Definition Of Curvature Research Articles

Berndtsson-Lempert-Szőke fields associated to proper holomorphic families of vector bundles

Drawing on work of Berndtsson and of Lempert and Szőke, we define a complex analytic structure for families of (possibly finite-dimensional) Hilbert spaces that might not fit together to form a holomorphic vector bundle but nevertheless have a reasonable definition of curvature that agrees with the curvature of the Chern connection when the family of Hilbert spaces is locally trivial. As one consequence, we obtain a new proof of a theorem of Berndtsson on the curvature of direct images of semi-positively twisted relative canonical bundles (i.e., adjoint bundles), and of its higher-rank generalization by Liu-Yang.

On the definition of curvature in Regge calculus

Abstract Regge calculus defines a curvature for piecewise constant metrics on simplicial complexes subject to a partial continuity requirement. We prove that if a part of the complex is embedded in a Euclidean space, by a piecewise affine map, and we perform smoothing by convolution there, then the smoothed metrics have a densitized scalar curvature that converges, in the sense of measures, to that defined by Regge, as the smoothing parameter goes to zero.

New approaches in choosing a suitable growth model: Mean Curvature and Arc Length Values

Logistic, Gompertz and Bertalanffy sigmoid growth models are widely used to study the growth dynamics of populations such as living plants, animals and bacteria. Appropriate model selection and parameter estimation are very important as these models will be used to make practical inferences. Because different growth models are modeled biologically, regardless of whether the parameters are definable or not. Applications that do not take into account parameter identifiability can lead to unreliable parameter estimates and misleading interpretations. Therefore, first the most suitable model should be determined and then the parameters should be defined. In this study, two new suitable model determination criteria such as mean curvature and arc length are proposed. For this, firstly, the definition of curvature was given. Then, the mean curvature and arc length values of the data belonging to two different species (kangal dog growth and eucalyptus plant growth) were calculated. For this purpose, a comparison was made with model selection criteria available in the literature such as coefficient of determination, error sum of squares and Akaike information criterion (AIC). It has been determined that the results obtained from the mean curvature and arc length values are in accordance with the existing criteria. In the two datasets, it was seen that the fit model ranking for both the existing criteria and the criteria we proposed was the same. For this reason, it is thought that the mean curvature and arc length values can be accepted as suitable model selection criteria.

FEVO-LOAM: Feature Extraction and Vertical Optimized Lidar Odometry and Mapping

Simultaneous Localization and Mapping (SLAM) is a significant research topic in robotics since it is one of the key technologies for robot automation. Although lidar-based SLAM methods have achieved promising performance, traditional lidar SLAM methods still produce large vertical errors. To address this issue, we propose a feature extraction and vertical optimized lidar odometry and mapping approach. Firstly, we optimize the feature extraction. Specifically, we propose a more accurate ground segmentation approach and a new curvature definition, which is used to extract more discriminative features. Additionally, we propose a lidar mapping approach, which adds new vertical residuals and pitch residuals to the objective function. Then a two-step Levenberg-Marquardt method is used to solve the pose transformation. Finally, we evaluate the proposed method in public datasets and real environments. Experiments show that compared with other state-of-the-art methods, our method achieves better accuracy and reduces the vertical error with a similar computational expense.

Curvature of differentiable Hilbert modules and Kasparov modules

In this paper we introduce the curvature of densely defined universal connections on Hilbert C⁎-modules relative to a spectral triple (or unbounded Kasparov module), obtaining a well-defined curvature operator. Fixing the spectral triple, we find that modulo junk forms, the curvature only depends on the represented form of the universal connection. We refine our definition of curvature to factorizations of unbounded Kasparov modules. Our definition recovers all the curvature data of a Riemannian submersion of compact manifolds, viewed as a KK-factorization.

Plant-Pollinator Specialization: Origin and Measurement of Curvature.

AbstractA feature of biodiversity is the abundance of curves displayed by organs and organisms. Curvature is a widespread, convergent trait that has important ecological and evolutionary implications. In pollination ecology, the curvature of flowers and pollinator mouthparts (e.g., hummingbird bills) along the dorsiventral plane has been associated with specialization, competition, and species coexistence. Six differing methods have historically been used to measure curvature in pollination systems; we provide a solution to this inconsistency by defining curvature using well-established concepts from differential geometry. Intuitively, curvature is the degree to which a line is not straight, but more formally it is the rate at which the tangent of a curve changes direction with respect to arc length. Here, we establish a protocol wherein a line is fitted against landmarks placed on an image of a curved organ or organism, then curvature is computed at many points along the fitted line and the sum taken. The protocol is demonstrated by studying the development of nectar spur curvature in the flowering plant genus Epimedium (Berberidaceae). By clarifying the definition of curvature, our aim is to make the language of comparative morphology more precise and broadly applicable to capture other curved structures in nature.

Ollivier-Ricci curvature convergence in random geometric graphs

Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional definition of curvature of a Riemannian manifold. Here we show that Ollivier curvature of random geometric graphs in any Riemannian manifold converges in the continuum limit to Ricci curvature of the underlying manifold, but only if the definition of Ollivier graph curvature is properly generalized to apply to mesoscopic graph neighborhoods. This result establishes the first rigorous link between a definition of curvature applicable to networks and a traditional definition of curvature of smooth spaces.

Relativity Isoframes—A Useful and Potentially Unifying Conceptual Framework

This brief note introduces the conceptual framework of special and general relativity isoclocks and isoframes. Isoclocks and isoframes, as defined herein, can be used to create geometrical maps of space and time (“space-time”) with and without matter embedded. They are useful for having a mental picture of space-time relationships without having to picture 4-dimensional manifolds, which very few students and scientists are able to do. With the aid of the optical lensing definition of curvature as inverse radius, a new gravitational force equation is derived, which also incorporates Einstein’s mass/energy relation in the mx term. Thus, one may see how it is that gravitational force correlates with its time-embedded curvature-squared (Cx2) space in a more accurate formulation than could be envisioned by Newton. This becomes more apparent in high gamma fields, such as found near a black hole horizon. It is hoped that probability theories, such as quantum field theories in curved space-time, might be adaptable to the general relativity isoframe concept introduced herein.

Interpolated corrected curvature measures for polygonal surfaces

AbstractA consistent and yet practically accurate definition of curvature onto polyhedral meshes remains an open problem. We propose a new framework to define curvature measures, based on the Corrected Normal Current, which generalizes the normal cycle: it uncouples the positional information of the polyhedral mesh from its geometric normal vector field, and the user can freely choose the corrected normal vector field at vertices for curvature computations. A smooth surface is then built in the Grassmannian ℝ3 × 𝕊2 by simply interpolating the given normal vector field. Curvature measures are then computed using the usual Lipschitz–Killing forms, and we provide closed‐form formulas per triangle. We prove a stability result with respect to perturbations of positions and normals. Our approach provides a natural scale‐space for all curvature estimations, where the scale is given by the radius of the measuring ball. We show on experiments how this method outperforms state‐of‐the‐art methods on clean and noisy data, and even achieves pointwise convergence on difficult polyhedral meshes like digital surfaces. The framework is also well suited to curvature computations using normal map information.

Noncommutativity of Finite Rotations and Definitions of Curvature and Torsion

The geometry of a space curve, including its curvature and torsion, can be uniquely defined in terms of only one parameter which can be the arc length parameter. Using the differential geometry equations, the Frenet frame of the space curve is completely defined using the curve equation and the arc length parameter only. Therefore, when Euler angles are used to describe the curve geometry, these angles are no longer independent and can be expressed in terms of one parameter as field variables. The relationships between Euler angles used in the definition of the curve geometry are developed in a closed-differential form expressed in terms of the curve curvature and torsion. While the curvature and torsion of a space curve are unique, the Euler-angle representation of the space curve is not unique because of the noncommutative nature of the finite rotations. Depending on the sequence of Euler angles used, different expressions for the curvature and torsion can be obtained in terms of Euler angles, despite the fact that only one Euler angle can be treated as an independent variable, and such an independent angle can be used as the curve parameter instead of its arc length, as discussed in this paper. The curve differential equations developed in this paper demonstrate that the curvature and torsion expressed in terms of Euler angles do not depend on the sequence of rotations only in the case of infinitesimal rotations. This important conclusion is consistent with the definition of Euler angles as generalized coordinates in rigid body dynamics. This paper generalizes this definition by demonstrating that finite rotations cannot be directly associated with physical geometric properties or deformation modes except in the cases when infinitesimal-rotation assumptions are used.

A discrete method of sphere-pipe intersecting curve for robot welding by offline programming

Abstract The sphere-pipe intersecting curve is a kind of spatial complex curve, which is controlled by robot in the sphere-pipe welding. In this paper, a discrete method for robot welding by offline programming is put forward. To move the robot with the specified curve trajectory, it need to be discretized to generate the robot motion instructions. As the core innovation of this paper, a discrete method of sphere-pipe intersecting curve is given according to the chord error limit and principle of equal arc length, which guarantees the accuracy of robot welding. To achieve this goal, this paper has done the following related works. First, the geometrical model of the sphere-pipe intersection is established, which can cover most of the intersection modes. Second, this paper gives the parameter equations in two different coordinate systems because the expression is very complex. This operation will simplify the subsequent calculations. Although the curve expression is simplified, it is very complicated to find the maximum based on the definition of curvature. After experimental analysis, this paper presents a simple and effective algorithm for the calculation of maximum curvature. Finally, this paper takes the Puma560 robot for simulation, and the simulation results are used to verify the feasibility and effectiveness of our method.

Positivities and vanishing theorems on complex Finsler manifolds

In this paper, we study the relations between curvatures and geometries of a compact complex Finsler manifold. We first introduce various definitions of curvatures and then vanishing theorems are established under some positivity assumptions of curvatures. In particular, we get some criteria of a compact complex Finsler manifold to be of negative Kodaira dimension.

Automatic Side Branch Ostium Detection and Main Vascular Segmentation in Intravascular Optical Coherence Tomography Images.

Intravascular optical coherence tomography is the state-of-the-art imaging modality in percutaneous coronary intervention planning and evaluation, in which side branch ostium and main vascular measurements play critical roles. However, manual measurement is time consuming and labor intensive. In this paper, we propose a fully automatic method for side branch ostium detection and main vascular segmentation to make up manual deficiency. In our method, side branch ostium points are first detected and subsequently used to divide the lumen contour into side branch and main vascular regions. Based on the division, main vascular contour is then smoothly fitted for segmentation. In side branch ostium detection, our algorithm creatively converts the definition of curvature into the calculation of the signed included angles in global view, and originally applies a differential filter to highlight the feature of side branch ostium points. A total of 4618 images from 22 pullback runs were used to evaluate the performance of the presented method. The validation results of side branch detection were TPR = 82.8 $\%$, TNR = 98.7$\%$ , PPV = 86.8$\%$, NPV = 98.7$\%$. The average ostial distance error (ODE) was 0.22mm, and the DSC of main vascular segmentation was 0.96. In conclusion, the qualitative and quantitative evaluation indicated that the presented method is effective and accurate.

Curvature Expressions for the Large Displacement Analysis of Planar Beam Motions

While several curvature expressions have been used in the literature, some of these expressions differ from basic geometry definitions and lead to kinematic coupling between bending and shear deformations. This paper uses three different elastic force formulations in order to examine the effect of the curvature definition in the large displacement analysis of beams. In the first elastic force formulation, a general continuum mechanics approach (method 1) based on the nonlinear strain–displacement relationship is used. The second approach (method 2) is based on a classical nonlinear beam theory, in which a curvature expression consistent with differential geometry and independent of the shear deformation is used. The third elastic force formulation (method 3) employs a curvature expression that depends on the shear angle. In order to examine numerically the effect of using different curvature definitions, three different planar beam elements are used. The first element (element I) is the fully parameterized absolute nodal coordinate formulation (ANCF) shear deformable beam element. The second element (element II) is an ANCF consistent rotation-based formulation (CRBF) shear deformable beam element obtained from element I by consistently replacing the position gradient vectors by rotation parameters. The third element (element III) is a low-order bilinear ANCF/CRBF finite element in which nonzero differential geometry-based curvature definition cannot be obtained because of the low order of interpolation. Numerical results are obtained using the three elastic force formulations and the three finite elements in order to shed light on the definition of bending and shear in the large displacement analysis of beams. The results obtained in this investigation show that the use of method 2, with a penalty formulation that restricts the excessive cross section deformation, can improve significantly the convergence of the ANCF finite element.

On the notion of curvature and its mechanical meaning in a geometrically exact plane beam theory

With the aim to answer to the question about what is the correct notion for curvature to adopt in constitutive relationships, this paper considers a geometrically exact beam theory built on very basic kinematic assumptions. The theory is developed in a consistent way, by deducing equations of motion from the Principle of Virtual Work. Further, by stipulating a relation between internal work for one- and three-dimensional beams, relationships among internal forces and stresses are found. Constitutive equations, which end up to be strongly coupled and nonlinear, are written in explicit form for a specific material model with linear behavior. The role of different notions of curvature to adopt in analysis of beams is investigated and linear, linearized and nonlinear equations for bending moment are considered. In particular, uncoupled linear approximation provides indication about the more suitable curvature definition. Further, a mechanical interpretation of generalized internal stresses is also given and benchmark numerical examples enlighten some features of the model.

Consequences of different definitions of bending curvature on nonlinear dynamics of beams

Beam theories may be grouped in two broad categories, namely induced or intrinsic theories. In the former, beam models are obtained as exact consequences of three-dimensional theory, making use either of asymptotic expansions in a slenderness parameter or projections of three-dimensional elasticity on certain function spaces, while beams are inherently one-dimensional bodies in the latter category. Although induced theories show a clear connection between three- and one-dimensional representations, they are often more demanding with respect to intrinsic ones, in which a finite number of strain parameters, depending on just one space variable, characterizes the motion of beams in an internally consistent way and without a direct linkage to three-dimensional material properties. Hence, as a consequence, intrinsic theories do not provide any structure for constitutive equations and, at least in principle, different choices can be allowed. A typical example of this fact is represented by the one-dimensional relationship between the bending moment and the beam curvature, since for this latter two notions are admissible. Indeed, both are adopted in the literature and, apparently, preferring one to the other is only related to the predictive capability of the ensuing model. The arising question is about possible differences in both static and dynamic responses of beams, when one or the other definition of curvature is selected.

Comparing Nonlinear Free Vibrations of Timoshenko Beams with Mechanical or Geometric Curvature Definition

The nonlinear free oscillations of a planar, initially straight Timoshenko beam are investigated by means of the asymptotic development method. Attention is focus on the difference in considering the “mechanical” vs the “geometric” curvature of the axis of the beam, which are different for extensible beams. A comparison of the results obtained by the two models is proposed, and it is shown when they are equivalent and when they give different nonlinear behaviours. A parametric analysis showing the effects of the slenderness (arbitrary, not necessarily large) and of the stiffness of the right-end axial spring is performed.

Topics in sub-Riemannian geometry

Sub-Riemannian geometry is the geometry of spaces with non- holonomic constraints. This paper presents an informal survey of some topics in this area, starting with the construction of geodesic curves and ending with a recent definition of curvature. Bibliography: 28 titles.

On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

For a finitely triangulated closed surface M2, let αx be the sum of angles at a vertex x. By the well-known combinatorial version of the 2- dimensional Gauss-Bonnet Theorem, it holds Σx(2π - αx) = 2πχ(M2), where χ denotes the Euler characteristic of M2, αx denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ(τ) as the normed dihedral angle defect around a simplex τ. Our main result is Στ (-1)dim(τ)δ(τ) = χ(K), where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature κ(x) at a vertex and we prove the vertex-version Σx∈K0 κ(x) = χ(K) of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial (n + 1)-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B: $$\chi \left( W \right) - \frac{1}{2}\chi \left( B \right) = \sum {_{\tau \in W - B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}} + \sum {_{\tau \in B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}}\rho \left( \tau \right)$$ .

Distributed mean curvature on a discrete manifold for Regge calculus

The integrated mean curvature of a simplicial manifold is well understood in both Regge Calculus and Discrete Differential Geometry. However, a well motivated pointwise definition of curvature requires a careful choice of the volume over which to uniformly distribute the local integrated curvature. We show that hybrid cells formed using both the simplicial lattice and its circumcentric dual emerge as a remarkably natural structure for the distribution of this local integrated curvature. These hybrid cells form a complete tessellation of the simplicial manifold, contain a geometric orthonormal basis, and are also shown to give a pointwise mean curvature with a natural interpretation as the fractional rate of change of the normal vector.