Discrete Geodesic Nets for Modeling Developable Surfaces

Rabinovich, Michael; Hoffmann, Tim; Sorkine‐Hornung, Olga

doi:10.1145/3180494

Cited by 66 publications

(89 citation statements)

References 33 publications

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“…Note, however, that this increases the number of free variables substantially. In Figure 9, we show different time-discrete geodesics in NRIC where we use the quadratic deformation energy (15) in (23). In particular, we compare for end shapes being two oppositely bent plates our NRIC geodesic (orange) to the linear interpolation (green) in the embedding space R 2|E| , which corresponds to a naive transfer of the projection approach by Fröhlich and Botsch [2].…”

Section: Numerical Experiments and Comparisonsmentioning

confidence: 99%

Geometric optimization using nonlinear rotation-invariant coordinates

Sassen

Heeren

Hildebrandt

et al. 2020

Computer Aided Geometric Design

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Geometric optimization problems are at the core of many applications in geometry processing. The choice of a representation fitting an optimization problem can considerably simplify solving the problem. We consider the Nonlinear Rotation-Invariant Coordinates (NRIC) that represent the nodal positions of a discrete triangular surface with fixed combinatorics as a vector that stacks all edge lengths and dihedral angles of the mesh. It is known that this representation associates a unique vector to an equivalence class of nodal positions that differ by a rigid body motion. Moreover, integrability conditions that ensure the existence of nodal positions that match a given vector of edge lengths and dihedral angles have been established. The goal of this paper is to develop the machinery needed to use the NRIC for solving geometric optimization problems. First, we use the integrability conditions to derive an implicit description of the space of discrete surfaces as a submanifold of an Euclidean space and a corresponding description of its tangent spaces. Secondly, we reformulate the integrability conditions using quaternions and provide explicit formulas for their first and second derivatives facilitating the use of Hessians in NRIC-based optimization problems. Lastly, we introduce a fast and robust algorithm that reconstructs nodal positions from almost integrable NRIC. We demonstrate the benefits of this approach on a collection of geometric optimization problems. Comparisons to alternative approaches indicate that NRIC-based optimization is particularly effective for problems involving near-isometric deformations.

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Section: Numerical Experiments and Comparisonsmentioning

confidence: 99%

Geometric optimization using nonlinear rotation-invariant coordinates

Sassen

Heeren

Hildebrandt

et al. 2020

Computer Aided Geometric Design

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show abstract

“…Our work is based on modeling a developable surface as a discrete orthogonal geodesic net (DOG) [Rabinovich et al 2018a], a model that has been shown both theoretically and empirically to avoid extrinsic and intrinsic deformation locking. We further rely on [Rabinovich et al 2018b] to explore the shape space of DOGs, but we replace their Laplacian ow based deformation with a sequential quadratic programming (SQP) based algorithm, detailed in Sec.…”

Section: Related Work 21 Modeling Developable Surfacesmentioning

confidence: 99%

“…5 for an illustration. Following [Rabinovich et al 2018a] we maintain continuity of curve points along edges while penalizing deviation of the edge lengths across duplicated quads.…”

Section: Modelmentioning

confidence: 99%

Modeling curved folding with freeform deformations

Rabinovich

Hoffmann²,

Sorkine‐Hornung

2019

ACM Trans. Graph.

Self Cite

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We present a computational framework for interactive design and exploration of curved folded surfaces. In current practice, such surfaces are typically created manually using physical paper, and hence our objective is to lay the foundations for the digitalization of curved folded surface design. Our main contribution is a discrete binary characterization for folds between discrete developable surfaces, accompanied by an algorithm to simultaneously fold creases and smoothly bend planar sheets. We complement our algorithm with essential building blocks for curved folding deformations: objectives to control dihedral angles and mountain-valley assignments. We apply our machinery to build the first interactive freeform editing tool capable of modeling bending and folding of complicated crease patterns.

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“…1.1.2 Geometric Modeling Approaches. Geometric modeling techniques have been widely developed for computing flat panels that can be assembled into a given 3D shape -e.g., segmentation-based methods [Julius et al 2005;Shatz et al 2006;Wang 2008], strip-based methods [Mitani and Suzuki 2004;Schüller et al 2018]), methods based on developable surfaces [Kilian et al 2008;Liu et al 2006;Rabinovich et al 2018;Rose et al 2007] and structures for selfdeformation [Guseinov et al 2017]. However, they cannot be directly used in the computation of fabric formwork as the shape of a fabric container will be deformed significantly after pouring in the liquid plaster.…”

Section: Related Workmentioning

confidence: 99%

Computational design of fabric formwork

et al. 2019

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We present an inverse design tool for fabric formwork - a process where flat panels are sewn together to form a fabric container for casting a plaster sculpture. Compared to 3D printing techniques, the benefit of fabric formwork is its properties of low-cost and easy transport. The process of fabric formwork is akin to molding and casting but having a soft boundary. Deformation of the fabric container is governed by force equilibrium between the pressure forces from liquid fill and tension in the stretched fabric. The final result of fabrication depends on the shapes of the flat panels, the fabrication orientation and the placement of external supports. Our computational framework generates optimized flat panels and fabrication orientation with reference to a target shape, and determines effective locations for external supports. We demonstrate the function of this design tool on a variety of models with different shapes and topology. Physical fabrication is also demonstrated to validate our approach.

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Discrete Geodesic Nets for Modeling Developable Surfaces

Cited by 66 publications

References 33 publications

Geometric optimization using nonlinear rotation-invariant coordinates

Geometric optimization using nonlinear rotation-invariant coordinates

Modeling curved folding with freeform deformations

Computational design of fabric formwork

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