The Design Space of Plane Elastic Curves

The Design Space of Plane Elastic Curves (Teaser Image)



ACM Transactions on Graphics 40(4) (SIGGRAPH 2021)


Elastic bending of initially flat slender elements allows the realization and economic fabrication of intriguing curved shapes. In this work, we derive an intuitive but rigorous geometric characterization of the design space of plane elastic rods with variable stiffness. It enables designers to determine which shapes are physically viable with active bending by visual inspection alone. Building on these insights, we propose a method for efficiently designing the geometry of a flat elastic rod that realizes a target equilibrium curve, which only requires solving a linear program. We implement this method in an interactive computational design tool that gives feedback about the feasibility of a design, and computes the geometry of the structural elements necessary to realize it within an instant. The tool also offers an iterative optimization routine that improves the fabricability of a model while modifying it as little as possible. In addition, we use our geometric characterization to derive an algorithm for analyzing and recovering the stability of elastic curves that would otherwise snap out of their unstable equilibrium shapes by buckling. We show the efficacy of our approach by designing and manufacturing several physical models that are assembled from flat elements.


Video Presentation (20 min)

Submission Video


  author    = {Hafner, Christian and Bickel, Bernd},
  title     = {The Design Space of Plane Elastic Curves},
  journal   = {ACM Transactions on Graphics (TOG)},
  number    = {4},
  volume    = {40},
  year      = {2021},
  publisher = {ACM}


We thank the anonymous reviewers for their generous feedback, and Michal Piovarči for his help in producing the supplemental video. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 715767).