Shape approximation by developable wrapping

Abstract

We present an automatic tool to approximate curved geometries with piece-wise developable surfaces. At the center of our work is an algorithm that wraps a given 3D input surface with multiple developable patches, each modeled as a discrete orthogonal geodesic net. Our algorithm features a global optimization routine for effectively finding the placement of the developable patches. After wrapping the mesh, we use these patches and a non-linear projection step to generate a surface that approximates the original input, but is also amendable to simple and efficient fabrication techniques thanks to being piecewise developable. Our algorithm allows users to steer the trade-off between approximation power and the number of developable patches used. We demonstrate the effectiveness of our approach on a range of 3D shapes. Compared to previous approaches, our results exhibit a smaller or comparable error with fewer patches to fabricate.

Data to Physicalization: A Survey of the Physical Rendering Process

“…Another possible solution in the fabrication of physicalizations is a papercraft and origami that is tailored and controlled through programming 3D CAD for fast and accurate folding and assembly purposes [STL06, IRHS20]. We refer to this fabrication technique as digital papercraft .…”

Developable Approximation via Gauss Image Thinning

“…In Figures 12 and 6 we show a comparison with the most recent works on developable surface approximation: Stein et al [SGC18], Ion et al [IRHS20] and Sellán et al [SAJ20] (the latter works only on height fields). Our results are sometimes smoother and tend to reflect the symmetries of the input geometry better than Ion et al [IRHS20], because we entirely avoid their segmentation step, which may introduce random symmetry breaking. Additionally, similar to Stein et al [SGC18], our method is able to produce a network of seams and open seam curves, unlike only closed seams that delineate disk topology patches as in [IRHS20].…”

“…More recently the developable approximation of more general surfaces, and with a wider variety of developable pieces has received attention. Ion et al [IRHS20] approximate general input meshes by wrapping them with developable patches, represented as discrete orthogonal geodesic nets (DOGs) [RHSH18]. They initialize the DOGs by selecting a sparse set of geodesic curves and aligning the DOGs coordinate curves to them.…”

“…There is a substantial and diverse body of literature on developable surfaces in mathematics and geometric modeling. We touch on the most relevant related works here and refer the interested reader to the detailed surveys in [Rab20,SAJ20,IRHS20].…”

“…Different developability criteria can be optimized to make the input surface more developable. Wang and Tang [WT04] propose to minimize Gauss curvature (angle deficit), which works well when the input surface is already almost developable, but suffers from instability on general inputs [SGC18,IRHS20]. A number of works use a common property of cones, cylinders and planar patches, i.e., developables with constant slope: their surface normals have a constant angle with a certain axis vector [JKS05,DJW∗06,JHR∗15,GSP19].…”