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1
Bartkowiak, Tomasz, and Christopher A. Brown. "Multiscale 3D Curvature Analysis of Processed Surface Textures of Aluminum Alloy 6061 T6." Materials 12, no. 2 (January 14, 2019): 257. http://dx.doi.org/10.3390/ma12020257.
Повний текст джерелаАнотація:
The objectives of this paper are to demonstrate the viability, and to validate, in part, a multiscale method for calculating curvature tensors on measured surface topographies with two different methods of specifying the scale. The curvature tensors are calculated as functions of scale, i.e., size, and position from a regular, orthogonal array of measured heights. Multiscale characterization of curvature is important because, like slope and area, it changes with the scale of observation, or calculation, on irregular surfaces. Curvatures can be indicative of the topographically dependent behavior of a surface and, in turn, curvatures are influenced by the processing and use of the surface. Curvatures of surface topographies have not been well- characterized yet. Curvature has been used for calculations in contact mechanics and for the evaluation of cutting edges. Manufactured surfaces are studied for further validation of the calculation method because they provide certain expectations for curvatures, which depend on scale and the degree of curvature. To study a range of curvatures on manufactured surfaces, square edges are machined and honed, then rounded progressively by mass finishing; additionally, a set of surfaces was made by turning with different feeds. Topographic measurements are made with a scanning laser confocal microscope. The calculations use vectors, normal to the measured surface, which are calculated first, then the eigenvalue problem is solved for the curvature tensor. Plots of principal curvatures as a function of position and scale are presented. Statistical analyses show expected interactions between curvature and these manufacturing processes.
2
Bartkowiak, Tomasz, and Christopher Brown. "Multi-scale curvature tensor analysis of machined surfaces." Archives of Mechanical Technology and Materials 36, no. 1 (December 1, 2016): 44–50. http://dx.doi.org/10.1515/amtm-2016-0009.
Повний текст джерелаАнотація:
Abstract This paper demonstrates the use of multi-scale curvature analysis, an areal new surface characterization technique for better understanding topographies, for analyzing surfaces created by conventional machining and grinding. Curvature, like slope and area, changes with scale of observation, or calculation, on irregular surfaces, therefore it can be used for multi-scale geometric analysis. Curvatures on a surface should be indicative of topographically dependent behavior of a surface and curvatures are, in turn, influenced by the processing and use of the surface. Curvatures have not been well characterized previously. Curvature has been used for calculations in contact mechanics and for the evaluation of cutting edges. In the current work two parts were machined and then one of them was ground. The surface topographies were measured with a scanning laser confocal microscope. Plots of curvatures as a function of position and scale are presented, and the means and standard deviations of principal curvatures are plotted as a function of scale. Statistical analyses show the relations between curvature and these two manufacturing processes at multiple scales.
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Jenmalm, Per, Antony W. Goodwin, and Roland S. Johansson. "Control of Grasp Stability When Humans Lift Objects With Different Surface Curvatures." Journal of Neurophysiology 79, no. 4 (April 1, 1998): 1643–52. http://dx.doi.org/10.1152/jn.1998.79.4.1643.
Повний текст джерелаАнотація:
Jenmalm, Per, Antony W. Goodwin, and Roland S. Johansson. Control of grasp stability when humans lift objects with different surface curvatures. J. Neurophysiol. 79: 1643–1652, 1998. In previous investigations of the control of grasp stability, humans manipulated test objects with flat grasp surfaces. The surfaces of most objects that we handle in everyday activities, however, are curved. In the present study, we examined the influence of surface curvature on the fingertip forces used when humans lifted and held objects of various weights. Subjects grasped the test object between the thumb and the index finger. The matching pair of grasped surfaces were spherically curved with one of six different curvatures (concave with radius 20 or 40 mm; flat; convex with radius 20, 10, or 5 mm) and the object had one of five different weights ranging from 168 to 705 g. The grip force used by subjects (force along the axis between the 2 grasped surfaces) increased with increasing weight of the object but was modified inconsistently and incompletely by surface curvature. Similarly, the duration and rate of force generation, when the grip and load forces increased isometrically in the load phase before object lift-off, were not influenced by surface curvature. In contrast, surface curvature did affect the minimum grip forces required to prevent frictional slips (the slip force). The slip force was smaller for larger curvatures (both concave and convex) than for flatter surfaces. Therefore the force safety margin against slips (difference between the employed grip force and the slip force) was higher for the higher curvatures. We conclude that surface curvature has little influence on grip force regulation during this type of manipulation; the moderate changes in slip force resulting from changes in curvature are not fully compensated for by changes in grip force.
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Tanaka, Minoru, and Kei Kondo. "The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces." Nagoya Mathematical Journal 209 (March 2013): 23–34. http://dx.doi.org/10.1017/s0027763000010679.
Повний текст джерелаАнотація:
AbstractWe construct distinctive surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we prove that a complete noncompact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary if the manifold M is not less curved than a noncompact model surface of revolution and if the total curvature of the model surface is finite and less than 2π. By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete noncompact Riemannian manifolds whose sectional curvatures are bounded from below by a constant.
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Lipnickas, Arūnas, and Vidas Raudonis. "Contour Representation by Clustering Curvatures of the 3D Objects." Solid State Phenomena 147-149 (January 2009): 633–38. http://dx.doi.org/10.4028/www.scientific.net/ssp.147-149.633.
Повний текст джерелаАнотація:
The purpose of this work is to segment large size triangulated surfaces and the contours extraction of the 3D object by the use of the object curvature value. The curvatures values allow categorizing the type of the local surface of the 3D object. In present work the curvature was estimated for the free-form surfaces obtained by the 3D range scanner. A free-form surface is the surface such that the surface normal is defined and continuous everywhere, except at sharp corners and edges [2, 5]. Two types of distance measurements functions based on Euclidian distance, bounded box and topology of surface were used for the curvature estimation. Clustering technique has been involved to cluster the values of the curvature for 3D object contour representation. The described technique was applied to the 3D objects with free-form surfaces such as the human foot and cube.
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WANG, DAN, YAJUN YIN, JIYE WU, and ZHENG ZHONG. "THE INTERACTION POTENTIAL BETWEEN MICRO/NANO CURVED SURFACE BODY WITH NEGATIVE GAUSS CURVATURE AND AN OUTSIDE PARTICLE." Journal of Mechanics in Medicine and Biology 15, no. 06 (December 2015): 1540055. http://dx.doi.org/10.1142/s0219519415400552.
Повний текст джерелаАнотація:
Based on the negative exponential pair potential ([Formula: see text]), the interaction potential between curved surface body with negative Gauss curvature and an outside particle is proved to be of curvature-based form, i.e., it can be written as a function of curvatures. Idealized numerical experiments are designed to test the accuracy of the curvature-based potential. Compared with the previous results, it is confirmed that the interaction potential between curved surface body and an outside particle has a unified expression of curvatures regardless of the sign of Gauss curvature. Further, propositions below are confirmed: Highly curved surface body may induce driving forces, curvatures and the gradient of curvatures are the essential factors forming the driving forces.
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Wang, Dan, Zhili Hu, Gang Peng, and Yajun Yin. "Surface Energy of Curved Surface Based on Lennard-Jones Potential." Nanomaterials 11, no. 3 (March 9, 2021): 686. http://dx.doi.org/10.3390/nano11030686.
Повний текст джерелаАнотація:
Although various phenomena have confirmed that surface geometry has an impact on surface energy at micro/nano scales, determining the surface energy on micro/nano curved surfaces remains a challenge. In this paper, based on Lennard-Jones (L-J) pair potential, we study the geometrical effect on surface energy with the homogenization hypothesis. The surface energy is expressed as a function of local principle curvatures. The accuracy of curvature-based surface energy is confirmed by comparing surface energy on flat surface with experimental results. Furthermore, the surface energy for spherical geometry is investigated and verified by the numerical experiment with errors within 5%. The results show that (i) the surface energy will decrease on a convex surface and increase on a concave surface with the increasing of scales, and tend to the value on flat surface; (ii) the effect of curvatures will be obvious and exceed 5% when spherical radius becomes smaller than 5 nm; (iii) the surface energy varies with curvatures on sinusoidal surfaces, and the normalized surface energy relates with the ratio of wave height to wavelength. The curvature-based surface energy offers new insights into the geometrical and scales effect at micro/nano scales, which provides a theoretical direction for designing NEMS/MEMS.
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Kong, Ling Ye, Qiu Sheng Yan, Jun Hui Song, and Ya Nan Song. "Research on Uniform Surface Roughness in Grinding of Revolving Curved Surface." Key Engineering Materials 416 (September 2009): 113–17. http://dx.doi.org/10.4028/www.scientific.net/kem.416.113.
Повний текст джерелаАнотація:
When grinding the revolving curved surface with Arc Envelope Grinding Method, the different curvatures in the convex and concave surfaces make a great difference in the surface roughness. In order to solve this problem, the relationship among envelope height, feeding rate, rotational speed and curvature of workpiece was analyzed based on equal-envelope-height grinding method. The results presented that, low feeding rate of grinding wheel and high rotational speed of workpiece were helpful to obtain smaller envelope height. And the smaller the radius of workpiece curvature, the more different the surface roughness. Besides, it was an effective method to solve this problem by changing feeding rate. The feeding rate should be changed directly proportionally to radius of workpiece curvature. Then, the experimental results indicate that, the fluctuation ratio of surface roughness with variable feeding rate is reduced to 4.896% from 26.17% with constant feeding rate. It proves the validity of hypothesis.
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Abdel-Baky, Rashad A., Nadia Alluhaibi, Akram Ali, and Fatemah Mofarreh. "A study on timelike circular surfaces in Minkowski 3-space." International Journal of Geometric Methods in Modern Physics 17, no. 06 (May 2020): 2050074. http://dx.doi.org/10.1142/s0219887820500747.
Повний текст джерелаАнотація:
This paper studies a smooth one-parameter family of standard Lorentzian circles with fixed radius. Such a surface is called a timelike circular surface with constant radius. We call each circle a generating circle. A new type of timelike circular surfaces was identified and coined as the timelike tangent circular surface. The new timelike tangent circular surface has the property of all generating circles being lines of curvature and its Gaussian and mean curvatures being independent of the geodesic curvature of the spherical indicatrix.
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Milin Šipuš, Željka, and Blaženka Divjak. "Surfaces of Constant Curvature in the Pseudo-Galilean Space." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–28. http://dx.doi.org/10.1155/2012/375264.
Повний текст джерелаАнотація:
We develop the local theory of surfaces immersed in the pseudo-Galilean space, a special type of Cayley-Klein spaces. We define principal, Gaussian, and mean curvatures. By this, the general setting for study of surfaces of constant curvature in the pseudo-Galilean space is provided. We describe surfaces of revolution of constant curvature. We introduce special local coordinates for surfaces of constant curvature, so-called the Tchebyshev coordinates, and show that the angle between parametric curves satisfies the Klein-Gordon partial differential equation. We determine the Tchebyshev coordinates for surfaces of revolution and construct a surface with constant curvature from a particular solution of the Klein-Gordon equation.
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Bandyopadhyay, Promode R., and Anwar Ahmed. "Turbulent boundary layers subjected to multiple curvatures and pressure gradients." Journal of Fluid Mechanics 246 (January 1993): 503–27. http://dx.doi.org/10.1017/s0022112093000242.
Повний текст джерелаАнотація:
The effects of abruptly applied cycles of curvatures and pressure gradients on turbulent boundary layers are examined experimentally. Two two-dimensional curved test surfaces are considered: one has a sequence of concave and convex longitudinal surface curvatures and the other has a sequence of convex and concave curvatures. The choice of the curvature sequences were motivated by a desire to study the asymmetric response of turbulent boundary layers to convex and concave curvatures. The relaxation of a boundary layer from the effects of these two opposite sequences has been compared. The effect of the accompanying sequences of pressure gradient has also been examined but the effect of curvature dominates. The growth of internal layers at the curvature junctions have been studied. Measurements of the Górtler and corner vortex systems have been made. The boundary layer recovering from the sequence of concave to convex curvature has a sustained lower skin friction level than in that recovering from the sequence of convex to concave curvature. The amplification and suppression of turbulence due to the curvature sequences have also been studied.
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Van-Brunt, B., and K. Grant. "Hyperbolic Weingarten surfaces." Mathematical Proceedings of the Cambridge Philosophical Society 116, no. 3 (November 1994): 489–504. http://dx.doi.org/10.1017/s0305004100072765.
Повний текст джерелаАнотація:
AbstractWeingarten surfaces which can be represented locally as solutions to second order hyperbolic partial differential equations are examined in this paper. In particular, the geometry of the families of curves corresponding to characteristics on these surfaces is investigated and the relationships of these curves with other curves on the surface such as asymptotic lines and lines of curvature are explored. It is shown that singularities in the lines of curvature, i.e. umbilic points, correspond to singularities in the families of characteristics, and that lines of curvature are non-characteristic curves. If there is a linear relation between the Gaussian and mean curvatures and real characteristics exist, then the characteristics form a Tchebychef net on the corresponding Weingarten surface.
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Teh, Yee Meng, R. U. Gobithaasan, Kenjiro T. Miura, Diya’ J. Albayari, and Wen Eng Ong. "The Development of Log Aesthetic Patch and Its Projection onto the Plane." Mathematics 10, no. 1 (January 5, 2022): 160. http://dx.doi.org/10.3390/math10010160.
Повний текст джерелаАнотація:
In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify its versatility, we approximated the hyperbolic paraboloid to LAP using the information of lines of curvature (LoC). The outer part of the LoCs, which play a role as the boundary of the hyperbolic paraboloid, is replaced with LACs before constructing the LAP. Since LoCs are essential in shipbuilding for hot and cold bending processes, we investigated the LAP in terms of the LoC’s curvature, derivative of curvature, torsion, and Logarithmic Curvature Graph (LCG). The numerical results indicate that the LoCs for both surfaces possess monotonic curvatures. An advantage of LAP approximation over its original hyperbolic paraboloid is that the LoCs of LAP can be approximated to LACs, and hence the first derivative of curvatures for LoCs are monotonic, whereas they are non-monotonic for the hyperbolic paraboloid. This confirms that the LAP produced is indeed of high quality. Lastly, we project the LAP onto a plane using geodesic curvature to create strips that can be pasted together, mimicking hot and cold bending processes in the shipbuilding industry.
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Çi̇mdi̇ker, Muradi̇ye, and Yasi̇n Ünlütürk. "The construction of the space-like surface of constant breadth." International Journal of Geometric Methods in Modern Physics 16, no. 04 (April 2019): 1950060. http://dx.doi.org/10.1142/s0219887819500609.
Повний текст джерелаАнотація:
The objective of this study is to define an ovaloid surface on the convex closed space-like surfaces of constant breadth when principal curvatures of these surfaces are continuous, non-vanishing functions, and to obtain some special geometrical properties of this ovaloid surface by using the radius of curvature, diameter of the surface in [Formula: see text].
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Chen, N. "Curvatures and Sliding Ratios of Conjugate Surfaces." Journal of Mechanical Design 120, no. 1 (March 1, 1998): 126–32. http://dx.doi.org/10.1115/1.2826664.
Повний текст джерелаАнотація:
A new approach for curvatures of conjugate surfaces is provided in this paper. The main characteristic of the approach is that relative curvatures and geodesic torsions of the conjugate surfaces are directly calculated in terms of the normal curvatures and geodesic torsions of the generating surface on two nonorthogonal tangents of surface curvilinears in the global surface system. Based on the curvature equations, sliding velocities and sliding ratios of the conjugate surfaces are studied. The approach is illustrated by a numerical example of a plane enveloping globoidal worm-gear drive.
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Gao, Bo, Rui Yu, Guangcai Hu, Cheng Liu, Xin Zhuang, and Peng Zhou. "Development Processes of Surface Trucking and Partial Discharge of Pressboards Immersed in Mineral Oil: Effect of Tip Curvatures." Energies 12, no. 3 (February 11, 2019): 554. http://dx.doi.org/10.3390/en12030554.
Повний текст джерелаАнотація:
The pressboard surface is the electric weak link of the oil-paper insulation in transformers, and long-term partial discharge (PD) erosion is the dominant cause of degradation in pressboard. To explore the development processes of surface tracking under the effect of tip curvature, the typical needle-plate model was selected to initiate an electric field with a high tangential component on pressboard surface under needle tip curvature of 4~42 μm. With the help of a high-speed camera and a PD detecting system, the development processes of surface tracking and PD were recorded under a sustained AC voltage. A profound difference between surface tracking under different curvatures was discussed. Pressboard surfaces after tests were observed under a scanning electron microscope (SEM), and the damage degree of cellulose fibers was dependent on the tip curvature.
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Carretero, Paula, and Ildefonso Castro. "A New Approach to Rotational Weingarten Surfaces." Mathematics 10, no. 4 (February 12, 2022): 578. http://dx.doi.org/10.3390/math10040578.
Повний текст джерелаАнотація:
Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first-order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones.
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Mazeron, Paul, and Stéphane Muller. "Curvature, surface fields and scattering." Journal of Optics 28, no. 1 (February 1997): 13–19. http://dx.doi.org/10.1088/0150-536x/28/1/004.
Повний текст джерела
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Çı̇mdı̇ker Aslan, Muradı̇ye, and Gülşah Aydın Şekerci̇. "Dual curves associated with the Bonnet ruled surfaces." International Journal of Geometric Methods in Modern Physics 17, no. 13 (October 12, 2020): 2050204. http://dx.doi.org/10.1142/s0219887820502047.
Повний текст джерелаАнотація:
An interest problem arises to determine the surfaces in the Euclidean three space, which admit at least one nontrivial isometry that preserves the principal curvatures. This leads to a class of surface known as a Bonnet surface. The intention of this study is to examine a Bonnet ruled surface in dual space and to calculate the dual geodesic trihedron of the dual curve associated with the Bonnet ruled surface and derivative equations of this trihedron by the dual geodesic curvature. Also, we find that the dual curvature, the dual torsion for the dual curves associated with the Bonnet ruled surface which are different from any dual curves. Moreover, some examples are obtained about the Bonnet ruled surface.
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Cevallos, Carlos, Peter Kovac, and Sharon J. Lowe. "Application of curvatures to airborne gravity gradient data in oil exploration." GEOPHYSICS 78, no. 4 (July 1, 2013): G81—G88. http://dx.doi.org/10.1190/geo2012-0315.1.
Повний текст джерелаАнотація:
We apply equipotential surface curvatures to airborne gravity gradient data. The mean and differential curvature of the equipotential surface, the curvature of the gravity field line, the zero contour of the Gaussian curvature, and the shape index improve the understanding and geologic interpretation of gravity gradient data. Their use is illustrated in model data and applied to FALCON airborne gravity gradiometer data from the Canning Basin, Australia.
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Choi, WooSeok, and Sungchan Yun. "Symmetry-Breaking Drop Bouncing on Superhydrophobic Surfaces with Continuously Changing Curvatures." Polymers 13, no. 17 (August 31, 2021): 2940. http://dx.doi.org/10.3390/polym13172940.
Повний текст джерелаАнотація:
Controlling the residence time of drops on the solid surface is related to a wide spectrum of engineering applications, such as self-cleaning and anti-icing. The symmetry-breaking dynamics induced by the initial drop shape can promote drop bouncing. Here, we study the bouncing features of spherical and ellipsoidal drops on elliptical surfaces that continuously change curvatures inspired by natural succulent leaves. The bounce characteristics highly depend on the geometric relations between the ellipsoidal drops and curved surfaces. Numerical results show that ellipsoidal shapes of the drops amplify asymmetries of the mass and momentum in synergy with an influence of the surface curvature during the impact, which is verified by experiments. Effects of the surface anisotropy and drops’ ellipticity on the residence time are investigated under various surface morphologies and Weber numbers. The residence time is closely associated with an initial surface curvature at the apex. The underlying principle of modifying the residence time via the drops’ ellipticity and initial surface curvature is elucidated based on momentum asymmetry. The understanding of the bouncing features on curved surfaces will offer practical implications for enhanced heat transfer performances and controlled water repellency, etc.
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HAYWARD, SEAN A. "INVOLUTE, MINIMAL, OUTER AND INCREASINGLY TRAPPED SURFACES." International Journal of Modern Physics D 20, no. 03 (March 2011): 401–11. http://dx.doi.org/10.1142/s0218271811018718.
Повний текст джерелаАнотація:
Eight different refinements of trapped surfaces are proposed, of three basic types, each intended as potential stability conditions. Minimal trapped surfaces are strictly minimal with respect to the dual expansion vector. Outer trapped surfaces have positivity of a certain curvature, related to surface gravity. Increasingly (future, respectively past) trapped surfaces generate surfaces which are more trapped in a (future, respectively past) causal variation, with three types: in any such causal variation; along the expansion vector; and in some such causal variation. This suggests a definition of doubly outer trapped surface involving two independent curvatures. This in turn suggests a definition of involute trapped surface. Adding a weaker condition, the eight conditions form an interwoven hierarchy, with four independent relations which assume the null energy condition, and another holding in a special case of symmetric curvature.
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Lemesle, Julie, Frederic Robache, Gaetan Le Goic, Alamin Mansouri, Christopher A. Brown, and Maxence Bigerelle. "Surface Reflectance: An Optical Method for Multiscale Curvature Characterization of Wear on Ceramic–Metal Composites." Materials 13, no. 5 (February 25, 2020): 1024. http://dx.doi.org/10.3390/ma13051024.
Повний текст джерелаАнотація:
Surface gradient characterization by light reflectance (SGCLR) is used for the first time for multiscale curvature calculations and discrimination of worn surfaces on six damaged ceramic–metal composites. Measurements are made using reflectance transformation imaging (RTI). Slope and curvature maps, generated from RTI, are analyzed instead of heights. From multiscale decompositions, bootstrapping, and analysis of variance (ANOVA), a strong correlation (R² = 0.90) is found between the density of furrows of Mehlum curvatures, with a band pass filter at 5.4 µm, present in ceramic grains and their mechanical properties. A strong correlation is found between the mean curvatures of the metal and the ceramics, with a high pass filter at 1286 µm.
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Bartkowiak, Tomasz, Johan Berglund, and Christopher A. Brown. "Multiscale Characterizations of Surface Anisotropies." Materials 13, no. 13 (July 7, 2020): 3028. http://dx.doi.org/10.3390/ma13133028.
Повний текст джерелаАнотація:
Anisotropy can influence surface function and can be an indication of processing. These influences and indications include friction, wetting, and microwear. This article studies two methods for multiscale quantification and visualization of anisotropy. One uses multiscale curvature tensor analysis and shows anisotropy in horizontal coordinates i.e., topocentric. The other uses multiple bandpass filters (also known as sliding bandpass filters) applied prior to calculating anisotropy parameters, texture aspect ratios (Str) and texture directions (Std), showing anisotropy in horizontal directions only. Topographies were studied on two milled steel surfaces, one convex with an evident large scale, cylindrical form anisotropy, the other nominally flat with smaller scale anisotropies; a µEDMed surface, an example of an isotropic surface; and an additively manufactured surface with pillar-like features. Curvature tensors contain the two principal curvatures, i.e., maximum and minimum curvatures, which are orthogonal, and their directions, at each location. Principal directions are plotted for each calculated location on each surface, at each scale considered. Histograms in horizontal coordinates show altitude and azimuth angles of principal curvatures, elucidating dominant texture directions at each scale. Str and Std do not show vertical components, i.e., altitudes, of anisotropy. Changes of anisotropy with scale categorically failed to be detected by traditional characterization methods used conventionally. These multiscale methods show clearly in several representations that anisotropy changes with scale on actual surface measurements with markedly different anisotropies.
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Wu, D., and S. S. Law. "Sensitivity of Uniform Load Surface Curvature for Damage Identification in Plate Structures." Journal of Vibration and Acoustics 127, no. 1 (February 1, 2005): 84–92. http://dx.doi.org/10.1115/1.1857918.
Повний текст джерелаАнотація:
In this paper a new sensitivity-based method of using measured modal parameters to locate and quantify damage is developed for plate-like structures. With the measured incomplete modal data for only the few lower modes in both the intact and damaged states, the two-dimensional distributed curvatures of uniform load surface (ULS) over the plate are approximated using the Chebyshev polynomials. Instead of directly comparing the curvatures before and after damage, like many existing damage localization methods using curvature techniques, e.g., mode shape curvature and flexibility curvature, the proposed method analytically studies the sensitivity of the ULS curvature with respect to the element-by-element stiffness parameters. The changes in the elemental stiffness parameters due to damage give the location and magnitude of the damaged plate elements. Based on the first-order Taylor series approximation, the inverse problem is modeled as a linear equation system and solved iteratively using truncated SVD technique. Numerical simulations are performed to verify the effectiveness of the proposed method with different support conditions, measurement noise, and sensor sparsity.
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Liang, Dawei, Tomohiro Onodera, Masanari Hamasaki, Ryosuke Hishimura, Kentaro Homan, Liang Xu, Yuan Tian, Satoshi Kanai, and Norimasa Iwasaki. "Quantification of Cartilage Surface Degeneration by Curvature Analysis Using 3D Scanning in a Rabbit Model." CARTILAGE 13, no. 2_suppl (November 20, 2021): 1734S—1741S. http://dx.doi.org/10.1177/19476035211059597.
Повний текст джерелаАнотація:
Objective Accurate analysis to quantify cartilage morphology is critical for evaluating degenerative conditions in osteoarthritis (OA). Three-dimensional (3D) optical scanning provides 3D data for the entire cartilage surface; however, there is no consensus on how to quantify it. Our purpose was to validate a 3D method for evaluating spatiotemporal alterations in degenerative cartilages in a rabbit OA model by analyzing their curvatures at various stages of progression. Design Twelve rabbits underwent anterior cruciate ligament transection (ACLT) unilaterally and were divided into 4 groups: 4 weeks control, 4 weeks OA, 8 weeks control, and 8 weeks OA. 3D scanning, India ink staining, and histological assessments were performed in all groups. In 3D curvature visualization, the surfaces of the condyles were divided into 8 areas. The standard deviations (SD) of mean curvatures from all vertices of condylar surfaces and subareas were calculated. Results Regarding the site of OA change, curvature analysis was consistent with India ink scoring. The SD of mean curvature correlated strongly with the India ink Osteoarthritis Research Society International (OARSI) score. In curvature histograms, the curvature distribution in OA was more scattered than in control. Of the 8 areas, significant OA progression in the posterolateral part of the lateral condyle (L-PL) was observed at 4 weeks. The histology result was consistent with the 3D evaluation in terms of representative section. Conclusions This study demonstrated that 3D scanning with curvature analysis can quantify the severity of cartilage degeneration objectively. Furthermore, the L-PL was found to be the initial area where OA degeneration occurred in the rabbit ACLT model.
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Gobithaasan, R. U., Yee Meng Teh, Kenjiro T. Miura, and Wen Eng Ong. "Lines of Curvature for Log Aesthetic Surfaces Characteristics Investigation." Mathematics 9, no. 21 (October 24, 2021): 2699. http://dx.doi.org/10.3390/math9212699.
Повний текст джерелаАнотація:
Lines of curvatures (LoCs) are curves on a surface that are derived from the first and second fundamental forms, and have been used for shaping various types of surface. In this paper, we investigated the LoCs of two types of log aesthetic (LA) surfaces; i.e., LA surfaces of revolution and LA swept surfaces. These surfaces are generated with log aesthetic curves (LAC) which comprise various families of curves governed by α. First, since it is impossible to derive the LoCs analytically, we have implemented the LoC computation numerically using the Central Processing Unit (CPU) and General Processing Unit (GPU). The results showed a significant speed up with the latter. Next, we investigated the curvature distributions of the derived LoCs using a Logarithmic Curvature Graph (LCG). In conclusion, the LoCs of LA surface of revolutions are indeed the duplicates of their original profile curves. However, the LoCs of LA swept surfaces are LACs of different shapes. The exception to this is when this type of surface possesses LoCs in the form of circle involutes.
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Cheshkova, M. A. "Examples of surfaces of constant mean curvature." Differential Geometry of Manifolds of Figures, no. 50 (2019): 148–54. http://dx.doi.org/10.5922/0321-4796-2019-50-17.
Повний текст джерелаАнотація:
A surface in E3 is called parallel to the surface M if it consists of the ends of constant length segments, laid on the normals to the surfaces M at points of this surface. The tangent planes at the corresponding points will be parallel. For surfaces in E3 the theorem of Bonnet holds: for any surface M that has constant positive Gaussian curvature, there exists a surface parallel to it with a constant mean curvature. Using Bonnet's theorem for a surfaces of revolution of constant positive Gaussian curvature, surfaces of constant mean curvature are constructed. It is proved that they are also surfaces of revolution. A family of plane curvature lines (meridians) is described by means of elliptic integrals. The surfaces of constant Gaussian curvature are also described by means of elliptic integrals. Using the mathematical software package, the surfaces under consideration are constructed.
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Liu, Haiming, and Xiawei Chen. "Lorentzian Approximations and Gauss–Bonnet Theorem for E 1,1 with the Second Lorentzian Metric." Journal of Mathematics 2022 (October 28, 2022): 1–12. http://dx.doi.org/10.1155/2022/5402011.
Повний текст джерелаАнотація:
In this paper, we consider the Lorentzian approximations of rigid motions of the Minkowski plane E L 2 1,1 . By using the method of Lorentzian approximations, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surface, and the intrinsic Gaussian curvature of Lorentzian surface in E 1,1 with the second Lorentzian metric away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss–Bonnet theorem for the Lorentzian surface in E 1,1 with the second left-invariant Lorentzian metric g 2 .
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Cheshkova, М. A., and A. A. Pavlova. "Example of Bianchi Transformation of Kuen’s Surface." Izvestiya of Altai State University, no. 1(117) (March 17, 2021): 126–28. http://dx.doi.org/10.14258/izvasu(2021)1-22.
Повний текст джерелаАнотація:
The work is devoted to the study of the Bianchi transformation for surfaces of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, and the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuen’s surface and the Dini’s surface. Studying the surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. Geometric characteristics of pseudospherical surfaces are found to be related to the theory of networks, the theory of solitons, nonlinear differential equations, and sin-Gordon equations. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transformation for the Kuen’s surface is constructed using a mathematical software package.
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Pegna, Joseph, and Franz-Erich Wolter. "Geometrical Criteria to Guarantee Curvature Continuity of Blend Surfaces." Journal of Mechanical Design 114, no. 1 (March 1, 1992): 201–10. http://dx.doi.org/10.1115/1.2916918.
Повний текст джерелаАнотація:
Computer Aided Geometric Design (CAGD) of surfaces sometimes presents problems that were not envisioned in classical differential geometry. This paper presents mathematical results that pertain to the design of curvature continuous blending surfaces. Curvature continuity across normal continuous surface patches requires that normal curvatures agree along all tangent directions at all points of the common boundary of two patches, called the linkage curve. The Linkage Curve theorem proved here shows that, for the blend to be curvature continuous when it is already normal continuous, it is sufficient that normal curvatures agree in one direction other than the tangent to a first order continuous linkage curve. This result is significant for it substantiates earlier works in computer aided geometric design. It also offers simple practical means of generating second order blends for it reduces the dimensionality of the problem to that of curve fairing, and is well adapted to a formulation of the blend surface using sweeps. From a theoretical viewpoint, it is remarkable that one can generate second order smooth blends with the assumption that the linkage curve is only first order smooth. The geometric criteria presented may be helpful to the designer since curvature continuity is a technical requirement in hull or cam design problems. The usefulness of the linkage curve theorem is illustrated with a second order blending problem whose implementation will not be detailed here.
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Mohd Kamarudzaman, Anis Solehah, Nurul Huda Mohamad Nasir, and Md Yushalify Misro. "Gaussian and Mean Curvature of Biquintic Trigonometric Bézier Surface." Pertanika Journal of Science and Technology 30, no. 2 (March 31, 2022): 1717–38. http://dx.doi.org/10.47836/pjst.30.2.46.
Повний текст джерелаАнотація:
Bézier curves and surfaces are very important in many areas, especially the manufacturing and aerospace. Surface inspection through visualisation is required to create high-quality surfaces and reduce unwanted products. The smoothness of the surface can be quantified using curvature. In this research, different surfaces types will be generated using the quintic trigonometric Bézier basis function. All the surfaces will be evaluated and analysed using Gaussian and mean curvature. Finally, curvature for each surface type will be mapped using colour-coded mapping and can be further characterised based on their positive and negative curvature values. This insight can also help the designer produce a smooth surface and develop quality products.
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Bandyopadhyay, P. R. "Review—Mean Flow in Turbulent Boundary Layers Disturbed to Alter Skin Friction." Journal of Fluids Engineering 108, no. 2 (June 1, 1986): 127–40. http://dx.doi.org/10.1115/1.3242552.
Повний текст джерелаАнотація:
Recent developments in methods of reducing drag in turbulent boundary layers have been briefly reviewed. The behavior of the mean flow in several drag reducing boundary-layer flows of current interest, viz., those over longitudinal surface riblets, outer-layer devices (OLD’s), and longitudinal convex surface curvature, has been examined. The boundary layer on a surface with longitudinal concave curvature has been studied to complement the results of convex curvature. The riblets alter the flow in their vicinity only and cause no drag penalty. However, the OLD’s disturb the entire boundary layer, and it is the slow downstream (≃150 δ0) relaxation back to the equilibrium state that produces a region of lower skin friction; a net drag reduction results when the wall-drag reduction exceeds the drag penalty due to the device. The net drag reduction achieved by the riblets and OLD’s remains a modest 10 percent compared with the more spectacular levels reached by polymer addition and microbubble injection in water. Over mild convex curvatures, the outer-boundary-layer response is a function of the curvature ratio (δ0/R), and the relaxation rate after a length of convex curvature is a function of the curved length ratio (Δs0/δi). Boundary layers exhibit an asymmetric response to streamwise surface curvatures; the response is slower to a concave curvature than to a convex. Detailed turbulence and accurate wall shear stress measurements in the altered boundary layers are needed to understand the drag-reducing mechanisms involved.
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Meyer, Dominik C., Norman Espinosa, Urs Lang, and Peter P. Koch. "A New Methodology to Determine the Anatomical Center and Radius of Curved Joint Surfaces." Journal of Medical Devices 1, no. 2 (August 27, 2006): 173–75. http://dx.doi.org/10.1115/1.2735973.
Повний текст джерелаАнотація:
This study describes a mechanical tool which allows us to determine the radius and center of curved joint surfaces both intraoperatively and in vitro. The tool is composed of longitudinal parallel hinges, connected with cross bars on one end. In the middle of each cross bar, one needle is attached at an angle of 90deg to both the hinges and the cross bars. When the parallel hinges are held against a curved surface, they will adapt to the curvature and the needles on the cross bars will cross each other. The crossing point of two needles represents the mean center of the curvature within the plane spanned by the needles. The radius is the distance between the center of curvature and the joint surface. The proposed tool and method allow us to determine the mean center of convex or concave curvatures, which often represent the isometric point of a corresponding curved joint surface. Knowing the radius and center of curvature may facilitate various surgical procedures such as collateral or cruciate ligament reconstruction. Appropriate adaptations of the tool appear to be a useful basis for biomechanical and anatomical joint analyses in the laboratory.
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Johnston, Alan, and Peter J. Passmore. "Shape from Shading. I: Surface Curvature and Orientation." Perception 23, no. 2 (February 1994): 169–89. http://dx.doi.org/10.1068/p230169.
Повний текст джерелаАнотація:
The human visual system makes effective use of shading alone in recovering the shape of objects. Pictures of sculptures are readily interpreted—a situation where shading provides virtually the sole cue to shape. However, shading has been considered a poor cue to depth in comparison with retinal disparity and kinetic cues. Curvature discrimination thresholds were measured with the use of a surface-alignment task for a range of surface curvatures from 0.16 cm−1 to 1.06 cm−1. Weber fractions were around 0.1, demonstrating considerable precision in this task. Weber fractions did not vary substantially as a function of surface curvature. Rotation of the light source around the line of sight had no effect on curvature discrimination but rotation towards the viewer increased discrimination thresholds. In contrast, slant discrimination declined with rotation of the light-source vector towards the viewpoint. When a band-limited random grey-level texture was mapped onto the sphere, curvature discrimination thresholds increased gradually as a function of texture contrast, even though texture and shading provided consistent cues to depth. Adding texture also increased slant discrimination thresholds, demonstrating that texture can act as a source of noise in shape-from-shading tasks. The psychophysical findings have been used to evaluate whether current algorithms for shape from shading in computer vision could serve as models of human three-dimensional shape analysis and to highlight low-level intramodular interactions between depth cues. It is demonstrated that, in the case of surfaces defined by shading, curvature descriptions are primary and do not depend upon the prior encoding of surface orientation, and Koenderink's local-shape index is suggested as an alternative intermediate representation of surface shape in the human visual system.
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Shen, Xiang, Theodosios Korakianitis, and Eldad Avital. "Numerical Investigation of Surface Curvature Effects on Aerofoil Aerodynamic Performance." Applied Mechanics and Materials 798 (October 2015): 589–95. http://dx.doi.org/10.4028/www.scientific.net/amm.798.589.
Повний текст джерелаАнотація:
The prescribed surface curvature distribution blade design (CIRCLE) method optimises aerofoils and blades by controlling curvature continuity and slope of curvature distribution along their surfaces. The symmetrical NACA0012 exhibits a surface curvature discontinuity at the leading edge point, and the non-symmetrical E387 exhibits slope-of-curvature discontinuities in the surface. The CIRCLE method is applied to both aerofoils to remove both surface curvature and slope-of-curvature discontinuities. Computational fluid dynamics analyses are used to investigate the curvature effects on aerodynamic performance of the original and modified aerofoils. These results are compared with experimental data obtained from tests on the original aerofoil geometry. The computed aerodynamic advantages of the modified aerofoil are analysed in different operating conditions. The leading edge singularity of NACA0012 is removed and it is shown that the surface curvature discontinuity affects the aerodynamic performance near the stalling angle of attack. The discontinuous slope-of-curvature distribution of E387 influences the size of the laminar separation bubble at lower Reynolds numbers, and it affects the inherent profile of the aerofoil at higher Reynolds numbers. It is concluded that the surface curvature distribution of aerofoils has a significant effect on aerofoil aerodynamic performance, which can be improved by redesigning the surface curvature distribution of the original aerofoil geometry.
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Enomoto, Kazuyuki. "Umbilical points on surfaces in RN." Nagoya Mathematical Journal 100 (December 1985): 135–43. http://dx.doi.org/10.1017/s002776300000026x.
Повний текст джерелаАнотація:
Let ϕ: M → RN be an isometric imbedding of a compact, connected surface M into a Euclidean space RN. ψ is said to be umbilical at a point p of M if all principal curvatures are equal for any normal direction. It is known that if the Euler characteristic of M is not zero and N = 3, then ψ is umbilical at some point on M. In this paper we study umbilical points of surfaces of higher codimension. In Theorem 1, we show that if M is homeomorphic to either a 2-sphere or a 2-dimensional projective space and if the normal connection of ψ is flat, then ψ is umbilical at some point on M. In Section 2, we consider a surface M whose Gaussian curvature is positive constant. If the surface is compact and N = 3, Liebmann’s theorem says that it must be a round sphere. However, if N ≥ 4, the surface is not rigid: For any isometric imbedding Φ of R3 into R4 Φ(S2(r)) is a compact surface of constant positive Gaussian curvature 1/r2. We use Theorem 1 to show that if the normal connection of ψ is flat and the length of the mean curvature vector of ψ is constant, then ψ(M) is a round sphere in some R3 ⊂ RN. When N = 4, our conditions on ψ is satisfied if the mean curvature vector is parallel with respect to the normal connection. Our theorem fails if the surface is not compact, while the corresponding theorem holds locally for a surface with parallel mean curvature vector (See Remark (i) in Section 3).
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Bowers, Daniel T., and Justin L. Brown. "Nanofiber curvature with Rho GTPase activity increases mouse embryonic fibroblast random migration velocity." Integrative Biology 13, no. 12 (December 2021): 295–308. http://dx.doi.org/10.1093/intbio/zyab022.
Повний текст джерелаАнотація:
Abstract Mechanotransduction arises from information encoded in the shape of materials such as curvature. It induces activation of small GTPase signaling affecting cell phenotypes including differentiation. We carried out a set of preliminary experiments to test the hypothesis that curvature (1/radius) would also affect cell motility due to signal pathway crosstalk. High molecular weight poly (methyl methacrylate) straight nanofibers were electrospun with curvature ranging from 41 to 1 μm−1 and collected on a passivated glass substrate. The fiber curvature increased mouse mesenchymal stem cell aspect ratio (P < 0.02) and decreased cell area (P < 0.01). Despite little effect on some motility patterns such as polarity and persistence, we found selected fiber curvatures can increase normalized random fibroblastic mouse embryonic cell (MEF) migration velocity close to 2.5 times compared with a flat surface (P < 0.001). A maximum in the velocity curve occurred near 2.5 μm−1 and may vary with the time since initiation of attachment to the surface (range of 0–20 h). In the middle range of fiber curvatures, the relative relationship to curvature was similar regardless of treatment with Rho-kinase inhibitor (Y27632) or cdc42 inhibitor (ML141), although it was decreased on most curvatures (P < 0.05). However, below a critical curvature threshold MEFs may not be able to distinguish shallow curvature from a flat surface, while still being affected by contact guidance. The preliminary data in this manuscript suggested the large low curvature fibers were interpreted in a manner similar to a non-curved surface. Thus, curvature is a biomaterial construct design parameter that should be considered when specific biological responses are desired. Statement of integration, innovation, and insight Replacement of damaged or diseased tissues that cannot otherwise regenerate is transforming modern medicine. However, the extent to which we can rationally design materials to affect cellular outcomes remains low. Knowing the effect of material stiffness and diameter on stem cell differentiation, we investigated cell migration and signaling on fibrous scaffolds. By investigating diameters across orders of magnitude (50–2000 nm), we identified a velocity maximum of ~800 nm. Furthermore, the results suggest large fibers may not be interpreted by single cells as a curved surface. This work presents insight into the design of constructs for engineering tissues.
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TENENBLAT, KETI. "On Ribaucour transformations and applications to linear Weingarten surfaces." Anais da Academia Brasileira de Ciências 74, no. 4 (December 2002): 559–75. http://dx.doi.org/10.1590/s0001-37652002000400001.
Повний текст джерелаАнотація:
We present a revised definition of a Ribaucour transformation for submanifolds of space forms, with flat normal bundle, motivated by the classical definition and by more recent extensions. The new definition provides a precise treatment of the geometric aspect of such transformations preserving lines of curvature and it can be applied to submanifolds whose principal curvatures have multiplicity bigger than one. Ribaucour transformations are applied as a method of obtaining linear Weingarten surfaces contained in Euclidean space, from a given such surface. Examples are included for minimal surfaces, constant mean curvature and linear Weingarten surfaces. The examples show the existence of complete hyperbolic linear Weingarten surfaces in Euclidean space.
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ÇALIŞKAN, Abdussamet. "Characterizations of Unit Darboux Ruled Surface with Quaternions." Journal of New Theory, no. 42 (March 31, 2023): 43–54. http://dx.doi.org/10.53570/jnt.1194990.
Повний текст джерелаАнотація:
This paper presents a quaternionic approach to generating and characterizing the ruled surface drawn by the unit Darboux vector. The study derives the Darboux frame of the surface and relates it to the Frenet frame of the base curve. Moreover, it obtains the quaternionic shape operator and its matrix representation using the normal and geodesic curvatures to provide a more detailed analysis. To illustrate the concepts discussed, the paper offers a clear example that will help readers better understand the concepts and showcases the quaternionic shape operator, Gauss curvature, mean curvature, and rotation matrix. Finally, it emphasizes the need for further research on this topic.
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Kaya, Onur, and Mehmet Önder. "Generalized normal ruled surface of a curve in the Euclidean 3-space." Acta Universitatis Sapientiae, Mathematica 13, no. 1 (August 1, 2021): 217–38. http://dx.doi.org/10.2478/ausm-2021-0013.
Повний текст джерелаАнотація:
Abstract In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space E3. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.
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Abdel Rahman Abdel Gadir, Abdel Radi, Abdelhalim Zaied Elawad Faread, Adel AhmedHassan Kubba, and Mohammed Iesa Mohammed Abker. "THE FORM OF THE COMPLETE SURFACES OF CONSTANT MEAN CURVATURE." International Journal For Research In Mathematics And Statistics 8, no. 9 (September 22, 2022): 1–4. http://dx.doi.org/10.53555/ms.v8i9.2127.
Повний текст джерелаАнотація:
We explained and classified the complete surfaces of constant mean curvature in addition to construct the first examples of complete surface of positive curvature, properly embedded minimal surfaces and we prove that every complete connected immersed surface with positive extrinsic curvature in must be properly embedded, homeomorphic to a sphere or a plane. We followed the analytical mathematical method and we found that the complete surface of positive curvature has multi applications in different fields of science specially in physics.
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Lou, Hsin-Ya, Wenting Zhao, Xiao Li, Liting Duan, Alexander Powers, Matthew Akamatsu, Francesca Santoro, et al. "Membrane curvature underlies actin reorganization in response to nanoscale surface topography." Proceedings of the National Academy of Sciences 116, no. 46 (October 7, 2019): 23143–51. http://dx.doi.org/10.1073/pnas.1910166116.
Повний текст джерелаАнотація:
Surface topography profoundly influences cell adhesion, differentiation, and stem cell fate control. Numerous studies using a variety of materials demonstrate that nanoscale topographies change the intracellular organization of actin cytoskeleton and therefore a broad range of cellular dynamics in live cells. However, the underlying molecular mechanism is not well understood, leaving why actin cytoskeleton responds to topographical features unexplained and therefore preventing researchers from predicting optimal topographic features for desired cell behavior. Here we demonstrate that topography-induced membrane curvature plays a crucial role in modulating intracellular actin organization. By inducing precisely controlled membrane curvatures using engineered vertical nanostructures as topographies, we find that actin fibers form at the sites of nanostructures in a curvature-dependent manner with an upper limit for the diameter of curvature at ∼400 nm. Nanotopography-induced actin fibers are branched actin nucleated by the Arp2/3 complex and are mediated by a curvature-sensing protein FBP17. Our study reveals that the formation of nanotopography-induced actin fibers drastically reduces the amount of stress fibers and mature focal adhesions to result in the reorganization of actin cytoskeleton in the entire cell. These findings establish the membrane curvature as a key linkage between surface topography and topography-induced cell signaling and behavior.
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Kalikakis, Dimitrios E. "On the curvature of nonregular saddle surfaces in the hyperbolic and spherical three-space." Abstract and Applied Analysis 7, no. 3 (2002): 113–23. http://dx.doi.org/10.1155/s1085337502000799.
Повний текст джерелаАнотація:
This paper proves that any nonregular nonparametric saddle surface in a three-dimensional space of nonzero constant curvaturek, which is bounded by a rectifiable curve, is a space of curvature not greater thankin the sense of Aleksandrov. This generalizes a classical theorem by Shefel' on saddle surfaces in𝔼3.
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Boriek, A. M., S. Liu, and J. R. Rodarte. "Costal diaphragm curvature in the dog." Journal of Applied Physiology 75, no. 2 (August 1, 1993): 527–33. http://dx.doi.org/10.1152/jappl.1993.75.2.527.
Повний текст джерелаАнотація:
The curvature of the midcostal region of the diaphragm in seven dogs was determined at functional residual capacity (FRC) and end inspiration during spontaneous breathing and mechanical ventilation and at total lung capacity in the prone and supine positions. Metallic markers were attached to muscle fibers on the abdominal surface of the diaphragm, and the dog was allowed to recover from surgery. The three-dimensional positions of the markers were determined by biplane videofluoroscopy. A quadratic surface was fit to the bead positions. The principal axes of the quadratic surface lie nearly along and perpendicular to the muscle fibers. In both the supine and prone positions, the values of the principal curvatures were similar at FRC and end inspiration during spontaneous breathing, when muscle tension and transdiaphragmatic pressure both increase with increasing lung volume, and during mechanical ventilation and passive inflation to total lung capacity, when both decrease relative to their magnitude at FRC. No abrupt change of curvature, which might be expected at the edge of the zone of apposition, was apparent. The curvature along the muscle fiber was 0.35 +/- 0.07 cm-1; the curvature perpendicular to the muscle fiber was much smaller, 0.06 +/- 0.01 cm-1. The costal region of the diaphragm displaces and shortens as lung volume increases, but its shape, as described by its curvatures, does not change substantially.
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Wu, J. Z., and R. G. Dong. "Analysis of the contact interactions between fingertips and objects with different surface curvatures." Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 219, no. 2 (February 1, 2005): 89–103. http://dx.doi.org/10.1243/095441105x9327.
Повний текст джерелаАнотація:
Previous experimental observations indicated that the contact interactions between finger and tool handle interfere with the grasp stability, affecting the comfort and manipulations of handheld tools. From a biomechanical point of view, the curvature of the contact surface should affect the contact pressure and contact area, and thereby the comfort and manipulations of hand tools. The current authors analysed, via a finite element model, the contact interactions between fingertips and objects with different curvatures. The effects of the curvature on the contact stiffness, fingertip deformations, contact pressure distributions, and stress/strain distributions within the soft tissues were analysed. The simulation results indicated that the curvature of the contact interface influences the contact characteristics significantly. For a given contact force, the contact area and the contact stiffness increase but the contact pressure and the fingertip deformation decrease with the decrease of the contact surface curvature. The present simulation results will be useful for ergonomic designers in their aim to improve the design of tool handles.
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Hokkyo, Tatsuya, Hideki Aoyama, and Noriaki Sano. "Method to Decide Paths and Postures of Flat End Mill for 5-Axis Control Machining Based on Minimum Cusp Height." Key Engineering Materials 523-524 (November 2012): 380–85. http://dx.doi.org/10.4028/www.scientific.net/kem.523-524.380.
Повний текст джерелаАнотація:
Generally, ball end mills are used for free-form surface machining. When machining curved surfaces with large curvature change using ball end mills, it is necessary to use tools with larger curvature of the cutting edges than the maximum curvature of the surface and minute pick feeds or to change tools for fitting the curvature of one part of the surface. However this causes poor machining efficiency. The curvature of the cutting edge of a flat end mill can be fitted to the curvature of a point on machined surfaces by adjusting the tool posture. Therefore, flat end mills can efficiently cut almost all curvature curved surfaces without tool change. This paper proposes two methods for deciding tool posture and tool path for 5-axis control machining based on minimum cusp height. To decide the tool path, one method defines tool paths along isoparametric curved lines, while the other defines tool paths along curved lines along the minimum curvature direction. The basic system was constructed based on the proposed method, and the effectiveness of the proposed method was verified.
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Tsai, Ming June, Jing Jing Fang, and Jian Feng Huang. "Automatic Polishing of Super Accuracy Mirror Mold with Free-Form Surface by Curvature Analysis." Materials Science Forum 505-507 (January 2006): 547–52. http://dx.doi.org/10.4028/www.scientific.net/msf.505-507.547.
Повний текст джерелаАнотація:
This paper proposed a polishing path planning method of super accuracy mirror mold with free-form surface by curvature analysis. First, IGES files of free-form surfaces are read and the mold geometry is regenerated as B-spline surface by the Automatic Mold Polishing System (AMPS). By using the derivative properties of B-spline surface, normal vector and principal curvatures at any point of the surface are calculated. In addition, the effective contact width between polishing tool and mold surface based on the grain size and the principal radii of curvature is also determined. The minimum contact width in 3-D is mapped onto the (u, v) parameters of B-spline surface. Then a modified Peano fractal path with weaving function is calculated based on the effective contact width in the (u, v) coordinate. This Peano-weaving path was tested on an optical mold with free-form surface. The polishing result shows the method is very effective and achieves the level of mirror surface with roughness Ra 29nm.
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KUCUKARSLAN YUZBASİ, Zuhal, and Sevinç TAZE. "On Parametric Surfaces with Constant Mean Curvature Along Given Smarandache Curves in Lie Group." Journal of New Theory, no. 40 (September 30, 2022): 82–89. http://dx.doi.org/10.53570/jnt.1165809.
Повний текст джерелаАнотація:
This paper finds sufficient conditions to determine a surface whose mean curvature along a given Smarandache curve is constant in a three-dimensional Lie group. This is accomplished by using the Frenet frames of the specified curve to express surfaces that span the TN, NB, and TB Smarandache curves parametrically. In terms of the curvatures of given Smarandache curves, marching scale functions, and their partial derivatives, the mean curvatures of these surfaces along the given TN, NB, and TB Smarandache curves are determined. Sufficient conditions are found to maintain the provided mean curvatures of the resulting surfaces at a constant value. Finally, some examples are provided.
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Vashpanova, N., O. Lesechko, and T. Podousova. "INFINITESIMAL DEFORMATIONS OF SURFACES WITH A GIVEN CHANGE OF THE RICCI TENSOR." Mechanics And Mathematical Methods 5, no. 1 (June 30, 2023): 97–109. http://dx.doi.org/10.31650/2618-0650-2023-5-1-97-109.
Повний текст джерелаАнотація:
In three-dimensional Euclidean space, we study the problem of the existence of an infinitesimal first-order deformation of single-connected regular surfaces with a predetermined change in the Ricci tensor. It is shown that for surfaces of nonzero Gaussian curvature, this problem is reduced to the study and solution of a system of seven equations (including differential equations) with respect to seven unknown functions, each solution of which determines a vector field that is a univariate function (with an accuracy of a constant vector) and can be interpreted as a moment-free stress state of equilibrium of a loaded shell. For regular surfaces of non-zero Gaussian and mean curvatures, the problem is reduced to finding solutions to one second-order partial differential equation with respect to two unknown functions. Given one of these functions, the resulting equation will in general be a nonhomogeneous second-order partial differential equation (nonhomogeneous Weingarten differential equation). It is proved that any regular surface of positive Gaussian and non-zero mean curvature admits an infinitesimal first-order deformation with a given change in the Ricci tensor in a sufficiently small region. In this case, the tensor fields will be represented by an arbitrary and predefined regular function. By considering the Neumann problem, it is shown that a single-connected regular surface of elliptic type of positive Gaussian and negative mean curvature with a regular boundary under a certain boundary condition admits, in general, an infinitesimal first-order deformation with a predetermined change in the Ricci tensor. In this case, the tensor fields will be determined uniquely. For surfaces of negative Gaussian and non-zero mean curvature, the resulting inhomogeneous partial differential equation with second-order partial differentials will be of hyperbolic type with known coefficients and right-hand side. The Darboux problem is considered for this equation. It is proved that any regular surface of negative Gaussian and non-zero mean curvature admits an infinitesimal first-order deformation with a given change in the Ricci tensor. Tensor fields are expressed through a given function of two variables and through two arbitrary regular functions of one variable. Keywords: infinitesimal deformation, Ricci tensor, tensor fields, Gaussian curvature, mean curvature.