International audienceWe present the first method in computer graphics to animate sheets of paper at interactive rates while automatically generating a plausible set of sharp features when the sheet is crumpled. Our hybrid, geometric and physical, model is based on a high-level understanding of the physical constraints that act on real sheets of paper, and of their geometric counterparts. This understanding enables us to use an adaptive mesh carefully representing the main geometric features of paper in terms of singular points and developability

Cani, Marie-Paule

Hahmann, Stefanie

Rohmer, Damien

Schreck, Camille

Hal - Université Grenoble Alpes

Interactively animating crumpling paperCamille Schreck, Damien Rohmer, Stefanie Hahmann, Marie-Paule CaniTo cite this version:Camille Schreck, Damien Rohmer, Stefanie Hahmann, Marie-Paule Cani. Interactively animat-ing crumpling paper. womEncourage 2015, Sep 2015, Uppsala, Sweden. <hal-01146700>HAL Id: hal-01146700https://hal.inria.fr/hal-01146700Submitted on 22 Dec 2015HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Interactively animating crumpling paperCamille SCHRECKUniv. Grenoble-Alpes,CNRS (LJK),Inria, FranceDamien ROHMERUniv. Grenoble-Alpes & CPELyonCNRS (LJK),Inria, FranceStefanie HAHMANNUniv. Grenoble-Alpes,CNRS (LJK),Inria, FranceMarie-Paule CANIUniv. Grenoble-Alpes,CNRS (LJK),Inria, FranceABSTRACTWe present the first method in computer graphics to ani-mate sheets of paper at interactive rates while automaticallygenerating a plausible set of sharp features when the sheetis crumpled. Our hybrid, geometric and physical, modelis based on a high-level understanding of the physical con-straints that act on real sheets of paper, and of their geo-metric counterparts.This understanding enables us to use an adaptive mesh care-fully representing the main geometric features of paper interms of singular points and developability.Categories and Subject DescriptorsI.3.7 [Computer Graphics]: Three-Dimensional Graphicsand RealismKeywordsComputer Animation, Geometric Modeling, Interactive De-formation, Paper Crumpling, Developable Surface1. INTRODUCTIONComputer animations of crumbled paper sheets are almostabsent from virtual environments although it is a very com-mon material in everyday life. This is due to the extremedifficulty of efficiently capturing the dynamic, non-smoothdevelopable geometry of paper sheets. Indeed the micro-scopic fibers structure significantly affects the macroscopicbehavior and makes it challenging to reproduce. The goalof this work is to achieve the animation of paper crumplingprocesses at interactive rates. This may greatly simplify thework of artists who need to model this phenomenon. Letus list the macroscopic features that need to be capturedin order to crumple a sheet of paper in a visually plausibleway:.Length preservation. The assumption can be made thatthe surface of a sheet of paper preserves lengths with re-spect to an original 2D pattern. It can be flattened onto aplane without distortion (stretching or compressing). Thisis a property which is called developability.Sharp features. Crumpled sheets of paper exhibit singu-lar points, and possibly some sharp creases.Plastic behavior and shape memory. In real life, themicroscopic fibers of paper or more precisely the bonds be-tween them may break upon deformation, causing irreversibledamages to the structure.The modeling of paper as been studied in numerous pre-vious works that can be divided into two main approaches :Geometric modeling. Bending and creasing virtual pa-per can be expressed through geometric deformation [7]. Aninteractive tool is presented in [18] to fold thin sheets of ma-terials. Other methods addresse interactive origami model-ing [15, 9, 14] where the folds are explicitly defined by theuser. A geometric way to create a virtual sheet of paper withsharp features from a 3D boundary curves and a 2D patternis proposed in [11] but cannot be extended to animation.Physically based modeling. Thin sheet materials aretypically modeled in computer graphics using 2D finite ele-ments with a linear elastic behavior. This approach has beenused to bend sheets of paper using the thin plate theory [5,2]. However, this method cannot handle the dynamic for-mation of singular points and sharp edges. Kang et al. [6]propose to model the plastic behavior of the paper usingbreakable springs, but the approach suffers from visual ar-tifacts as edge-split and singularities can only occur alongpreexisting coarse triangle edges. Recently Narain et al. [10]obtained high-quality visual simulation of paper by com-bining this inelastic deformation model with an adaptiveremeshing method. Yet, this approach is computationallyexpensive due to the large number of triangles in the foldedregions, so it cannot be used in interactive applications.The key idea of this work is to obtain interactive rates andautomatic generation of sharp feature by interleaving stan-dard physically-based simulation steps with procedural gen-eration of a piecewise continuous developable surface.Our contributions include a hybrid physically-based and ge-ometric model, a realistic modeling of singular points, an op-timal adaptive isometric meshing and a validation throughcomparison with real paper.2. A NEW HYBRID MODEL FOR PAPERAs already mentioned, a sheet of paper can be considered asa developable surface, which can be defined as a ruled sur-face with zero Gaussian curvature (product of the principalcurvatures) everywhere. As a consequence, if a smooth partof the paper is bent such that one principal curvature is notzero, then the other principal curvature in the direction ofthe fold is necessarily zero. Thus the surface becomes rigidin the direction of zero-curvature, leading discontinuities toappear instead of having the surface bend if a compressionis applied in the orthogonal direction.The appearance of singularities has also been studied inthe mechanical literature: due to the fact that the bendingrigidity of thin plates is much smaller than their stretch-ing rigidity, the paper mechanical transformations are fa-vorable to bending [3]. In addition, thin sheets have thespecific behavior of concentrating elastic energy when be-ing constrained [17]. When the thickness tends to 0, thisenergy concentrates into singular points, which are calledd-cones [1, 8, 4].These d-cones denote developable generalized cones definedby a fixed point called the apex and a one-parameter familyof straight lines (rulings) passing through the apex. Outsideof the influence of the d-cones, the surface is smooth andexhibits a low distribution of elastic energy [13]: it can bedeformed according to standard linear models for continu-ous mechanics laws.We choose thus to have the motion of the surface guided byan usual physic simulation based on continuous mechanics.The surface model used to represent the shape of a sheet ofpaper must be highly deformable, as sharp features may ap-pear anywhere during deformation. This usually requires touse dense meshes, which provide a large number of degreesof freedom and reduce visual artifacts due to badly orientededges. Instead we choose to dynamically adapt the mesh tothe surface using a dedicated geometric model in order tohave the necessary degrees of freedom for surface deforma-tion while keeping a coarse mesh. This reduces most of thetime usually spent in physically-based simulation.The observations made above lead us to use the followingmodel to get an optimized mesh (represented in Figure 1).The surface of the paper is approximated by a set a curvedregions (in magenta), represented by generalized cones, andplanar regions (in green). The mesh used by the physically-based simulation is composed of the triangles and quadran-gles formed by the rulings of generalized cones and of a verycoarse triangulation of the planar regions. In this way, thecurved regions are rigid along the direction of the rulings,as for real paper, while planar regions are still able to bendin any direction.The sharp creases of the paper are modeled by singularpoints (red dots) which can only be apices of d-cones. Theyare generated in the regions where conflicts between twobending directions arise. Their positions are computed us-ing a probability law in the conflict’s region. As the appari-tion of sharp points is a plastic event, we record the positionof these vertices on the 2D-pattern.We interleave the updating of the mesh with the simulationsteps in order to obtain a simplified, yet visually realisticpaper behavior. The optimized mesh stays coarse such thatthe animation can be achieved in interactive time. More-over, during the whole animation process, we are able tomaintain an approximative isometric mapping between thepaper surface S and its pattern S defined in the 2D param-eter domain.Figure 1: Photograph of a sheet of paper bent in several direc-tions (top left). Our geometric surface representation (pattern onthe bottom left, and 3D surface on the bottom right) is divided intocurved regions (magenta) and planar regions (green). Singular ver-tices (red dots) are apices of generalized cones.We propose the following algorithm, a more detailed de-scription can be found in [12].Initialization. The input is a developable surface S and itsisometric mapping to a 2D pattern S. Additionally, somehandles (black dots in Figure 1), i.e. a finite set of pointsthat the user can manipulate, are defined. They act as hardconstraints and are used to govern the deformation.Animation loop. (see Figure 2)• Step 1: Elastic deformation. A state-of-the-art physically-based simulation is applied for efficiently deforming the sur-face mesh in a plausible way but without taking care ofpaper plasticity at this stage. Isometry to S, and thereforedevelopability, may thus be lost.• Step 2: Modeling bending and crumpling. The connec-tivity of the mesh is modified to update the position of thecurved regions and the flat regions. Notably the conflictsthat may appear between two regions bent in different di-rections are solved by introducing new singular vertices.• Step 3: Developable and isometric tracking. The curvedregions are segmented into quasi developable regions by lo-cally computing the best approximation by a set of general-ized cones. Developability is then geometrically enforced byaligning the mesh edges along these rulings. The isometrypreservation is then optimized by constraining edge lengthsusing a method similar to [16]. Finally the mesh is displayed.3. RESULTS AND VALIDATIONValidation. We used experiments involving real pieces ofpaper to validate our virtual model. Note that contraryto previous approaches, we specifically conduct comparisonsin the critical situation when the first few singular verticesappear. Indeed, when and where these first sharp featuresappear is very noticeable and capturing it accurately is akey element towards realism.The first experiment, shown in Figure 3, was used to validatethe way a new singular vertex appears: in real, we movedFigure 2: Overview of the algorithmthe two opposite sides of an smooth flat square sheet of pa-per (10cm×10cm) closer to each other by a factor of 10%.Then we applied an orthogonal constraint on one of the bor-ders until a singular point appeared (see Figure 3-left). Werepeated the experiment 20 times with other sheets of thesame thickness and then 20 times using our method.The resulting distributions of the position of the first sin-gular point are depicted in Figure 3 (center and right), fortwo different types of paper. Our results show a good accor-dance between real measurements and our virtual model.Figure 3: Comparison of the positions of the firstappearing singular vertex. Left: deformations applied tothe sheet of paper. Right: the measurements on real paper (blackcrosses) and the positions generated by our method (red dots) for a70 g/m2 paper (center) and a 90 g/m2 paper (right). We use dif-ferent standard deviation for computing the position of the singularpoints in order to take into account the different type of paper.Visual comparison between real paper and our results arealso provided in Figure 4. The top row illustrates the defor-mation of a smooth paper strip continuously twisted into aMoebius strip. The second row compares the behavior of asheet of paper after a singular vertex is inserted: we get thesame effect of the sheet straightening when the constrainedpoints at the back are unloaded.Comparison with other methods. Figure 5 comparesour method with a model using a standard finite elementmethod applied to a fine mesh of fixed connectivity, andwith one using adaptive remeshing from Narain et al. [10].All three methods use the same physical FEM solver, andare able to obtain comparable visual results. Our method re-quires approximately ten times less triangles than the adap-Figure 4: Comparing our results with real paperdeformations.tive remeshing method in Narain et al. and twenty timesless than the standard FEM approach with fixed connectiv-ity for similar visual result, leading to far better time rates(see Table 1).Figure 5: Comparison with other methods. The Stan-dard FEM (left) and Narain et al. [10] method (middle) required avery dense mesh in order to get a result of similar visual quality asour method (right).Other results. Table 2 provides the average length dis-tortion over all the animation frames and the computationaltimes for the different examples. Figure 6 gathers other re-Figure 6: Some other results.Table 1: Average computation times.Our standard Narainmethod FEM et al.Remeshing time (ms) 1 - 450Simulation time (ms) 100 14,700 4,640Nb. of triangles 89 1668 788Computed for the example presented in Fig. 5.Table 2: Measures of error in length and Computa-tional time analysis.Models cm Nt tsim (ms) tgeom (ms)Fig. 4 (top) 0.17 117 294 3.3Fig. 4 (bottom) 0.19 87 60 1.4Fig. 6 (left) 1.47 156 247 2.8cm is the mean percentage of compression over all the model edgesand Nt is the average number of triangles of the mesh through theanimation. tsimu and tgeom are the respective times spent inaverage in the simulation steps versus in the geometrical steps.sults of our method.4. CONCLUSIONWe get a variety of results that qualitatively look as realpaper, indicating that our method has reached most of itsgoals.We would like to extend our work to have creased curvesand not only singular points as sharp features, as shown inmechanical studies. Also, although it is much faster thanstate-of-the-art methods, our prototype currently runs at 1to 10 fps. As future work, we would like to improve thismethod to reach real-time performances.Acknowledgements: This research was partially fundedby the ERC advanced grant no. 291184 EXPRESSIVE.5. REFERENCES[1] M. B. Amar and Y. Pomeau. Crumpled paper.Proceedings of the Royal Society. Mathematical,Physical and Engineering Sciences., 1997.[2] R. Burgoon, E. Grinspun, and Z. Wood. Discreteshells origami. Proceedings of Computers And TheirApplications, pages 180–187, 2006.[3] A. D. Cambou and N. Menon. Three-dimensionalstructure of a sheet crumpled into a ball. Proceedingsof the National Academy of Sciences., 2011.[4] E. Cerda and L. Mahadevan. Conical surfaces andcrescent singularities in crumpled sheets. PhysicalReview Letters, 80(11), 1998.[5] E. Grinspun, A. N. Hirani, M. Desbrun, , andP. Shro¨der. Discrete shells. Symposium on ComputerAnimation (SCA), 27:62–67, 2003.[6] Y.-M. Kang, H.-G. Zhang, and H.-G. Cho. Plausiblevirtual paper for real-time application. ComputerAnimation and Social Agents (CASA), Short Paper,2009.[7] Y. Kergosien, H. Gotoda, and T. Kunii. Bending andcreasing virtual paper. IEEE Computer Graphics andApplications, 14(1), 1994.[8] L. Mahadevan and E. Cerda. Conical surfaces andcrescent singularities in crumpled sheets. PhysicalReview Letters, 80(11), 1998.[9] J. Mitani and T. Igarashi. Interactive design of planarcurved folding by reflection. Pacific Graphics (shortpaper), 2011.[10] R. Narain, T. Pfaff, and J. F. O’Brien. Folding andcrumpling adaptive sheets. ACM Trans. Graph. (Proc.of SIGGRAPH), 32(4), 2013.[11] D. Rohmer, M.-P. Cani, S. Hahmann, and B. Thibert.Folded paper geometry from 2d pattern and 3dcontour. Eurographics Short Paper, pages 21–24, 2011.[12] C. Schreck, D. Rohmer, S. Hahmann, M.-P. Cani,S. Jin, C. C. Wang, and J.-F. Bloch. Non-smoothdevelopable geometry for interactively animatingpaper crumpling. ACM Trans. Graph., 2015. to bepublished.[13] R. D. Schroll, E. Katifori, and B. Davidovitch. Elasticbuilding blocks for confined sheets. Physical ReviewLetters, 106, 2011.[14] J. Solomon, E. Vouga, M. Wardetzky, andE. Grinspun. Flexible developable surfaces. ComputerGraphics Forum (CGF), Proc. of Symposium onGeometry Processing (SGP), 31(5), 2012.[15] T. Tachi. Origamizing polyhedral surfaces. IEEETransactions on Visualization and ComputerGraphics, 16(2):298–311, 2010.[16] C. Wang. Towards flattenable mesh surfaces. Comput.Aided Des., 40(1):109–122, 2008.[17] T. Witten. Stress focusing in elastic sheets. Review ofModern Physics, 79, 2007.[18] L. Zhu, T. Igarashi, and J. Mitani. Soft folding.Computer Graphics Forum (Pacific Graphics),32(7):167–176, 2013.

Interactively animating crumpling paper

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