It is rare to see an animated character with bouncy, curly hair, since computer animators don't have a simple mathematical means for describing it, researchers said.
Researchers at the Massachusetts Institute of Technology (MIT) and the Universite Pierre et Marie Curie in Paris provide the first detailed model for the 3-D shape of a strand of curly hair.
This work could have applications in the computer animation film industry, and also be used by engineers to predict the curve that long steel pipes, tubing, and cable develop after being coiled around a spool for transport.
"Our work doesn't deal with the collisions of all the hairs on a head, which is a very important effect for animators to control a hairstyle," said Pedro Reis from MIT.
"But it characterises all the different degrees of curliness of a hair and describes mathematically how the properties of the curl change along the arc length of a hair," Reis said.
The team identified the main parameters for curly hair and simplified them into two dimensionless parameters for curvature (relating to the ratio of curvature and length) and weight (relating to the ratio of weight and stiffness). Given curvature, length, weight, and stiffness, their model will predict the shape of a hair, steel pipe, or Internet cable suspended under its own weight.
As a strand of hair curls up from the bottom, its 2-D hook grows larger until it reaches a point where it becomes unstable under its own weight and falls out of plane to become a 3-D helix.
Reis and co-authors describe the 3-D curl as a localised helix, where only a portion of the strand is curled, or a global helix, if the curliness extends the entire length up to the head. A curl can change phase - from 2-D to 3-D local helix to 3-D global helix, and back again - if its parameters change.
Because a strand of hair is weighted from the bottom by gravity, the top of the strand has more weight under it than the tip, which has none. Thus, if the weight on a hair is too great for its innate curliness, the curl will fail and become either straight or helical, depending on the strand's length and stiffness.
Miller created flexible, thin rods using molds as small as a bottle of Tabasco sauce and as large as the columns. He injected a rubber-like material inside hollow flexible tubing wrapped around these molds. Once the rubber material cured and the tubing was cut away, Miller and Reis had flexible polyvinyl thin rods whose natural curvature was based on the size of the object around which they had been wrapped. The study appears in the journal Physical Review Letters.