Flexible Developable Surfaces

Abstract

We introduce a discrete paradigm for developable surface modeling. Unlike previous attempts at interactive developable surface modeling, our system is able to enforce exact developability at every step, ensuring that users do not inadvertently suggest configurations that leave the manifold of admissible folds of a flat two-dimensional sheet. With methods for navigation of this highly nonlinear constraint space in place, we show how to formulate a discrete mean curvature bending energy measuring how far a given discrete developable surface is from being flat. This energy enables relaxation of user-generated configurations and suggests a straightforward subdivision scheme that produces admissible smoothed versions of bent regions of our discrete developable surfaces.

Lattice mechanics of origami tessellations

“…Controlled actuation of thin materials via patterned folds has led to a variety of self-assembly strategies in polymer gels [8] and shape-memory materials [4], as well elastocapillary self-assembly [9], leading to the design of a new category of shape-transformable materials inspired by origami design. The origami repertoire itself, buoyed by advances in the mathematics of folding and the burgeoning field of computational geometry [10], is no longer limited to designs of animals and children's toys that dominate the art in popular consciousness, but now includes tessellations, corrugations, and other non-representational structures whose mechanical properties are of interest from a scientific perspective. These properties originate from the confluence of geometry and mechanical constraints that are an intrinsic part of origami, and ultimately allow for the construction of mechanical meta-materials using origami-based design [1-4, 6, 11-13].…”

Analysis of displacements in beam structures and shells with middle developable surfaces

“…Since this type of surfaces is widely used in manufacturing [10][11][12][13][14][15] in many fields such as shipbuilding [16,17], engineering [18] and architecture [19,20] and products are mostly obtained by the means of bending, the problem of plastic deformations should be taken into account. However, most works about developable surfaces do not concern to this point [21][22][23][24][25][26][27].…”

String Actuated Curved Folded Surfaces

“…Kilian et al [2008] reconstruct scanned paper surfaces as Origami patterns using curved folds, utilizing the reference surface to estimate ruling directions. Later, Solomon et al [2012] presented a subdivision based modeling approach involving curved folds assuming the crease pattern and corresponding crease angles as input. Tang et al [2015] encode developability conditions as nonlinear constraints and solve for curved folded surfaces using a projection-based efficient constraint solver for interactive modeling.…”