Geodesic Curvature Research Articles

Two-Stage Sector Partition Path Planning Method for Automated Fiber Placement on Complex Surfaces

Automated fiber placement (AFP) is an advanced composite manufacturing technology which is widely used in the production of complex shaped aircraft components. However, on the surface with large curvature, the fiber compaction performance is poor, and the direction deviation and the geodesic curvature of the path increase sharply with the offset distance, resulting in various placement defects. To solve this problem, the influence of the surface curvature on the path performance is studied, and a two-stage sector partition path planning method for AFP on complex surface is proposed in this paper. The constraints of the direction deviation, turning radius and the compaction state of the fiber are comprehensively considered in this method. By analyzing the compaction feature, the distance between the roller and the surface is used to evaluate the compaction condition and an initial partition criterion is established according to the normal curvature of the surface along the roller axis. Then, the quantitative relationship between path performance and offset distance is derived, and the precise sector is determined under the constraints of placement direction and turning radius. The path performance constraints are fully considered in the path generation process, so the iterative process of evaluation–modification is effectively avoided. Simulations and experiments are then carried out on the winglet surface and the results show that compared with the parallel method, the proposed method can significantly improve the direction deviation, turning radius and compaction quality of the placement paths at the cost of gaps/overlaps. Satisfactory placement paths can be obtained with reasonable constraints on the complex surface.

Voss Surfaces: A Design Space for Geodesic Gridshells

The design of envelopes with complex geometries often leads to construction challenges. To overcome these difficulties, resorting to discrete differential geometry proved successful by establishing close links between mesh properties and the existence of good fabrication, assembling and mechanical properties. In this paper, the design of a special family of structures, called geodesic shells, is addressed using Voss nets, a family of discrete surfaces. The use of discrete Voss surfaces ensures that the structure can be built from simply connected, initially straight laths, and covered with flat panels. These advantageous constructive properties arise from the existence of a conjugate network of geodesic curves on the underlying smooth surface. Here, a review of Voss nets is presented and particular attention is given to the projection of normal vectors on the unit sphere. This projection, called Gauss map, creates a dual net which unveils the remarkable characteristics of Voss nets. Then, based on the previous study, two generation methods are introduced. One enables the exploration and the deformation of Voss nets while the second provides a more direct computational technique. The application of theses methodologies is discussed alongside formal examples.

A simple differential geometry for complex networks

AbstractWe introduce new definitions of sectional, Ricci, and scalar curvatures for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in general metric spaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicable to unweighted or weighted and undirected or directed networks and are more intuitive and easier to compute than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature allow us to give a network analogue of the classical local Gauss–Bonnet theorem. Furthermore, we propose even simpler and more intuitive proxies for the Haantjes curvature that allow for even faster and easier computations in large-scale networks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures. Lastly, we also investigate the behavior, both on model and real-world networks, of the curvatures introduced herein with more established notions of Ricci curvature and other widely used network measures.

A Lagrangian approach to extremal curves on Stiefel manifolds

<p style='text-indent:20px;'>A unified framework for studying extremal curves on real Stiefel manifolds is presented. We start with a smooth one-parameter family of pseudo-Riemannian metrics on a product of orthogonal groups acting transitively on Stiefel manifolds. In the next step Euler-Langrange equations for a whole class of extremal curves on Stiefel manifolds are derived. This includes not only geodesics with respect to different Riemannian metrics, but so-called quasi-geodesics and smooth curves of constant geodesic curvature, as well. It is shown that they all can be written in closed form. Our results are put into perspective to recent related work where a Hamiltonian rather than a Lagrangian approach was used. For some specific values of the parameter we recover certain well-known results.

Mannheim Partner Curves According to Darboux Frame in the Euclidean 3-space E^3

In this study, we consider Mannheim partner curves according to Darboux Frame. Characterization of Mannheim partner curves lying on the surface are examined, and some curves are obtained. The characteristics of these curves are given by the geodesic curvatures, normal curvatures and geodesic torsions of these curves..............................................................................................................................................................................................................................................................

Dual curves associated with the Bonnet ruled surfaces

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.

Admissible surfaces in progressive addition lenses.

Progressive addition lenses (PALs) contain a surface of spatially varying curvature, which supplies variable optical power for different viewing areas over the lens. We derive complete compatibility equations providing the exact magnitude of a cylinder along lines of curvature on any arbitrary PAL smooth surface. These equations reveal that, contrary to current knowledge, the cylinder and its derivative depend not only on the principal curvature and its derivatives along the principal line but also on the geodesic curvature and its derivatives along the line orthogonal to the principal line. We quantify the relevance of the geodesic curvature through numerical computations. We also derive an extended and exact Minkwitz theorem restricted only to be applied along lines of curvature, but excluding umbilical points.

Maxwell equations in a curved spacetime: Spin optics approximation

We study propagation of high-frequency electromagnetic waves in a curved spacetime. We demonstrate how a modification of the standard geometric optics allows one to include the helicity dependent corrections into the equations of motion of circularly polarized beams of radiation. As a result, polarized light rays are still null but not geodesic curves. To achieve these results we construct null frames associated with a set of (non-geodesic) null rays and use these frames for description of the high-frequency wave propagation. We call this approach spin optics approximation. It is completely covariant and it can be used in an arbitrary time-dependent gravitational field.

Incomplete Orientation Mapping for Teleoperation With One DoF Master-Slave Asymmetry

Teleoperation systems require a human-centered approach in which the kinematic mapping is intuitive and straightforward for the operators. However, a mismatch in the degrees-of-freedom (DoFs) between master and slave could result in an asymmetrical teleoperation system. That is an obstacle for intuitive kinematic mapping. In particular, it is even more challenging when the missing DoF is a pure rotational DoF, since the rotation group SO(3) is a nonlinear Riemannian manifold. This letter is concerned with an asymmetric teleoperation system, where the master subsystem can provide 6-DoF pose sensing while the slave subsystem only has 5 DoFs. The rotation along the missing DoF, which is configuration-dependent, is mapped to a geodesic curve in SO(3). We define and prove the closed-form solution of the perpendicular curve to the geodesic curve. Based on the perpendicular curve, we develop a novel Incomplete Orientation Mapping (IOM) approach to avoid the motion in the missing DoF. By comparing with two baseline methods, the experimental results demonstrate that the proposed method can discard the rotational motion along the missing DoF for all configurations, while preserving the remaining rotations.

Closed ideal planar curves

In this paper we use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the $L^2$ sense. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a multiply-covered circle. Moreover, we show that curves in any homotopy class with initially small $L^3\lVert k_s\rVert_2^2$ enjoy a uniform length bound under the flow, yielding the convergence result in these cases.

Path planning in a Riemannian manifold using optimal control

We consider the motion planning of an object in a Riemannian manifold where the object is steered from an initial point to a final point utilizing optimal control. Considering Pontryagin Minimization Principle we compute the Optimal Controls needed for steering the object from initial to final point. The Optimal Controls were solved with respect to time [Formula: see text] and shown to have norm [Formula: see text] which should be the case when the extremal trajectories, which are the solutions of Pontryagin Principle, are arc length parametrized. The extremal trajectories are supposed to be the geodesics on the Riemannian manifold. So we compute the geodesic curvature and the Gaussian curvature of the Riemannian structure.

The Geometry of $$C^1$$ Regular Curves in Sphere with Constrained Curvature

In this article, we study $$C^1$$ regular curves in the 2-sphere that start and end at given points with given directions, whose tangent vectors are Lipschitz continuous, and their a.e. existing geodesic curvatures have essentially bounds in an open interval. Especially, we show that a $$C^1$$ regular curve is such a curve if and only if the infimum of its lower curvature and the supremum of its upper curvature are constrained in the same interval.

Surface Family With A Common Bertrand-B Isogeodesic Curve

Bu çalışmada 3-boyutlu Öklid uzayında parametrik denklemi ile verilen yüzey üzerinde eğriliği sıfırdan farklı olan bir eğrinin Bertrand B-çiftinin bu yüzey üzerinde isogeodezik olması için gerekli ve yeterli şartlar elde edilerek, ortak Bertrand-B isogeodezik eğrili yüzey aileleri problemi ele alınmıştır. Bu çalışmada 3-boyutlu Öklid uzayında parametrik denklemi ile verilen yüzey üzerinde eğriliği sıfırdan farklı olan bir eğrinin Bertrand B-çiftinin bu yüzey üzerinde isogeodezik olması için gerekli ve yeterli şartlar elde edilerek, ortak Bertrand-B isogeodezik eğrili yüzey aileleri problemi ele alınmıştır.

Curves on a Smooth Surface with Position Vectors Lie in the Tangent Plane

The present paper deals with a study of curves on a smooth surface whose position vector always lies in the tangent plane of the surface and it is proved that such curves remain invariant under isometry of surfaces. It is also shown that length of the position vector, tangential component of the position vector and geodesic curvature of a curve on a surface whose position vector always lies in the tangent plane are invariant under isometry of surfaces.

Smarandache curves according to the Sabban frame belonging to spherical indicatrix curve of the Salkowski curve

In this study, Smarandache curves are defined according to the Sabban frame by means of the spherical indicatrix curve of the Salkowski curve. Then using these curves we calculated geodesic curvatures. Finally each curve is drawn with maple program.

Geometry of entanglement

In the context of the surface-state correspondence we propose the geodesic curvature of a convex curve as a local measure of factorization of the dual CFT state. Its integral will be interpreted as computing the bipartite entanglement among degrees of freedom with support on the chosen domain. We will derive results through application of the Gauss-Bonnet theorem and show quantitative agreement with computations using the MERA tensor network and the formalism of entanglement density.

Directional spherical indicatrices of timelike space curve

In this work, the directional spherical indicatrices of a timelike space curve using tangent, quasi-normal and quasi-binormal vectors with q-frame are introduced. Then we work on the condition, that a timelike space curve to be slant helix, by using the geodesic curvature of the directional normal spherical indicatrix. Finally, an application of the results is given.

A computational model of protein induced membrane morphology with geodesic curvature driven protein-membrane interface

Continuum or hybrid modeling of bilayer membrane morphological dynamics induced by embedded proteins necessitates the identification of protein-membrane interfaces and coupling of deformations of two surfaces. In this article we developed (i) a minimal total geodesic curvature model to describe these interfaces, and (ii) a numerical one-to-one mapping between two surface through a conformal mapping of each surface to the common middle annulus. Our work provides the first computational tractable approach for determining the interfaces between bilayer and embedded proteins. The one-to-one mapping allows a convenient coupling of the morphology of two surfaces. We integrated these two new developments into the energetic model of protein-membrane interactions, and developed the full set of numerical methods for the coupled system. Numerical examples are presented to demonstrate (1) the efficiency and robustness of our methods in locating the curves with minimal total geodesic curvature on highly complicated protein surfaces, (2) the usefulness of these interfaces as interior boundaries for membrane deformation, and (3) the rich morphology of bilayer surfaces for different protein-membrane interfaces.

On elastic geodesic grids and their planar to spatial deployment

We propose a novel type of planar-to-spatial deployable structures that we call elastic geodesic grids. Our approach aims at the approximation of freeform surfaces with spatial grids of bent lamellas which can be deployed from a planar configuration using a simple kinematic mechanism. Such elastic structures are easy-to-fabricate and easy-to-deploy and approximate shapes which combine physics and aesthetics. We propose a solution based on networks of geodesic curves on target surfaces and we introduce a set of conditions and assumptions which can be closely met in practice. Our formulation allows for a purely geometric approach which avoids the necessity of numerical shape optimization by building on top of theoretical insights from differential geometry. We propose a solution for the design, computation, and physical simulation of elastic geodesic grids, and present several fabricated small-scale examples with varying complexity. Moreover, we provide an empirical proof of our method by comparing the results to laser-scans of the fabricated models. Our method is intended as a form-finding tool for elastic gridshells in architecture and other creative disciplines and should give the designer an easy-to-handle way for the exploration of such structures.

Finslerian geodesics on Frechet manifolds

We establish a framework, namely, nuclear bounded Fr\'{e}chet manifolds endowed with Riemann-Finsler structures to study geodesic curves on certain infinite dimensional manifolds such as the manifold of Riemannian metrics on a closed manifold. We prove on these manifolds geodesics exist locally and they are length minimizing in a sense. Moreover, we show that a curve on these manifolds is geodesic if and only if it satisfies a collection of Euler-Lagrange equations. As an application, without much difficulty, we prove that the solution to the Ricci flow on an Einstein manifold is not geodesic.