stiffness. Eventually such a tool could be incorporated into animation software, but other groups will first have to investigate how a full head of curly hair interacts with itself and with wind and other outside forces.
The model could also calculate curvature of steel pipes or other spooled material. “We were engineers trying to solve practical, useful problems from the start,” Reis says. “I'm not a professional hairstylist—I'm bald, actually.”