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The architects working with the shell use well-established geometry forms, which make up about 5-10 % of the number of known surfaces, in their projects. However, there is such a well-known surface of rotation, which from the 19th century to the present is very popular among mathematicians-geometers, but it is practically unknown to architects and designers, there are no examples of its use in the construction industry. This is a pseudosphere surface. For a pseudospherical surface with a pseudosphere rib radius, the Gaussian curvature at all points equals the constant negative number. The pseudosphere, or the surface of the Beltram, is generated by the rotation of the tracersis, evolvent of the chain line. The article provides an overview of known methods of calculation of pseudospherical shells and explores the strain-stress state of thin shells of revolution with close geometry parameters to identify optimal forms. As noted earlier, no examples of the use of the surface of the pseudosphere in the construction industry have been found in the scientific and technical literature. Only Kenneth Becher presented examples of pseudospheres implemented in nature: a gypsum model of the pseudosphere made by V. Martin Schilling at the end of the 19th century.

Biological membranes have distinct geometries that confer specific functions. However, the molecular mechanisms underlying the phenomenological geometry/function correlations remain elusive. We studied the effect of membrane geometry on the localization of membrane-bound proteins. Quantitative comparative experiments between the two most abundant cellular membrane geometries, spherical and cylindrical, revealed that geometry regulates the spatial segregation of proteins. The measured geometry-driven segregation reached 50-fold for membranes of the same mean curvature, demonstrating a crucial and hitherto unaccounted contribution by Gaussian curvature. Molecular-field theory calculations elucidated the underlying physical and molecular mechanisms. Our results reveal that distinct membrane geometries have specific physicochemical properties and thus establish a ubiquitous mechanistic foundation for unravelling the conserved correlations between biological function and membrane polymorphism., Cellular organelles display highly conserved morphologies, e.g., cylindrical (tubes) or spherical (vesicles), and here we show that their Gaussian curvature differences can regulate protein recruitment.

This article discusses geometric quantities associated with the concept of surfaces and the indicatrix of a surface in four-dimensional Galileo space. In this case, the second orderly line in the plane is presented as a surface indicatrix. It is shown that with the help of the Galileo space movement, the second orderly line can be brought to the canonical form. The movement in the Galileo space is radically different from the movement in the Euclidean space. Galileo movements include parallel movement, axis rotation, and sliding. Sliding gives deformation in the Euclidean space. The surface indicatrix is deformed by the Galileo movement. When the indicatrix is deformed, the surface will be deformed. In the classification of three-dimensional surface points in the four-dimensional Galileo phase, the classification of the indicatrix of the surface at this point was used. This shows the cyclic state of surface points other than Euclidean geometry. The geometric characteristics of surface curves were determined using the indicatrix test. It is determined what kind of geometrical meaning the identified properties have in the Euclidean phase. It is shown that the Galilean movement gives surface deformation in the Euclidean sense. Deformation of the surface is indicated by the fact that the Gaussian curvature remains unchanged.

An in-depth comparative study is conducted on four three-dimensional (3D) angularity indices, namely, the radius-based angularity index (AIr), gradient-based angularity index (AIg), deviation angle-based angularity index (AId), and Gaussian curvature-based angularity index (AIG). An innovative modification is carried out to ensure that all the indices equal zero for spheres and ellipsoids. Each AI is evaluated on particle surfaces reconstructed through spherical harmonic analyses with regular polyhedral meshes of different sizes. This comprehensive comparative study revealed that AIg and AIG better reflect the local morphological characteristics of the particles than AIr and AId. To obtain a convergent value of the overall AI of the particles, a relatively coarse mesh is sufficient for estimating AIr and AId, whereas a fine mesh is required to estimate AIG and AIg. Among the four indices, only AIG and AIg are well correlated, and their relationship can be described by a power function.

Made available in DSpace on 2022-04-29T08:31:15Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-11-01 Japan Society for the Promotion of Science Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) We study the geometry of bifurcation sets of generic unfoldings of D4±-functions. Taking blow-ups, we show each of the bifurcation sets of D4±-functions admits a parametrization as a surface in R3. Using this parametrization, we investigate the behavior of the Gaussian curvature and the principal curvatures. Furthermore, we investigate the number of ridge curves and subparabolic curves near their singular point. Department of Mathematics Graduate School of Science Kobe University, Rokkodai 1-1, Nada Departamento de Matemática Ibilce Universidade Estadual Paulista (Unesp), R. Cristóvão Colombo, 2265, Jd Nazareth Departamento de Matemática Ibilce Universidade Estadual Paulista (Unesp), R. Cristóvão Colombo, 2265, Jd Nazareth Japan Society for the Promotion of Science: 18K03301 FAPESP: 2019/10156-4

Continuous roll forming (CRF) is a new technology that combines continuous forming and multi-point forming to produce three-dimensional (3D) curved surfaces. Compared with other methods, the equipment of CRF is very simple, including only a pair of bendable work rolls and the corresponding shape adjustment and support assembly. By controlling the bending shapes of the upper and lower rolls and the size of the roll gap during forming, double curvature surfaces with different shapes can be produced. In this paper, a simplified expression of the exit velocity of the sheet is provided, and the formulas for the calculation of the longitudinal curvature radius are further derived. The reason for the discrepancy between the actual and predicted values of the longitudinal radius is deeply discussed from the perspective of the distribution of the exit velocity. By using the response surface methodology, the effects of the maximum compression ratio, the sheet width, the sheet thickness, and the transverse curvature radius on the longitudinal curvature radius are analyzed. Meanwhile, the correction coefficients of the predicted formulas for the positive and negative Gaussian curvature surfaces are obtained as 1.138 and 0.905, respectively. The validity and practicability of the modified formulas are verified by numerical simulations and forming experiments.

In this work we investigate the differential geometry of singular surfaces known as frontals. We prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the frontals in Euclidean 3- space. Also, we characterize in a simple way these singular surfaces and its fundamental forms with local properties in the differential of its parametrization and decompositions in the matrices associated to the fundamental forms. In particular we introduce new types of curvatures which can be used to characterize wave fronts. On the other hand, we investigate necessary and sufficient conditions for the extendibility and boundedness of Gaussian curvature, Mean curvature and principal curvatures near all types of singularities of fronts. Furthermore, we study the convergence to infinite limits of these geometrical invariants and we show how this is tightly related to a property of approximation of fronts by parallel surfaces. Neste trabalho investigamos a geometria diferencial de superfícies singulares conhecidas como frontais. Provamos um resultado semelhante ao teorema fundamental das superfícies regulares na geometria diferencial clássica, que estende o teorema clássico aos frontais no espaço Euclidiano. Além disso, caracterizamos de forma simples essas superfícies singulares e suas formas fundamentais com propriedades locais na diferencial de sua parametrização e decomposições nas matrizes associadas às formas fundamentais. Em particular, introduzimos novos tipos de curvaturas que podem ser usadas para caracterizar as frentes de onda. Por outro lado, investigamos as condições necessárias e suficientes para estender e delimitar a curvatura Gaussiana, curvatura média e curvaturas principais perto de todos os tipos de singularidades das frentes. Além disso, estudamos a convergência para limites infinitos desses invariantes geométricos e mostramos como isso está estreitamente relacionado a uma propriedade de aproximação de frentes por superfícies paralelas

Relevance. Architects and engineers, designing shells of revolution, use in their projects, as a rule, spherical shells, paraboloids, hyperboloids, and ellipsoids of revolution well proved themselves. But near hundreds of other surfaces of revolution, which can be applied with success in building and in machine-building, are known. Methods. Optimization problem of design of axisymmetric shell subjected to given external load is under consideration. As usual, the solution of this problem consists in the finding of shape of the meridian and in the distribution of the shell thickness along the meridian. In the paper, the narrower problem is considered. That is a selection of the shell shape from several known types, the middle surfaces of which can be given by parametrical equations. The results of static strength analyses of the domes of different Gaussian curvature with the same overall dimensions subjected to the uniformly distributed surface load are presented. Variational-difference energy method of analysis is used. Results. Comparison of results of strength analyses of six selected domes showed that a paraboloid of revolution and a dome with a middle surface in the form of the surface of rotation of the z = - a cosh( x/b ) curve around the Oz axis have the better indices of stress-strain state. These domes work almost in the momentless state and it is very well for thin-walled shell structures. New criterion of optimality can be called “minimum normal stresses in shells of revolution with the same overall dimensions, boundary conditions, and external load”.

Dilational structures can change in size without changing their shape. Current dilational designs are only suitable for specific shapes or curvatures and often require parts of the structure to move perpendicular to the dilational surface, thereby occupying part of the enclosed volume. Here, we present a general method for creating dilational structures from arbitrary surfaces (2-manifolds with or without boundary), where all motions are tangent to the described surface. The method consists of triangulating the target curved surface and replacing each of the triangular faces by pantograph mechanisms according to a tiling algorithm that avoids collisions between neighboring pantographs. Following this algorithm, any surface can be made to mechanically dilate and could, theoretically, scale from the fully expanded configuration down to a single point. We illustrate the method with three examples of increasing complexity and varying Gaussian curvature., Methods for dilation are currently limited to specific shapes and curvatures, with the potential for some of the structure's shape to encroach onto the final dilated volume. Here, the authors develop a method for creating dilational structures from arbitrary surfaces that avoids volume encroachment.

Reliance on π-diagrams and tangent plots to describe fold geometry has created an unwarranted status for conical folds at the expense of periclines. Analysis of virtual periclines using synthetic stereograms, tangent diagrams, and geologic curvature analysis demonstrates small portions of periclines are “cylindrical.” The more significant “non-cylindrical” portion of periclines allows for a mathematically permissible, but unrealistic, geometrical representation as portions of cones. We demonstrate cones are an extremely poor geometric representation of both the virtual periclines and examples of natural non-cylindrical folds. “Conical folds,” if they exist, terminate at a point; the amplitude to width ratio and the plunge of the crestal line must remain constant towards the fold terminus, with a Gaussian curvature of zero. In contrast, the amplitude to width ratio, plunge along the crestal line, and Gaussian curvature of periclines vary towards the fold terminus. These differences have important implications for the rheological modeling of folds, and while realistic dynamic models for periclines exist, models for conical fold formation remain conceptual. We suggest that in order to advance our understanding of how folds form, it may be “pointless” to continue the misconception of conical folds as an accurate geometric representation of how folds end.

Presented is a procedure to determine the extreme relative curvatures between conjugate gear flanks. This procedure is a modification of the third law of gearing which establishes the extreme relative curvatures between conjugate tooth flanks in terms of normal pressure angle, spiral angle, and hyperboloidal pitch surfaces. The original formulation for the third law of gearing was based on parabolic contact along a line of contact for the limiting condition of relative curvature between two gear flanks. Penetration or existence of negative Gaussian curvature for the relative flank curvature can occur for skew axis gearing with “conjugate” teeth. In such cases, reciprocity between the tooth normal and the twist defined by the gear rotation axes and instantaneous gear ratio is a necessary condition but not a sufficient condition to ensure suitable gear flanks in skew axis gearing. For cases with penetration, a transverse profile amendment is introduced to achieve parabolic contact (zero Gaussian curvature) for the relative curvature between meshing gear tooth flanks. First and second order reciprocity are applied in the transverse direction to determine profile amendment. Ease-off topography and unloaded transmission error (UTE) are used to assess conjugate action between mating gear flanks in direct contact.

The problem of elastic stability of the shells with special shapes with corrugated middle surfaces under external pressure is debated in the presented paper. Solution of the problem is based on FEM study. Corrugated barrelled, pseudo-barrelled, and cylindrical shells of constant mass are considered. Geometrical modification of the middle surface geometry is based on sine wave along principal directions. Middle surface of the corrugated shells are described referring to differential geometry of surfaces by parametric functions in three-dimensional Euclidean space. Linear and nonlinear buckling analyses are conducted. Examples of buckling modes are presented, which differ significantly from those typical for shells of revolution with positive or zero Gaussian curvature. It is proven that corrugation may lead to serious increase or decrease of critical load for all types of presented shells.

Arrested coalescence occurs in Pickering emulsions where colloidal particles adsorbed on the surface of the droplets become crowded and inhibit both relaxation of the droplet shape and further coalescence. The resulting droplets have a nonuniform distribution of curvature and, depending on the initial coverage, may incorporate a region with negative Gaussian curvature around the neck that bridges the two droplets. Here, we resolve the relative influence of the curvature and the kinetic process of arrest on the microstructure of the final state. In the quasistatic case, defects are induced and distributed to screen the Gaussian curvature. Conversely, if the rate of area change per particle exceeds the diffusion constant of the particles, the evolving surface induces local solidification reminiscent of jamming fronts observed in other colloidal systems. In this regime, the final structure is shown to be strongly affected by the compressive history just prior to arrest, which can be predicted from the extrinsic geometry of the sequence of surfaces in contrast to the intrinsic geometry that governs the static regime.

The formulation used by the most of studies on an elastic torus are either Reissner mixed formulation or Novozhilov's complex-form one, however, for vibration and some displacement boundary related problem of the torus, application of those formulations has encountered great difficulty. It is highly demanded to have a displacement-type formulation for the torus. In this paper, I will simulate some typical problems and free vibration of the torus. The numerical results are verified by both finite element analysis and H. Reissner's formulation. My investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. Regarding the free vibration of the torus, our analysis indicates that both initial in u and w direction must be included otherwise will cause big errors in eigenfrequency.

The formulation used by most of the studies on an elastic torus are either Reissner mixed formulation or Novozhilov's complex-form one, however, for vibration and some displacement boundary related problem of a torus, those formulations face a great challenge. It is highly demanded to have a displacement-type formulation for the torus. In this paper, I will carry on my previous work [ B.H. Sun, Closed-form solution of axisymmetric slender elastic toroidal shells. J. of Engineering Mechanics, 136 (2010) 1281-1288.], and with the help of my own maple code, I am able to simulate some typical problems and free vibration of the torus. The numerical results are verified by both finite element analysis and H. Reissner's formulation. My investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. Regarding the free vibration of a torus, our analysis indicates that both initial in u and w direction must be included otherwise will cause big errors in eigenfrequency. One of the most intestine discovery is that the crowns of a torus are the turning point of the Gaussian curvature at the crown where the mechanics' response of inner and outer torus is almost separated.

The Gol'denveizer problem of a torus can be described as follows: a toroidal shell is loaded under axial forces and the outer and inner equators are loaded with opposite balanced forces. Gol'denveizer pointed out that the membrane theory of shells is unable to predict deformation in this problem, as it yields diverging stress near the crowns. Although the problem has been studied by Audoly and Pomeau (2002) with the membrane theory of shells, the problem is still far from resolved within the framework of bending theory of shells. In this paper, the bending theory of shells is applied to formulate the Gol'denveizer problem of a torus. To overcome the computational difficulties of the governing complex-form ordinary differential equation (ODE), the complex-form ODE is converted into a real-form ODE system. Several numerical studies are carried out and verified by finite-element analysis. Investigations reveal that the deformation and stress of an elastic torus are sensitive to the radius ratio, and the Gol'denveizer problem of a torus can only be fully understood based on the bending theory of shells.

Liquid crystal elastomers (LCEs) can undergo large reversible contractions along their nematic director upon heating or illumination. A spatially patterned director within a flat LCE sheet thus encodes a pattern of contraction on heating, which can morph the sheet into a curved shell, akin to how a pattern of growth sculpts a developing organism. Here we consider, theoretically, numerically and experimentally, patterns constructed from regions of radial and circular director, which, in isolation, would form cones and anticones. The resultant surfaces contain curved ridges with sharp V-shaped cross-sections, associated with the boundaries between regions in the patterns. Such ridges may be created in positively and negatively curved variants and, since they bear Gauss curvature (quantified here via the Gauss-Bonnet theorem), they cannot be flattened without energetically prohibitive stretch. Our experiments and numerics highlight that, although such ridges cannot be flattened isometrically, they can deform isometrically by trading the (singular) curvature of the V angle against the (finite) curvature of the ridge line. Furthermore, in finite thickness sheets, the sharp ridges are inevitably non-isometrically blunted to relieve bend, resulting in a modest smearing out of the encoded singular Gauss curvature. We close by discussing the use of such features as actuating linear features, such as probes, tongues and limbs, and highlighting the similarities between these patterns of shape change and those found during the morphogenesis of several biological systems., Comment: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Journal of Applied Physics 129, 224701 (2021) and may be found at https://doi.org/10.1063/5.0044158

Triaxial weaving is a handicraft technique that has long been used to create curved structures using initially straight and flat ribbons. Weavers typically introduce discrete topological defects to produce nonzero Gaussian curvature, albeit with faceted surfaces. We demonstrate that, by tuning the in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be varied continuously, which is not feasible using traditional techniques. Further, we reveal that the shape of the physical unit cells is dictated solely by the in-plane geometry of the ribbons, not elasticity. Finally, we leverage the geometry-driven nature of triaxial weaving to design a set of ribbon profiles to weave smooth spherical, ellipsoidal, and toroidal structures., 6 pages, 4 figures; Supplementary information provided

This paper investigated the flattening effect of negative Gaussian curvature on simply supported (SS) thick asymmetrically laminated cross-ply panels in the absence of surface-parallel edge...

We present a geometric framework to model the optical effects of deformations of planar holographic optical elements (HOE) into curved surfaces, such as sphere segments. In particular, we consider deformations which do not preserve the Gaussian curvature., 23 pages, 12 Figures

Flat sheets encoded with patterns of contraction/elongation morph into curved surfaces. If the surfaces bear Gauss curvature, the resulting actuation can be strong and powerful. We deploy the Gauss-Bonnet theorem to deduce the Gauss curvature encoded in a pattern of uniform-magnitude contraction/elongation with spatially varying direction, as is commonly implemented in patterned liquid crystal elastomers. This approach reveals two fundamentally distinct contributions: a structural curvature which depends on the precise form of the pattern, and a topological curvature generated by defects in the contractile direction. These curvatures grow as different functions the contraction/elongation magnitude, explaining the apparent contradiction between previous calculations for simple +1 defects, and smooth defect-free patterns. We verify these structural and topological contributions by conducting numerical shell calculations on sheets encoded with simple higher-order contractile defects to reveal their activated morphology. Finally we calculate the Gauss curvature generated by patterns with spatially varying magnitude and direction, which leads to additional magnitude gradient contributions to the structural term. We anticipate this form will be useful whenever magnitude and direction are natural variables, including in describing the contraction of a muscle along its patterned fiber direction, or a tissue growing by elongating its cells., Final peer reviewed version

Given a surface $S$ in a 3D contact sub-Riemannian manifold $M$, we investigate the metric structure induced on $S$ by $M$, in the sense of length spaces. First, we define a coefficient $\widehat K$ at characteristic points that determines locally the characteristic foliation of $S$. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points., 24 pages, 15 figures

Liquid crystal elastomers and glasses can have significant shape change determined by their director patterns. Cones deformed from circular director patterns have non-trivial Gaussian curvature localised at tips, curved interfaces, and intersections of interfaces. We employ a generalised metric compatibility condition to characterize two families of interfaces between circular director patterns -- hyperbolic and elliptical interfaces, and find that the deformed interfaces are geometrically compatible. We focus on hyperbolic interfaces to design complex topographies and non-isometric origami, including n-fold intersections, symmetric and irregular tilings. The large design space of three-fold and four-fold tiling is utilized to quantitatively inverse design an array of pixels to display target images. Taken together, our findings provide comprehensive design principles for the design of actuators, displays, and soft robotics in liquid crystal elastomers and glasses., 13 pages, 11 figures