Summary:
February 12, 2014
Massachusetts Institute of
Technology
The heroes and villains in animated films tend
to be on opposite ends of the moral spectrum.
But they're often similar in their hair, which is
usually extremely rigid or -- if it moves at all
-- is straight and swings to and fro. It's rare to
see an animated character with bouncy, curly
hair, since computer animators don't have a
simple mathematical means for describing it.
But now, researchers have developed the first
detailed model for a 3-D strand of curly hair.
FULL RESEARCH
The heroes and villains in animated films
tend to be on opposite ends of the moral
spectrum. But they're often similar in
their hair, which is usually extremely rigid
or -- if it moves at all -- is straight and swings
to and fro. It's rare to see an animated
character with bouncy, curly hair, since
computer animators don't have a simple
mathematical means for describing it.
However, change may be coming soon to a
theater near you: In a paper appearing in the
Feb. 13 issue of Physical Review Letters ,
researchers at MIT and the Université 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, but it also
could be used by engineers to predict the
curve that long steel pipes, tubing, and cable
develop after being coiled around a spool for
transport. In the field, these materials often
act like a stubborn garden hose whose intrinsic
curves make it behave in unpredictable ways.
In engineering terminology, these items -- and
hair -- are all examples of a slender, flexible
rod.
Co-authors on the paper are Pedro Reis, an
assistant professor in MIT's Department of Civil
and Environmental Engineering and Department
of Mechanical Engineering; Basile Audoly and
Arnaud Lazarus, of the Université Pierre et
Marie Curie; and former MIT graduate student
James Miller, who is now a research associate
at Schlumberger-Doll Research. Miller worked
on this project as part of his doctoral thesis
research and is lead author of the paper.
"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," Reis says. "But it characterizes 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."
When Reis set out to investigate the natural
curvature in flexible rods, he wasn't thinking
of hair. But as he studied several small flexible,
curved segments of tubing suspended from a
structure in his lab, he realized they weren't so
different from strands of curly hair hanging on
a head. That's when he contacted Audoly, who
had previously developed a theory to explain
the 2-D shape of human hair.
Using lab experimentation, computer
simulation, and theory -- "the perfect triangle
of science," Reis says -- 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 localized 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.
For the curvature study, Miller created
flexible, thin rods using molds as small as a
bottle of Tabasco sauce and as large as the
columns in MIT's Lobby 7 (about a meter in
diameter). 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 researchers' use of dimensionless numbers
to describe innate curvature means the
equation will hold true at all scales. Even with
lengths measured in kilometers, the steel
piping used by the oil industry is flexible
enough to be spooled. "We think of steel pipes
as being nice and straight but usually at some
point they're getting wrapped around
something," Miller says. "And at large
dimensions, they're so flexible that it's like you
and I dealing with a limp spaghetti noodle."
"The mathematician [Leonhard] Euler first
derived the equation for a slender elastic body
-- like a hair strand -- in 1744," Audoly says.
"Even though the equations are well-known,
they have no explicit solution and, as a result,
it is challenging to connect these equations
with real shapes."
"The fact that I am bald and worked on this
problem for several years became a nice
running joke in our lab," Reis says. "But joking
aside, for me the importance of the work is
being able to take the intrinsic natural
curvature of rods into account for this class of
problems, which can dramatically affect their
mechanical behavior. Curvature can delay
undesirable instability that happens at higher
loads or torsion, and this is an effect that
engineers need to be able to understand and
predict."
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