We have grown accustomed to seeing many aspects of our everyday world depicted using computer graphics, but some phenomena remain difficult for even the most experienced animators. Hair, specifically the highly coiled hair that is most common to Black characters, remains a notoriously difficult digital challenge.

Part of this problem is the lack of algorithms. Scores of technical papers have been written over the last few decades proposing algorithms for hair, but they have focused on the features most closely associated with white characters: straight or wavy hair. The number of papers written for highly coiled hair (aka Black hair) is virtually zero.

A new paper will be presented at the SIGGRAPH Asia conference in December that is the first to examine the geometric properties of highly coiled hair and propose methods for replicating their unique visual properties (figure 1).

This is the first time a paper on this topic has ever appeared at the conference, which along with its sister conference, SIGGRAPH North America, has existed in various forms for the last 51 years. Considered the premier conference in the field of computer animation, it is attended by leaders in both academia and industry.

For the study, led by Yale computer science professor Theodore Kim, the research team identified a variety of unique visual phenomena that emerge in highly coiled hair. "When you stop thinking of hair as a parabola and instead as a high-frequency helix, lots of interesting things happen," Kim reported.

In collaboration with Black hair expert professor A.M. Darke from University of California, Santa Cruz, the team quickly realized that there's a lot more going on with hair near the scalp. The follicles in that region often form a thick, spongy layer, but as they travel further away from the scalp, they self-organize into helical curls. The paper has given this phenomenon a name, "phase locking," and proposes a Fourier method for computing it (figure 2).

"They don't form into a perfect curl, just straight out of the scalp," Kim said. "Every hair is already a helix, but they don't have a relationship with each other yet. But if you give them enough space as they grow, they phase lock into these coherent curls. It's amazing."

Then there are "switchbacks." That's when the direction of the coil changes direction partway through. These are the staple-shaped curves that occur everywhere in highly coiled hair, and might also be familiar from tangled Slinkies or coiled telephone cords (figure 3).

"We asked some of the top technical people in the industry, "Have you figured out how to get this shape?'" he said. "The answer was no. In fact, one of the top experts in digital Black hair said, "No, I've wanted that shape for a long time, and I thought it would be easy, but it turned out to be really hard, and I've had to make do without it.'"

A third phenomenon they identify is "period skipping," in which an individual hair temporarily breaks away from its coalesced curl (figure 4). Precisely capturing this feature, they note, is crucial to achieving the wide range of looks characteristic of highly coiled hair (figure 5).

In each of these cases, the researchers developed a computational method to efficiently depict these phenomena. Kim's graduate student, Haomiao Wu, relied on her physics background to develop an algorithm to create the shape of a "switchback." Botany was one of her starting points, particularly plants with spiral stems.

"No one had looked at how you compute the points in space that actually compose the actual shape," he said. "To our knowledge, her algorithm is the first one for directly computing this shape."

These are a few of the many details that have gone overlooked in the field of computer animation when it comes to animating highly coiled hair. Because the algorithms for hair have traditionally focused on straight hair, he said, "there's tons of stuff missing" when it comes to Afro-textured hair.

"None of these phenomena show up in straight hair," Kim said. "As a result, nobody ever thought to design an algorithm for them. Very common haircuts like natural fades had never been investigated before" (figure 6).

Kim noted that having Professor Darke, who is also a professional artist, as a collaborator on the project helped the team make observations that would have been impossible otherwise.

"This is how computer graphics algorithms have been generated since the very beginning," Kim said. "An artist and a scientist put their knowledge together and make something really interesting that neither could have done individually."

Ideally, Kim said, every type of hair will eventually get the same attention within the field that straight and wavy hair has.

"The assumption in the past has been that if we get straight hair done, then all the other types are also solved—but that's not true at all," he said. "I would really love to see papers on every single different type of hair. They don't all have to fit together into some big universal model. When people try to do that, it just ends up being the straight hair model with some half-hearted curly-hair add-ons."

While some in the field still don't see the need to diversify hair algorithms, there are signs that progress is being made. Besides the paper from Kim and his collaborators, a team from Microsoft will also be presenting at SIGGRAPH Asia on a similar topic.

"People are starting to get the message that when you treat different types of hair as first-class scientific questions, it opens the door to lots of interesting new research."

More information: Curly-Cue: Geometric Methods for Highly Coiled Hair. SIGGRAPH Asia 2024 Conference Papers (SA Conference Papers '24), December 3–6, 2024, Tokyo, Japan. ACM, www.cs.yale.edu/homes/wu-haomi … cation/curlyCue.html