Principal Curvature Directions Research Articles

The architectural application of shells whose boundaries subtend a constant solid angle

Surface geometry plays a central role in the design of bridges, vaults and shells, using various techniques for generating a geometry which aims to balance structural, spatial, aesthetic and construction requirements.In this paper we propose the use of surfaces defined such that given closed curves subtend a constant solid angle at all points on the surface and form its boundary. Constant solid angle surfaces enable one to control the boundary slope and hence achieve an approximately constant span-to-height ratio as the span varies, making them structurally viable for shell structures. In addition, when the entire surface boundary is in the same plane, the slope of the surface around the boundary is constant and thus follows a principal curvature direction. Such surfaces are suitable for surface grids where planar quadrilaterals meet the surface boundaries. They can also be used as the Airy stress function in the form finding of shells having forces concentrated at the corners.Our technique employs the Gauss-Bonnet theorem to calculate the solid angle of a point in space and Newton's method to move the point onto the constant solid angle surface. We use the Biot-Savart law to find the gradient of the solid angle. The technique can be applied in parallel to each surface point without an initial mesh, opening up for future studies and other applications when boundary curves are known but the initial topology is unknown.We show the geometrical properties, possibilities and limitations of surfaces of constant solid angle using examples in three dimensions.

Coordinated Motion of Epithelial Layers on Curved Surfaces.

Coordinated cellular movements are key processes in tissue morphogenesis. Using a cell-based modeling approach we study the dynamics of epithelial layers lining surfaces with constant and varying curvature. We demonstrate that extrinsic curvature effects can explain the alignment of cell elongation with the principal directions of curvature. Together with specific self-propulsion mechanisms and cell-cell interactions this effect gets enhanced and can explain observed large-scale, persistent, and circumferential rotation on cylindrical surfaces. On toroidal surfaces the resulting curvature coupling is an interplay of intrinsic and extrinsic curvature effects. These findings unveil the role of curvature and postulate its importance for tissue morphogenesis.

Recovery of Interlaminar Tensile Stresses in Curved Laminates Subject to Biaxial Flexure: The Case of an Elastically Induced Curvature Purposely Misaligned with the Principal Directions of Initial Curvature

Many literature contributions deal with the evaluation at FE post-processing stage of the interlaminar normal stress distribution in moderately thick single or doubly curved laminates subject to flexure, based on through-the-thickness equilibrium relations. Most of the contributions, however, discuss test cases in which the principal directions of the load-induced elastic curvatures are aligned with the principal directions of the initial curvature; this condition is not general. In the present work, a simple post-processing methodology for the recovery of the interlaminar tensile stress is presented.The procedure retrieves the Interlaminar Tensile Stresses (ILTS) distribution in a doubly curved laminate modelled through common wise quadrilateral shell Finite Elements and is based on customary FE sub-model which inputs are the displacements and rotation calculated on the shell structure. At the moment the sub-model can't retrieves Interlaminar Shear Stresses but can be further enhanced to do it.In this paper the methodology is successfully applied to an open thin walled profile in torsion test case, in which the principal directions of elastic curvature are strongly misaligned with the principal directions of initial curvature.

Crossed helical gears—simulative studies and experimental results on non-involute geometries

Crossed helical gear units generally consist of a steel worm and a plastic helical gear with an axis crossing angle. Due to the high transmission ratio in a small installation space, they are used in auxiliary and actuating gears such as in window regulators, assembly lines, household appliances and positioning tasks. This paper deals with a calculation approach for crossed helical gears to include non-involute flank geometries for new and optimized gearboxes. The geometry derivation for general flank contours based on a polynomial and extends the geometric possibilities beyond the special case of the involute form. The use of principal curvatures and principal curvature directions to describe freely curved surface geometries enables the calculation of flank pressures, efficiency and sliding paths in the entire contact area. Previous studies confirmed the advantages in the mentioned load characteristics for geometries deviating from the involute. Furthermore, parameter studies show the influences of the individual design parameters and demonstrate the effects on operating behavior, with those a design recommendation for crossed helical gear units with general flank geometries can be created. ZC and ZC‑S geometries are used as calculation examples. With regard to the use of these new geometries in applications, manufacturing is a crucial aspect. Grinding disc parameters as well as a three-roll size to control the worm manufacturing for general flank geometries are necessary and described below. In addition to the theoretical calculation, a ZC crossed helical gear geometry was designed, manufactured and compared to a ZI reference gear in test rigs. This confirmed the advantages in load-carrying capacity and efficiency increase resulting from the calculation and proved the benefits.

Fluid Flow-inspired Curvature-aware Print-paths from Hexahedral Meshes for Additive Manufacturing

Thanks to recent improvements, computational methods are able to produce high-quality hexahedral meshes, whose hexahedrons are arranged along the principal curvature directions of the input surfaces. Curvature-aware print-paths following these curvature directions have been recently proposed by Gunpinar and enable reduction in stair-stepping effect in the printed parts. However, crosswise contacts can exist between the print-paths (i.e., print-paths are quasi-perpendicular to each other), which is undesirable as failures may occur particularly at those contact regions. Therefore, the present work aims at generation of curvature-aware print-paths without crosswise contacts between them. To solve this problem, we inspire from fluid flow and imitate (laminar) streamlines for an inlet and an outlet of a duct for designing print-paths. A hexahedral mesh is decomposed into blocks (cuboid-like sub-volumes), each of which is covered with fluid flow-inspired print-paths. A multi-axis (collision-free) additive manufacturing (AM) planning technique is also proposed. As a proof of concept, fluid flow-inspired curvature-aware print-paths are validated using a multi-axis AM simulator and machine.

Convexity estimates for high codimension mean curvature flow

We consider the flow by mean curvature of smooth n-dimensional submanifolds of Rn+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{n+k}$$\\end{document}, k≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\ge 2$$\\end{document}, which are compact and quadratically pinched. We establish that such flows are asymptotically convex, that is, the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. This generalises the convexity estimate of Huisken–Sinestrari to higher codimensions. By combining our estimate with work of Naff, we conclude that singularity models for the flow are convex ancient solutions of codimension one.

Deformation and Springback Behavior of Sheet Metal With Convex-Shaped Surfaces in Heat-Assisted Incremental Bending Process Based on Minimum Energy Method

Abstract Three-dimensional sheet metal forming is one of the biggest challenges in the shipbuilding industry. The forming quality of the traditionally applied line heating method depends entirely on the skills of the technicians. Moreover, this approach is inefficient and error-prone. This calls for a need to develop new forming technologies that are easy to control and more efficient. In this paper, a novel forming method called heat-assisted incremental bending is presented. The minimum energy loading path of bidirectional curvature sheet along the direction of Gaussian curvature is adapted to deform the sheet metals with a convex shape surface. Besides, the auxiliary heating method and the auxiliary supporting are exclusively included in the forming process to improve the formability and accuracy. In this study, the convex-shaped sheet metal with large curvature is obtained along the two loading trajectories in two principal curvature directions, respectively. Results proved that using the minimum energy loading trajectory and including the auxiliary supports and heating can reduce the forming force to the greatest extent and effectively reduce the number of stamping points, thus improving the sheet metal forming efficiency and forming accuracy.

Exploring differential geometry in neural implicits

We introduce a neural implicit framework that exploits the differentiable properties of neural networks and the discrete geometry of point-sampled surfaces to approximate them as the level sets of neural implicit functions.To train a neural implicit function, we propose a loss functional that approximates a signed distance function, and allows terms with high-order derivatives, such as the alignment between the principal directions of curvature, to learn more geometric details. During training, we consider a non-uniform sampling strategy based on the curvatures of the point-sampled surface to prioritize points with more geometric details. This sampling implies faster learning while preserving geometric accuracy when compared with previous approaches.We also use the analytical derivatives of a neural implicit function to estimate the differential measures of the underlying point-sampled surface.

Computational Design of Self‐Actuated Surfaces by Printing Plastic Ribbons on Stretched Fabric

AbstractWe introduce a new mechanism for self‐actuating deployable structures, based on printing a dense pattern of closely‐spaced plastic ribbons on sheets of pre‐stretched elastic fabric. We leverage two shape‐changing effects that occur when such an assembly is printed and allowed to relax: first, the incompressible plastic ribbons frustrate the contraction of the fabric back to its rest state, forcing residual strain in the fabric and creating intrinsic curvature. Second, the differential compression at the interface between the plastic and fabric layers yields a bilayer effect in the direction of the ribbons, making each ribbon buckle into an arc at equilibrium state and creating extrinsic curvature. We describe an inverse design tool to fabricate low‐cost, lightweight prototypes of freeform surfaces using the controllable directional distortion and curvature offered by this mechanism. The core of our method is a parameterization algorithm that bounds surface distortions along and across principal curvature directions, along with a pattern synthesis algorithm that covers a surface with ribbons to match the target distortions and curvature given by the aforementioned parameterization. We demonstrate the flexibility and accuracy of our method by fabricating and measuring a variety of surfaces, including nearly‐developable surfaces as well as surfaces with positive and negative mean curvature, which we achieve thanks to a simple hardware setup that allows printing on both sides of the fabric.

Uniformity Prediction of Bending Stiffness of Composite Space Mirrors With Ply Angle Misalignments by Deflection Angle

Based on the classical lamination theory, the principal moment direction of the laminate is derived. It is 45° apart from the generalized ply angle, which is the orientation of the ply where the ply angle misalignment occurs nominally. After adding a fixed one-ply angle or random multiple ply angle misalignments to six quasi-isotropic and symmetric stacking sequences, out-of-plane deformations of mirrors were simulated and compared. The result shows an angle difference between the principal moment and curvature directions. It is referred to as the deflection angle, approximately appearing as a trigonometric curve, which varies with laminates and is dependent on the generalized ply angle within each laminate. The deflection angle directly reflects the consistency of bending stiffness or laminate bending anisotropy; the smaller the deflection angle, the better will be the uniformity of bending stiffness. Consequently, it is an efficient approach to obtain a laminate with more uniformity of bending stiffness by comparing the deflection angle. These results enrich the research of out-of-plane deformation and could objectively promote the study of quasi-isotropic laminates for bending stiffness.

Corrected trapezoidal rules for singular implicit boundary integrals

We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in Kublik et al. (2013) [7]. In this formulation, a boundary integral over a smooth, closed hypersurface is transformed into an equivalent volume integral defined in a sufficiently thin tubular neighborhood of the surface. The volumetric formulation makes it possible to use the simple trapezoidal rule on uniform Cartesian grids and relieves the need to use parameterization for developing quadrature. Consequently, typical point singularities in a layer potential extend along the surface's normal lines. We propose new higher-order corrections to the trapezoidal rule on the grid nodes around the singularities. This correction is based on local decompositions of the singularity and is dependent on the angle of approach to the singularity relative to the surface's principal curvature directions. The proposed decomposition, combined with the volumetric formulation, leads to a special quadrature error cancellation.

Coarse-grained curvature tensor on polygonal surfaces

Using concepts from integral geometry, we propose a definition for a local coarse-grained curvature tensor that is well-defined on polygonal surfaces. This coarse-grained curvature tensor shows fast convergence to the curvature tensor of smooth surfaces, capturing with accuracy not only the principal curvatures but also the principal directions of curvature. Thanks to the additivity of the integrated curvature tensor, coarse-graining procedures can be implemented to compute it over arbitrary patches of polygons. When computed for a closed surface, the integrated curvature tensor is identical to a rank-2 Minkowski tensor. We also provide an algorithm to extend an existing C++ package, that can be used to compute efficiently local curvature tensors on triangulated surfaces.

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input that connect the PQ strips. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling-based representations of developable surfaces while still obtaining developable, curvature-aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its isolines are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight isolines is non-linear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle-based interactive freeform modeling of developable shapes.

Characteristic parameterizations of surfaces with a constant ratio of principal curvatures

Motivated by applications in architectural geometry, we study and compute surfaces with a constant ratio of principal curvatures (CRPC surfaces) based on their characteristic parameterizations. For negative Gaussian curvature K, these parameterizations are asymptotic. For positive K they are conjugate and symmetric with respect to the principal curvature directions. CRPC surfaces are described by characteristic parameterizations whose parameter lines form a constant angle. We use them to derive characteristic parameterizations of rotational CRPC surfaces in a simple geometric way. Pairs of such surfaces with principal curvature ratio κ1/κ2=±a can be seen as equilibrium shapes and reciprocal force diagrams of each other. We then introduce discrete CRPC surfaces, expressed via discrete isogonal characteristic nets, and show how to efficiently compute them through numerical optimization. In particular, we derive discrete helical and spiral CRPC surfaces. We provide various ways how these and other special types of CRPC surfaces can serve as a basis for computational design of more general CRPC surfaces. Our computational tools may also serve as an experimental basis for mathematical studies of the largely unexplored class of CRPC surfaces.

Learning Direction Fields for Quad Mesh Generation

AbstractState of the art quadrangulation methods are able to reliably and robustly convert triangle meshes into quad meshes. Most of these methods rely on a dense direction field that is used to align a parametrization from which a quad mesh can be extracted. In this context, the aforementioned direction field is of particular importance, as it plays a key role in determining the structure of the generated quad mesh. If there are no user‐provided directions available, the direction field is usually interpolated from a subset of principal curvature directions. To this end, a number of heuristics that aim to identify significant surface regions have been proposed. Unfortunately, the resulting fields often fail to capture the structure found in meshes created by human experts. This is due to the fact that experienced designers can leverage their domain knowledge in order to optimize a mesh for a specific application. In the context of physics simulation, for example, a designer might prefer an alignment and local refinement that facilitates a more accurate numerical simulation. Similarly, a character artist may prefer an alignment that makes the resulting mesh easier to animate. Crucially, this higher level domain knowledge cannot be easily extracted from local curvature information alone. Motivated by this issue, we propose a data‐driven approach to the computation of direction fields that allows us to mimic the structure found in existing meshes, which could originate from human experts or other sources. More specifically, we make use of a neural network that aggregates global and local shape information in order to compute a direction field that can be used to guide a parametrization‐based quad meshing method. Our approach is a first step towards addressing this challenging problem with a fully automatic learning‐based method. We show that compared to classical techniques our data‐driven approach combined with a robust model‐driven method, is able to produce results that more closely exhibit the ground truth structure of a synthetic dataset (i.e. a manually designed quad mesh template fitted to a variety of human body types in a set of different poses).

3D B-Rep meshing for real-time data-based geometric parametric analysis

This paper presents an effective framework to automatically construct 3D quadrilateral meshes of complicated geometry and arbitrary topology adapted for parametric studies. The input is a triangulation of the solid 3D model’s boundary provided from B-Rep CAD models or scanned geometry. The triangulated mesh is decomposed into a set of cuboids in two steps: pants decomposition and cuboid decomposition. This workflow includes an integration of a geometry-feature-aware pants-to-cuboids decomposition algorithm. This set of cuboids perfectly replicates the input surface topology. Using aligned global parameterization, patches are re-positioned on the surface in a way to achieve low overall distortion, and alignment to principal curvature directions and sharp features. Based on the cuboid decomposition and global parameterization, a 3D quadrilateral mesh is extracted. For different parametric instances with the same topology but different geometries, the MEG-IsoQuad method allows to have the same representation: isotopological meshes holding the same connectivity where each point on a mesh has an analogous one into all other meshes. Faithful 3D numerical charts of parametric geometries are then built using standard data-based techniques. Geometries are then evaluated in real-time. The efficiency and the robustness of the proposed approach are illustrated through a few parametric examples.

Mode Surfaces of Symmetric Tensor Fields: Topological Analysis and Seamless Extraction.

Mode surfaces are the generalization of degenerate curves and neutral surfaces, which constitute 3D symmetric tensor field topology. Efficient analysis and visualization of mode surfaces can provide additional insight into not only degenerate curves and neutral surfaces, but also how these features transition into each other. Moreover, the geometry and topology of mode surfaces can help domain scientists better understand the tensor fields in their applications. Existing mode surface extraction methods can miss features in the surfaces. Moreover, the mode surfaces extracted from neighboring cells have gaps, which make their subsequent analysis difficult. In this paper, we provide novel analysis on the topological structures of mode surfaces, including a common parameterization of all mode surfaces of a tensor field using 2D asymmetric tensors. This allows us to not only better understand the structures in mode surfaces and their interactions with degenerate curves and neutral surfaces, but also develop an efficient algorithm to seamlessly extract mode surfaces, including neutral surfaces. The seamless mode surfaces enable efficient analysis of their geometric structures, such as the principal curvature directions. We apply our analysis and visualization to a number of solid mechanics data sets.

Patterns for gridshells with negligible geometrical torsion at nodes

Abstract Curved envelope structural building envelopes have been quite popular in architecture in the past decades, and pose many challenges in their design, manufacturing and planning. In gridshells, a popular structural morphology for curved structure, designers will often strive to orient beams such that their top face is parallel to the envelope surface. However, this tends to induce geometrical torsion along the beam centerline, which complexifies significantly the manufacturing of the connection nodes or of the beams themselves. It is well known that such issue can be avoided by aligning beams with principal curvature directions of the envelope surface, thus yielding a quadrangular paneling. In this article, we study how other types of patterns (non-quadrangular) can be used to design torsion-free grid-shells. Based on asymptotic considerations, we derive a set of geometrical rules which, if fulfilled by a pattern, insure that a surface can be covered by this pattern with negligible torsion and limited deviation of beams from surface normals. A wide variety of patterns fulfill these rules, offering interesting possibilities for the design of curved architectural envelopes (Figure 1) is shown. As these rules are based on first order asymptotic analysis, we perform global validation on case studies. One main application is for structures in which face planarity is not necessary, for example ones cladded with ETFE cushions.

Unsupervised Palm Vein Image Segmentation

In this article the new hybrid algorithm for palm vein image segmentation using convolutional neural network and principal curvatures is proposed. After palm vein image preprocessing vein structure is detected using unsupervised learning approach based on W-Net architecture, that ties together into a single autoencoder two fully convolutional neural network architectures, each simi-lar to the U-Net. Then segmentation results are improved using principal cur-vatures technique. Some vein points with highest maximum principal curva-ture values are selected, and the other vein points are found by moving from starting points along the direction of minimum principal curvature. To obtain the final vein image segmentation the result of intersection of the principal curvatures-based and neural network-based segmentations is taken. The evaluation of the proposed unsupervised image segmentation method based on palm vein recognition results using multilobe differential filters is given. Test results using CASIA multi-spectral palmprint image database show the effectiveness of the proposed segmentation approach.

CurveNet: Curvature-Based Multitask Learning Deep Networks for 3D Object Recognition

In computer vision fields, 3D object recognition is one of the most important tasks for many real-world applications. Three-dimensional convolutional neural networks (CNNs) have demonstrated their advantages in 3D object recognition. In this paper, we propose to use the principal curvature directions of 3D objects (using a CAD model) to represent the geometric features as inputs for the 3D CNN. Our framework, namely CurveNet, learns perceptually relevant salient features and predicts object class labels. Curvature directions incorporate complex surface information of a 3D object, which helps our framework to produce more precise and discriminative features for object recognition. Multitask learning is inspired by sharing features between two related tasks, where we consider pose classification as an auxiliary task to enable our CurveNet to better generalize object label classification. Experimental results show that our proposed framework using curvature vectors performs better than voxels as an input for 3D object classification. We further improved the performance of CurveNet by combining two networks with both curvature direction and voxels of a 3D object as the inputs. A Cross-Stitch module was adopted to learn effective shared features across multiple representations. We evaluated our methods using three publicly available datasets and achieved competitive performance in the 3D object recognition task.