Developability of Triangle Meshes

Oded Stein        Eitan Grinspun        Keenan Crane

Columbia University       Carnegie Mellon University

By encouraging discrete developability, a given mesh evolves toward a shape comprised of flattenable pieces separated by highly regular seam curves.


A surface is developable if it can be flattened into the plane without stretching or shearing - piecewise developable surfaces are desirable in contexts like manufacturing, where an object might be assembled from pieces of, \eg, thin plywood or sheet metal, or milled from a solid block using a cylindrical cutting tool.
We introduce a new notion of developability for triangle meshes which captures two key properties of smooth developable surfaces, namely flattenability and existence of straight ruling lines. This definition can be used as a starting point for algorithms in developable surface modeling - we consider a variational approach that drives a given mesh toward developable pieces separated by regular seam curves. Computation amounts to gradient descent on an energy with support in the vertex star, without the need to explicitly cluster patches or identify seams. We briefly explore applications to developable design and manufacturing.




author = {Stein, Oded and Grinspun, Eitan and Crane, Keenan},
title = {Developability of Triangle Meshes},
journal = {ACM Trans. Graph.},
volume = {37},
number = {4},
year = {2018},
publisher = {ACM},
address = {New York, NY, USA},


Thanks to Timothy Sun for preliminary discussion and investigation, and to Ryan Schmidt for performing the toolpath extraction and milling for our faucet example. This work was sponsored in part by NSF Awards 1717320 1409286, 1717268, and 1319483, and gifts from Adobe, Autodesk Research, Pixar, and SideFX Software. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.