Exploring the Geometry of the Space of Shells

Heeren, Behrend; Rumpf, Martin; Schröder, Peter; Wardetzky, Max; Wirth, Benedikt

doi:10.1111/cgf.12450

Cited by 51 publications

(56 citation statements)

References 33 publications

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“…Periodic spline D → E → F → D with K = 12 and different bending weights η = δ 2 , where δ represents the thickness of the thin sheet (see also Fig. in [HRS*14]). The discrete segments E → F and F → D are not shown due to the symmetry of the problem.…”

Section: Discussionmentioning

confidence: 99%

“…where describes the physical thickness of the material layer represented by the shell. As shown in [HRS*14], 풲 induces a proper Riemannian metric in the space of shells modulo rigid body motions.…”

Section: Review: Shortest Paths In Shell Spacementioning

confidence: 99%

“…In this case we obtain , where N is the number of vertices of one triangle mesh, and we identify a discrete shell with its vector of nodal positions s ∈ ℝ 3 N . To compute distances on shell space, we use 풲 given by the Discrete Shells energy [GHDS03], which induces a Riemannian metric on the space of triangle meshes modulo rigid motions [HRS*14].…”

Section: Splines On Shell Spacementioning

confidence: 99%

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Splines in the Space of Shells

Heeren

Rumpf

Schröder

et al. 2016

Computer Graphics Forum

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Cubic splines in Euclidean space minimize the mean squared acceleration among all curves interpolating a given set of data points. We extend this observation to the Riemannian manifold of discrete shells in which the associated metric measures both bending and membrane distortion. Our generalization replaces the acceleration with the covariant derivative of the velocity. We introduce an effective time‐discretization for this novel paradigm for navigating shell space. Further transferring this concept to the space of triangular surface descriptors—edge lengths, dihedral angles, and triangle areas—results in a simplified interpolation method with high computational efficiency.

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Section: Discussionmentioning

confidence: 99%

Section: Review: Shortest Paths In Shell Spacementioning

confidence: 99%

Section: Splines On Shell Spacementioning

confidence: 99%

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Splines in the Space of Shells

Heeren

Rumpf

Schröder

et al. 2016

Computer Graphics Forum

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“…They are also natural variables for computing and minimizing discrete shell energies [GHDS03], which are efficient geometric energies penalizing stretching and bending in a richer way than rigid energies can achieve. Note that the Riemannian geometry of the space of shells has been studied by Heeren et al [HRS*14], allowing them to extend concepts such as geodesics and parallel transport to the space of deformations governed by these energies.…”

Section: Synthesis and Modelingmentioning

confidence: 99%

A Survey on Data‐driven Dictionary‐based Methods for 3D Modeling

Lescoat

Ovsjanikov

Memari

et al. 2018

Computer Graphics Forum

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Dictionaries are very useful objects for data analysis, as they enable a compact representation of large sets of objects through the combination of atoms. Dictionary‐based techniques have also particularly benefited from the recent advances in machine learning, which has allowed for data‐driven algorithms to take advantage of the redundancy in the input dataset and discover relations between objects without human supervision or hard‐coded rules. Despite the success of dictionary‐based techniques on a wide range of tasks in geometric modeling and geometry processing, the literature is missing a principled state‐of‐the‐art of the current knowledge in this field. To fill this gap, we provide in this survey an overview of data‐driven dictionary‐based methods in geometric modeling. We structure our discussion by application domain: surface reconstruction, compression, and synthesis. Contrary to previous surveys, we place special emphasis on dictionary‐based methods suitable for 3D data synthesis, with applications in geometric modeling and design. Our ultimate goal is to enlight the fact that these techniques can be used to combine the data‐driven paradigm with design intent to synthesize new plausible objects with minimal human intervention. This is the main motivation to restrict the scope of the present survey to techniques handling point clouds and meshes, making use of dictionaries whose definition depends on the input data, and enabling shape reconstruction or synthesis through the combination of atoms.

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“…Algorithms have also been proposed to explicitly model shell structures [Grinspun et al 2003;Burgoon et al 2006;Heeren et al 2014]. Recently, modeling deformation behavior of paper in the context of thin shells has been explored by Narain et al [2013].…”

Section: Related Workmentioning

confidence: 99%

String Actuated Curved Folded Surfaces

2017

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e e e e e e e e e e e e e e e e e Fig. 1: The top row shows a crease pattern together with the strings S = {a, b, c, d, e} computed by our algorithm and simulation of the induced, string driven folding motion. The bottom row shows a physical realization of the same crease pattern as a thin aluminum sheet. Each string is realized as a strand of colored thread. Strings are collected at a single anchor point and folding is driven by pulling the threads.Curved folded surfaces, given their ability to produce elegant freeform shapes by folding flat sheets etched with curved creases, hold a special place in computational Origami. Artists and designers have proposed a wide variety of different fold patterns to create a range of interesting surfaces. The creative process, design as well as fabrication, is usually only concerned with the static surface that emerges once folding has completed. Folding such patterns, however, is difficult as multiple creases have to be folded simultaneously to obtain a properly folded target shape. We introduce string actuated curved folded surfaces that can be shaped by pulling a network of strings thus vastly simplifying the process of creating such surfaces and making the folding motion an integral part of the design. Technically, we solve the problem of which surface points to string together and how to actuate them by locally expressing a desired folding path in the space of isometric shape deformations in terms of novel string actuation modes. We demonstrate the validity of our apMartin Kilian was supported by the Erwin Schrödinger fellowship J-3678-N25 awarded by the Austrian Science Fund (FWF), ERC StG-2013-335373, and the DFGCollaborative Research Center, TRR 109, 'Discretization in Geometry and Dynamics' through grant I-706-N26 of FWF. Aron Monszpart was supported by a Google PhD Fellowship, the ERC Starting Grant SmartGeometry (StG-2013-335373), and Adobe Research. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. proach by computing string actuation networks for a range of well known crease patterns and testing their effectiveness on physical prototypes. All the examples in this paper can be downloaded for personal use from

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Exploring the Geometry of the Space of Shells

Cited by 51 publications

References 33 publications

Splines in the Space of Shells

Splines in the Space of Shells

A Survey on Data‐driven Dictionary‐based Methods for 3D Modeling

String Actuated Curved Folded Surfaces

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