Developability-driven piecewise approximations for triangular meshes

Abstract

We propose a novel method to compute a piecewise mesh with a few developable patches and a small approximation error for an input triangular mesh. Our key observation is that a deformed mesh after enforcing discrete developability is easily partitioned into nearly developable patches. To obtain the nearly developable mesh, we present a new edge-oriented notion of discrete developability to define a developability-encouraged deformation energy, which is further optimized by the block nonlinear Gauss-Seidel method. The key to successfully applying this optimizer is three types of auxiliary variables. Then, a coarse-to-fine segmentation technique is developed to partition the deformed mesh into a small set of nearly discrete developable patches. Finally, we refine the segmented mesh to reduce the discrete Gaussian curvature while keeping the patches smooth and the approximation error small. In practice, our algorithm achieves a favorable tradeoff between the number of developable patches and the approximation error. We demonstrate the feasibility and practicability of our method over various examples, including seventeen physical manufacturing models with paper.

Freeform Shape Fabrication by Kerfing Stiff Materials

“…Although not all works towards developable approximations specifically focus on papercraft, many still validate their results by building objects from paper. Examples for these are D‐Charts by [JKS05], a method to segment a mesh into quasi‐developable charts in an iterative manner, and in recent years methods based on local gauss image thinning [SGC18, BVHSH21, ZFO*22] and developable wrapping [IRHSH20]. [VVHSH22] discuss quadrilateral strip remeshing of already developable surfaces.…”

Fabricatable 90° Pop‐ups: Interactive Transformation of a 3D Model into a Pop‐up Structure

“…Instead of starting with a sheet of paper, a 3D shape can be generated by gluing multiple paper patches together [MS04b, STL06, ZFO*22]. The basic idea is to segment a 3D model into locally developable patches and glue them to create papercraft toys.…”