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The low cost, simplicity, and ease of use of paper microfluidic devices have made them valuable medical diagnostics for applications from pregnancy testing to COVID-19 screening. Meanwhile, the increasing complexity of paper-based microfluidic devices is driving the need to produce new tools and methodologies that enable more robust biological diagnostics and potential therapeutic applications. A new design framework is being used to facilitate both research and fabrication of paper-based microfluidic biological devices to accelerate the investigative process and reduce material utilization and manpower. In this work we present a methodology for this framework to dynamically place and route microfluidic components in a nondiscrete design space where fluid volume usage, surface area utilization, and the timing required to perform specified biological assays are accounted for and optimized while also accelerating the development of potentially lifesaving new devices.

A two-step framework to analyze local microstructure variations of paper sheets based on 3D image data is presented. First, a multi-stage workflow efficiently acquires a large set of highly resolved tomographic image data, which enables—in combination with statistical image analysis—the quantification of local variations and pairwise correlations of morphological microstructure characteristics on length scales ranging from micrometers to centimeters. Secondly, the microstructure is analyzed in terms of the local behavior of porosity, thickness, and further descriptors related to transportation paths. The power of the presented framework is demonstrated, showing that it allows one (i) to quantitatively reveal the difference in terms of local structural variations between a model paper before and after unidirectional compression via hard-nip calendering and that (ii) the field of view which is required to reliably compute the probability distributions of the considered local microstructure characteristics is at least 20 mm$^{2}$. The results elucidate structural differences related to local densification. In particular, it is shown how calendering transforms local variations in sheet thickness into marked local mass density variations. The obtained results are in line with experimental measurements of macroscopic properties (basis weight, Bekk smoothness parameters, thickness, and Gurley retention times).

Measured local paper structure—i.e. local basis weight, local thickness, local density and local fiber orientation—has been linked to local strain and local material failure (local temperature increase due to energy dissipation upon fiber–fiber bond failure) measured during tensile testing. The data has been spatially linked through data map registration delivering several thousand $$1\times 1\,\hbox{mm}^2$$ 1 × 1 mm 2 paper regions, each containing all measured properties. The relation between local paper structure and resulting local deformation and failure is studied with regression models. Multiple linear regression modeling was used to identify the paper structure related drivers for local concentrations of strain under load and local concentrations of material failure, which are both starting to occur considerably before rupture of the paper. Analyzing the development of local strain in paper we found that regions with higher basis weight and higher fiber orientation in load direction tend to exhibit considerably lower strain during tensile testing. Furthermore, the relation between local strain and local grammage can be predicted with the statistical theory of elasticity. Also regions with higher density have lower local strain, but not as pronounced. The findings for local fiber–fiber bond failure of paper are similar but not equivalent. The strongest correlation exists with local grammage. Local density and local fiber orientation show in turn weaker correlation with local bond failure. Local variations in paper thickness were not relevant in any case. These findings are highlighting the relevance of local fiber orientation and local density variations as structural mechanisms governing paper failure. In the past the focus has been mostly on paper formation. Together with local grammage (formation) they are responsible for the weak spots in paper, and thus cause local concentrations of paper strain and the initiation of failure under tensile load.

Looking at the past, among the main modalities of transmission of knowledge (alongside with manuscripts and printed treatises and with respect to the various ‘precepts’) we would have found also various professional figures. This is even truer with regard to the training and dissemination of knowledge among military engineers, who used to be real ‘living vehicles’ in national and international cultural systems. This is also true for what concern that circumscribed set of knowledge of geometrical-mathematical theories, including their application to the codes of architectural Representation and Design. This research aims to investigate the example of the military engineer Gaspare Beretta (Milan 1620–1703), whose theoretical production both on architectural graphic and on various applications of Geometry is still unpublished. With the aim of developing the relationship between geometry, architecture and the transmission of knowledge, we will go through his practices of the regular fortress tracing on ‘paper and ground’, in order to verify the resulting connections between geometric reconstruction and Author’s intentions.

Lampshades were developed as wind shield for light source when fire was used for light source. Though contemporary light sources, like LED lamp, fluorescent lamp and incandescent light, are working with electricity and wind shield is not required for light source, almost all light sources are still covered by lampshade to arrange or control luminous flux from light source. And combination of lampshades and light sources may give distinctive impression to surrounding people. Until now, paper folding method ‘SQEF’ (Skew Quadrilateral Elastic Folding) was proposed to construct shapes with non-flat texture in the field of lampshade design. With SQEF methods, final shape is constructed by skew quadrilateral unit with a curved surface. As each skew quadrilateral unit is composed by continuously curved surface, luminance distribution within each unit is continuous as well. The shapes of section appeared in proposed lampshades are convex and curved surface in each unit is concave. The purpose of the research described in this paper is expansion of the horizontal sections of lampshades to nonconvex shape to generate convex unit. The buffer part was inserted between concave parts to combine them. An example of the lampshade including concave parts was manufactured. And finally, luminance distribution on the lampshade was examined.

A general method is proposed to predict the shape of a paper model of a brassiere cup. A brassiere cup consists of several cloth and wire parts and the shapes of cloth parts are determined by repeating creation of a paper cup model, check of its 3D shape, and modification of 2D shapes of parts. For efficient design of a brassiere cup, prediction of its 3D shape with a simulation is required. The deformed shape of a paper part is represented as a single or multiple developable surfaces. So, a model that can represent a part both as a single surface and as multiple surfaces is proposed. Which case is selected depends on the magnitude of the potential energy of the part in each case. The potential energy of the part and geometric constraints imposed on the part are formulated based on the model. Minimizing the potential energy under geometric constraints derives the stable shape of the part in either case. Furthermore, our proposed method can be applied to prediction of the paper cup model consisting of parts with complex shapes.

Miura-ori is well-known for its capability of flatly folding a sheet of paper through a tessellated crease pattern made of repeating parallelograms. Many potential applications have been based on the Miura-ori and its primary variations. Here we are considering how to generalize the Miura-ori: what is the collection of rigid-foldable creased papers with a similar quadrilateral crease pattern as the Miura-ori? This paper reports some progress. We find some new variations of Miura-ori with less symmetry than the known rigid-foldable quadrilateral meshes. They are not necessarily developable or flat-foldable, and still only have single degree of freedom in their rigid folding motion. This article presents a classification of the new variations we discovered and explains the methods in detail., Comment: Main text 24 pages, 9 figures, 2 tables, Supplementary Material 13 pages, 2 figures; submitted to the Proceedings of the Royal Society A, in minor revision

We can obtain a two-fold symmetric figure by folding a square sheet of paper in the middle and then cutting along some curves drawn on the paper. By making two perpendicular folds through the centre of the paper and then cutting, we can obtain a four-fold symmetric figure. We can also get an eight-fold symmetric figure by making a fold bisecting an angle made by the two perpendicular folds before cutting. But it is not possible to obtain a three-fold, five-fold or six-fold symmetric figure in this way; we need to make more folds before cutting. Making a three-fold (respectively five-fold and six-fold) figure involves the division of the angle at the centre (360°) of a square sheet of a paper into six (respectively ten and twelve) equal parts. In other words, we need to construct the angles 60°, 36° and 30°. But these angles cannot be obtained by repeated bisections of 180° by simple folding as in the making of two-fold, four-fold and eight-fold figures. In [1], we see that each of the constructions of 60° and 30° applies the fact that sin 30° = ½ and takes only a few simple folding steps. The construction of 36° is more tedious (see, for example, [2] and [3]) as sin 36° is not a simple fraction but an irrational number. In this Article, we show how to make, by paper-folding and cutting a regular pentagon, a five-pointed star and create any five-fold figure as we want. The construction obtained by dividing the angle at the centre of a square paper into ten equal parts is called a pentagon base. We gained much insight from [2] and [3] when developing the method for making the pentagon base to be presented below.

The key operation in tissue production is the creping at the Yankee cylinder. The tissue sheet is adhered to a Yankee cylinder and then detached from the surface with a blade. As a result a strong microstructure - crepe folds - is generated on the web. In this work the structure of creping pattern in CD and MD of tissue paper was studied. The samples available in this work consisted of bath tissue and towel grades. The significant difference between the samples was in the number of crepe folds. The tissue sheets were analyzed with an imaging system. The variation of crepe folds in images was studied in spatial frequency space through 2D Welch spectrum. It was noticed that the difference in creping pattern structure between the samples can be seen clearly in spectra. The creping pattern was simulated based on the creping pattern variation found from the Welch spectra of tissue paper images. The simulation was based on a set of sinusoidal terms representing several orientations and amplitudes. The amplitudes of sinusoidal terms located in regular grid in spectra were approximated with 2D normal distribution. Thus, the structure of creping pattern can be represented with three parameters originating from the deviations and location of 2D normal distribution. The creping pattern reconstructed with simulations contains the relevant features in the creping pattern of true tissue images.

In this paper we correct a mistake in [C. Ciliberto, F. Russo, On surfaces with two apparent double points. Adv. Geom. 5 (2005), 223–235].

We reformulate the original model of Hatchell and Bourne and Roste, Stovas and Landro that couples fractional velocity change to subsurface strain via a fundamental constant R. The new model combines elastic compressibility of a dual-porosity system for a sand–shale mixture with horizontal planes of inter-granular weakness. The majority of observed R-factor magnitudes from post-stack 4D seismic data in both the reservoir and overburden can thus be explained. R is predicted to depend strongly on lithology and also initial strain state. The model is also extended to predict the observed angle-dependence of time-lapse time-shifts from pre-stack data. An expression for the gradient of time-shift with incidence angle is obtained in terms of the background VP/VS, and also the ratio of tangential to normal compliances BT/BN representing loss or creation of inter-granular coupling. If accurately estimated from data, this compliance ratio can be used as an additional parameter to assess the post-production state of the overburden. It is concluded that whilst R remains the over-arching parameter controlling the magnitude of time-shifts measured from 4D seismic data, BT/BN is a subtler parameter that may also prove of future value.

We derive a one-dimensional model that describes pressing of water saturated paper in the press-section of the paper machine. The model involves two nonlinear diffusion equations which are coupled across an internal boundary. Existence and uniqueness as a number of qualitative properties are demonstrated. Further, computational results for a concrete case are discussed.

Two different methods of analysis were developed in order to clarify the phenomenon of paper flutter. One of these is a flutter simulation using a Navier–Stokes code (N–S simulation). N–S simulation was used to determine the unsteady lift force, the amplitude of flutter, and the air-flow around a paper sheet, by means of a time-marching scheme. The other form of analysis is based on a potential-flow analysis of an oscillating thin airfoil via an eigenvalue analysis for determining stability. The flutter speeds and flutter modes obtained by each method are consistent. Aspects of the behavior of paper flutter that had not been clarified in the experimental analysis were clarified by potential flow analysis. From the results of this study, it was shown that potential flow analysis is very convenient and adequate for a parametric study of this problem.

A machine that transports paper sheet by using a pair of flat belts, handles various paper sheet with regard to size, thickness and texture. This often causes the gap between the sheets to change when it is transported over a long distance. If the gap becomes shorter than a certain threshold value, it causes sheet jam. We have therefore developed a gap control system that determines the transport distance of sheets and measures the gap change before the sheets reach the gap controller. Using thses information, it estimates the gap change after the gap controller and sets a target value. The contoller varies the transport velocity in an optimum pattern and controls the gap to the target value. We applied the system to a transport route and it reduced the standard deviation of gap into 2/3.

In the field of lampshade design, paper folding method has been highly utilized as the non-flat texture by paper folding method can generate beautiful luminance distribution. Especially, if the non-flat texture is composed of curved surface units, luminance distribution in the units is continuous and visually soft. And the method combining non-flat pattern and geometrical transmittance pattern has been already proposed to overlap luminance distribution caused by non-flat texture and that by transmittance pattern. In this paper, plane tessellations are introduced to expand diversity of lampshade design. Two methods were proposed with single tessellation and multiple tessellations. With single tessellation method, a tessellation was used for both paper folding and geometrical transmittance pattern. With multiple tessellation method, multiple tessellations were used for paper folding and geometrical transmittance pattern respectively. To verify the effect of the combination, lampshades with non-flat texture and geometrical transmittance pattern were manufactured and brightness distributions of them were measured.

This study aims to investigate the similarities and differences between prospective mathematics teachers’ creative thinking skills in paper-pencil test and on a Geogebra-supported environment in terms of problem-posing. This case study used purposive sampling method for determining the participants. Findings revealed that the activities carried out in the GeoGebra-supported environment were insufficient to produce creative problems, and GeoGebra’s main utility to prospective teachers was in identifying their mistakes related to mathematical concepts and discrepancies among numerical values of the problems posed. The reasons for the low achievement in posing problem were discussed: These were; (i) lack of problem-posing experience, (ii) the structure of problem-posing activity, and (iii) prospective teachers’ mathematical content knowledge.

Until Mandelbrot introduced the concept of fractal geometry and fractal dimension in early 1970s, it has been generally considered that the geometry of nature should be too complex and irregular to describe analytically or mathematically. Here fractal dimension indicates a non-integer number such as 0.5, 1.5, or 2.5 instead of only integers used in the traditional Euclidean geometry, i.e., 0 for point, 1 for line, 2 for area, and 3 for volume. Since his pioneering work on fractal geometry, the geometry of nature has been found fractal. Mandelbrot introduced the concept of fractal geometry. For example, fractal geometry has been found in mountains, coastlines, clouds, lightning, earthquakes, turbulence, trees and plants. Even human organs are found to be fractal. This suggests that the fractal geometry should be the law for Nature rather than the exception. Fractal geometry has a hierarchical structure consisting of the elements having the same shape, but the different sizes from the largest to the smallest. Thus, fractal geometry can be characterized by the similarity and hierarchical structure. A process requires driving energy to proceed. Otherwise, the process would stop. A hierarchical structure is considered ideal to generate such driving force. This explains why natural process or phenomena such as lightning, thunderstorm, earth quakes, and turbulence has fractal geometry. It would not be surprising to find that even the human organs such as the brain, the lung, and the circulatory system have fractal geometry. Until now, a normal frequency distribution (or Gaussian frequency distribution) has been commonly used to describe frequencies of an object. However, a log-normal frequency distribution has been most frequently found in natural phenomena and chemical processes such as corrosion and coagulation. It can be mathematically shown that if an object has a log-normal frequency distribution, it has fractal geometry. In other words, these two go hand in hand. Lastly, applying fractal principles is discussed, focusing on pulp and paper industry. The principles should be applicable to characterizing surface roughness, particle size distributions, and formation. They should be also applicable to wet-end chemistry for ideal mixing, felt and fabric design for papermaking process, dewatering, drying, creping, and post-converting such as laminating, embossing, and printing.

A technique to determine the permeability from X-ray microtomographic images was developed and applied to paper structures. The images are thresholded and a skeleton of the pore space is obtained from the images. The skeleton is represented by means of a network of resistances. The total resistance of the network along the principal directions is now determined both in the in-plane (X, Y) as well as through the plane (Z) directions. The network resistance for structures at each section is determined. From this data, a correlation length for the network resistance is obtained. Correlation length data are used for determining the effect of heterogeneities on transport properties of paper structures. From this analysis, it is possible to analyze and simulate the moisture transport processes such as wicking and drying in paper, reproducing smaller-scale variations. Permeabilities in the thickness direction were calculated from the X-ray microtomographic reconstructions of the pore space. A set of pape...

This paper studies geometric properties of a class of irreversible dynamical systems, referred to in the literature as metriplectic systems. This class of systems, related to generalized (or dissipative) Hamiltonian systems, are generated by a conserved component and a dissipative component and appear, for example, in non-equilibrium thermodynamics. In non-equilibrium thermodynamics, the two potentials generating the dynamics are interpreted as generalized energy and generalized entropy, respectively. Stability and stabilization results for metriplectic systems have been presented in the literature, however, some aspects are still poorly understood, in particular the existence of dynamical invariants such as periodic orbits. In this note, we study the properties of metriplectic systems by considering a lift from the n-dimensional state space to a (2n+1)-dimensional contact space, following an approach introduced in recent years to study irreversible control systems. This lift leads to a deeper geometric characterization of metriplectic systems in the extended space. An example is provided to illustrate the approach proposed in this paper.

Designing computer systems for educational purpose is a difficult task. While many of them have been developed in the past, their use in classrooms is still scarce. We make the hypothesis that this is because those systems take into account the needs of individuals and groups, but ignore the requirements inherent in their use in a classroom. In this work, we present a computer system based on a paper and tangible interface that can be used at all three levels of interaction: individual, group, and classroom. We describe the current state of the interface design and why it is appropriate for classroom orchestration, both theoretically and through two examples for teaching geometry.

A method is proposed to predict the shape of a paper model of a three piece brassiere cup, which consists of several cloth and wire parts. Currently, the shape of each part is determined by creating a paper model and then refining the model. This process is repeated until the desired shape is achieved. However, predicting the 3D shape with a simulation would improve design efficiency. As a model is made of paper, each part is assumed to be inextensible and its surface is represented as combination of developable surfaces. The potential energy of a cup and geometric constraints imposed on the cup are formulated by use of the normal curvature and the direction of a generatrix of each developable surface. Minimizing the potential energy under geometric constraints derives a stable shape of the cup model. The computed and measured cup shapes coincided qualitatively.

We have devised a paper model of the auricle that is simple to design and easily fabricated. An oval-shaped sheet of paper is folded so that the prominent and hollow parts center around a helical crus and concha. Most parts of the human auricle can be created by using this model. Studies of the model show that a large skin area is required for the helical crus. The model is considered to be useful in simulating total or partial auricular reconstruction. It also reveals new possibilities in reconstructive methods using tissues composed of cartilage and skin, which are flat and large. Furthermore, it is interesting that the formation of our paper model by forward rotation and folding is compatible with the process of auricle embryonic development.

We present some rigorous results about the bending behaviour of paper. By adapting these results to the Mobius strip, we obtain some qualitative properties of developable Mobius strips which minimize the bending energy. We also provide some numerical simulations which illustrate and strengthen the analytic results.

This paper proposes a free-viewpoint interface for mobile-robot teleoperation, which provides viewpoints that are freely configurable by the human operator head pose. The viewpoints are acquired by a head tracker equipped on a head mounted display. A real-time free-viewpoint image generation method based on view-dependent geometry and texture is employed by the interface to synthesize the scene presented to the operator. In addition, a computer graphics model of the robot is superimposed on the free-viewpoint images using an augmented reality technique. We developed a prototype system based on the proposed interface using an omnidirectional camera and depth cameras for experiments. The experiments under both virtual and physical environments demonstrated that the proposed interface can improve the accuracy of the robot operation compared with first- and third-person view interfaces, while the quality of the free-viewpoint images generated by the prototype system was satisfactory for expressing the potential advantages on operational accuracy.

A method to predict the shape of a paper model of a two piece brassiere cup is proposed. A brassiere cup consists of several parts made of cloth and wires. The shape of each part is determined by making a paper model of the cup and checking the 3D shape of the model. For efficient design of parts, it is required to predict the 3D shape of such model with simulation. As a model is made of paper, each part is inextensible and its deformed surface is developable. Then, the potential energy of a part can be represented as a function of the principal curvature and the principal direction along an edge of the part. Minimizing the potential energy under geometric constraints, the stable shape of the cup model is derived. The computed shape of the cup almost coincided with the measured shape of an actual paper model.

We describe a method for characterization of the microstructure of a paper, in terms of the statistical distribution of the effective radii and lengths of the channels between the paper's fibers, and the average coordination numbers that characterize the connectivity of such channels, both in the main planes of the paper and in the transverse direction perpendicular to these planes. The microstructure of the paper is then represented by a three-dimensional network of interconnected flow channels between the fibers. The geometrical characteristics of the channels, such as their effective radii and lengths, are selected from the statistical distributions obtained by the characterization method, as are the two average coordination numbers. Extensive computer simulations are carried out to compute the effective permeability tensor of the paper, and to investigate the effect of the various microstructural parameters on the tensor. These simulations indicate that the two average coordination numbers, as well as the spatial distribution of the fibers and their length, strongly influence the flow properties of a paper.

The present study treats a paper wrinkle appearing in a fixing roller unit of printers or copiers with electro-photographic process. An experiment shows that a wrinkle always occurs, when an A4-sized paper is tilted by 3 degrees against a horizontal fixing roller unit without a heating devise and is fed to a roller nip. The wrinkle appears in the middle of the paper width. Computer simulations were carried out to illustrate this wrinkle occurring process. The simulation model consists of a rigid cylindrical driving roller, a driven rubber roller, a paper, and a guide plate. The paper is assumed to be an isotropic elastic-plastic body. The results reveal the following wrinkle occurring process. When the tilted paper is fed to the rollers, the center part of the paper leading edge firstly enters the roller nip and is pulled by friction forces in the nip. Then paper wavy bending deformations appear in front of the nip. A steep slope of the deformation is crushed and yielded in the nip. After this, a neighbor area of the yielded point gathers together and a folded wrinkle is formed. Further, this type of wrinkle is liable to occur when the paper tilt angle, Young's modulus, or coefficient of friction is large, or when the paper yield stress is small.

Convergent plate boundaries around the globe show a high degree of structural complexity and variability in site-specific geometry and mass flux. The heterogeneity in the structural evolution, the interior regime as well as external architecture of individual margins is reflected in their seismic character, resulting in a segmentation along-strike as well as along-dip. Subduction zones generate more than 80 percent of global earthquakes above magnitude 8.0, but rupture characteristics are highly individual and linked to margin specific geometrical conditions. Major segments of subduction zones are commonly submerged in deep water and difficult to access at the majority of margins. Marine geophysical techniques, which are able to image the complex structures in these settings with sufficient coherency and depth penetration, have proven crucial to improve our knowledge on the geological framework of the different types of subduction zones. The aim of this review paper is to unravel the structural diversity of convergent margins and between individual subduction zone segments. Field data from different margins around the globe deliver images of the seafloor and subsurface in unprecedented resolution, which show segmentation to be far more complex than previously inferred. Along-strike segmentation results in accretionary segments contiguous to erosive segments along a single margin. Modes of mass transfer must hence be viewed as transient processes dependent on sediment supply and lower plate structure. Along-strike segment boundaries commonly correlate with underthrusting lower plate relief that controls the deep deformation of a subduction zone and the spatial and temporal variation in slip behavior. Examples of underthrusting oceanic basement relief at different stages of subduction elucidate their impact on the inner geometry of the margin. Lower plate heterogeneities occur at subduction zones worldwide and thus pose a common phenomenon, whose role as barriers to seismic rupture constitute a central control on subduction zone seismicity and segmentation.

Recent research on braided river morphology shows that the intensity (number of anabranches) of braiding channels increases with dimensionless discharge and (or) stream power. This variation in intensity reflects the adjustment of total sinuosity of the river to imposed gradient at a given discharge and grain size. Only a subset of channels is active at a given time and this active braiding intensity reflects the limited number of channels that can sustain bed load transport as the flow is divided. This is governed mechanistically by the dynamics of bifurcations and avulsions. Braided channel networks also have a characteristic length scale (or scales) related to the wavelength of the bars from which braiding develops and to the scale of the bars and confluence–bifurcation units within the braided network. The range of scales is limited by the size (and, therefore, number) of the active channels within the network and the width of the entire river.

We discuss the problem of splitting a given geometrical plane figure in two regions of equal area by drawing a line through a given point in the plane. We also discuss the possibility of splitting the figure into regions in proportions p% and (1 - p)% of the area of the given figure.

Han et al. [W.Z. Han, Z.F. Zhang, S.D. Wu, S.X. Li, Acta Mat. 55 (2007) 5889–5900] have published a paper about equal-channel angular extrusion (ECAE) testing of Al single crystals. It is shown in this work that their paper contains numerous errors in both presentation and interpretation of the results. The main experimental error concerns the initial orientations of the crystals that are far from their assumed orientations, while the theoretical discussion is faulty in terms of the deformation mechanisms that apply in ECAE processing. It is also shown that the use of the so-called “shear factors” can lead to erroneous results related to the identification of the active slip systems in simple shear.

To display the break of paper, we propose the new method for calculating transformation of paper and for representing a minute structure of paper. Paper is defined as a “spring-mass” cubic mesh model. And, to express the flow of the paper fiber, some cells of the mesh are combined to one group. By giving “spring-mass” parallelepiped covering the group, the bending stress is added to the model. For minuteness expression of paper, display data is created from the model by the following 2 steps. At first, thickness of the mesh model is compressed, then many points are added on separated edges of the mesh to display jagged shape. As the result of computer simulation in various conditions, realistic break lines of paper were represented.

We present a novel and effective method for modeling a developable surface to simulate paper bending in interactive and animation applications. The method exploits the representation of a developable surface as the envelope of rectifying planes of a curve in 3D, which is therefore necessarily a geodesic on the surface. We manipulate the geodesic to provide intuitive shape control for modeling paper bending. Our method ensures a natural continuous isometric deformation from a piece of bent paper to its flat state without any stretching. Test examples show that the new scheme is fast, accurate, and easy to use, thus providing an effective approach to interactive paper bending. We also show how to handle non-convex piecewise smooth developable surfaces.