# Curved Ribbon

**3D Weaving with Curved Ribbons**

##### YINGYING REN^{1}, JULIAN PANETTA^{2}, TIAN CHEN^{1}, FLORIN ISVORANU^{1}, SAMUEL POINCLOUX^{1}, CHRISTOPHER BRANDT^{1}, ALISON MARTIN^{3}, MARK PAULY^{1}

^{1}EPFL, Switzerland

^{1}EPFL, Switzerland^{2}UC Davis, USA

^{2}UC Davis, USA^{3}Weaver and Independent Researcher, Italy

^{3}Weaver and Independent Researcher, ItalyBasket weaving is a traditional craft for creating curved surfaces as an interwoven array of thin, flexible, and initially straight ribbons. The three-dimensional shape of a woven structure emerges through a complex interplay of the elastic bending behavior of the ribbons and the contact forces at their crossings.

Curvature can be injected by carefully placing topological singularities in the otherwise regular weaving pattern. However, shape control through topology is highly non-trivial and inherently discrete, which severely limits the range of attainable woven geometries.

Here, they demonstrate how to construct arbitrary smooth free-form surface geometries by weaving carefully optimized curved ribbons. They present an optimization-based approach to solving the inverse design problem for such woven structures.

Their algorithm computes the ribbons’ planar geometry such that their interwoven assembly closely approximates a given target design surface in equilibrium. They systematically validate our approach through a series of physical prototypes to show a broad range of new woven geometries that is not achievable by existing methods.

They anticipate our computational approach to significantly enhance the capabilities for the design of new woven structures. Facilitated by modern digital fabrication technology, they see potential applications in material science, bio- and mechanical engineering, art, design, and architecture.

In this paper, they generalize the traditional process of basket weaving using straight ribbons to arbitrarily curved ribbons and show how this generalization significantly enlarges the range and quality of attainable shapes.

On the other hand, determining the optimal curved shape of each individual ribbon in a globally coupled woven structure is highly non-trivial and typically beyond the capabilities of human designers. They therefore propose a computational design approach for curved weaving.

The core contribution of this paper is a computational pipeline for the inverse design of woven structures using curved planar ribbons. The key algorithmic novelty is a multi-stage optimization method to solve for the freeform 2D rest shape of the ribbons such that the equilibrium state of their woven ensemble best approximates a given input surface.

They show how a novel ribbon crossing model allows optimizing the contact forces acting on interlaced ribbons to improve the stability of the woven structure. Their approach enables the weaving of complex freeform surfaces that cannot be handled by any existing method.

They validate our optimization method through a series of physical prototypes that show excellent agreement with the simulation prediction.

Curved ribbons offer a rich design space for weaving 3D surface structures. Even though often not obvious on the final woven model, the geometry of the curved ribbons can be highly complex and unintuitive.

Effective design thus requires advanced computational methods to optimize ribbon geometry and accurately predict the final equilibrium structure.

Their inverse design algorithm provides a solution to this challenging problem, opening the door to the entirely new applications of woven geometries for industrial and consumer products, artistic installations, or architectural designs, for example.

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