Showing papers on "Tangent published in 2008"

Approximating Catmull-Clark subdivision surfaces with bicubic patches

Abstract: We present a simple and computationally efficient algorithm for approximating Catmull-Clark subdivision surfaces using a minimal set of bicubic patches. For each quadrilateral face of the control mesh, we construct a geometry patch and a pair of tangent patches. The geometry patches approximate the shape and silhouette of the Catmull-Clark surface and are smooth everywhere except along patch edges containing an extraordinary vertex where the patches are C0. To make the patch surface appear smooth, we provide a pair of tangent patches that approximate the tangent fields of the Catmull-Clark surface. These tangent patches are used to construct a continuous normal field (through their cross-product) for shading and displacement mapping. Using this bifurcated representation, we are able to define an accurate proxy for Catmull-Clark surfaces that is efficient to evaluate on next-generation GPU architectures that expose a programmable tessellation unit.

On boundary potential energies in deformational and configurational mechanics

Abstract: This contribution deals with the implications of boundary potential energies, i.e. in short surface, curve and point potentials, on deformational and configurational mechanics. Within the realm of deformational mechanics the surface/curve potentials are allowed in the most general case to depend on the deformation, the surface/curve deformation gradient and the spatial surface normal/curve tangent and are parametrised in the material placement and the material surface normal/curve tangent. The point potentials depend on the deformation and are parametrised in the material placement. From the configurational mechanics perspective the roles of fields and parametrisations are reversed. By considering variational arguments based on the kinematics of deforming surfaces/curves, in particular the relevant surface/curve stresses and distributed forces contributing to (localized) deformational and configurational force balances at surfaces/curves/points, which extend the common traction boundary conditions, are derived. Thereby, dissipative distributed configurational forces that are energetically conjugate to configurational changes are introduced as definitions. The (localized) force balances at surfaces/curves/points together with the contributing stresses and distributed forces within deformational and configurational mechanics display an intriguing duality. The resulting dissipative configurational tractions at the boundary are exemplified for some illustrative cases of boundary potentials.

The Milky Way Rotation Curve and Its Vertical Derivatives: Inside the Solar Circle

Abstract: We measure the Galactic rotation curve and its first two vertical derivatives in the first and fourth quadrants of the Milky Way using the 21 cm VGPS and SGPS. We find tangent velocities of the atomic gas as a function of Galactic longitude and latitude by fitting an analytic line profile to the edges of the velocity profiles. The shape of the analytic profile depends only on the tangent velocity and the velocity dispersion of the gas. We use two complementary methods to analyze the tangent velocities: a global model to fit typical parameter values and a local fitting routine to examine spatial variations. We confirm the validity of our fitting routines by testing simple models. Both the global and local fits are consistent with a vertical falloff in the rotation curve of – -->22 ± 6 km s−1 kpc−1 within 100 pc of the Galactic midplane. The magnitude of the falloff is several times larger than what would be expected from the change in the potential alone, indicating some other physical process is important. The falloff we measure is consistent in magnitude with that measured in the halo gas of other galaxies.

Numerical Approximation of Tangent Moduli for Finite Element Implementations of Nonlinear Hyperelastic Material Models

Abstract: Finite element (FE) implementations of nearly incompressible material models often employ decoupled numerical treatments of the dilatational and deviatoric parts of the deformation gradient. This treatment allows the dilatational stiffness to be handled separately to alleviate ill conditioning of the tangent stiffness matrix. However, this can lead to complex formulations of the material tangent moduli that can be difficult to implement or may require custom FE codes, thus limiting their general use. Here we present an approach, based on work by Miehe (Miehe, 1996, "Numerical Computation of Algorithmic (Consistent) Tangent Moduli in Large Strain Computational Inelasticity," Comput. Methods Appl. Mech. Eng., 134, pp. 223-240), for an efficient numerical approximation of the tangent moduli that can be easily implemented within commercial FE codes. By perturbing the deformation gradient, the material tangent moduli from the Jaumann rate of the Kirchhoff stress are accurately approximated by a forward difference of the associated Kirchhoff stresses. The merit of this approach is that it produces a concise mathematical formulation that is not dependent on any particular material model. Consequently, once the approximation method is coded in a subroutine, it can be used for other hyperelastic material models with no modification. The implementation and accuracy of this approach is first demonstrated with a simple neo-Hookean material. Subsequently, a fiber-reinforced structural model is applied to analyze the pressure-diameter curve during blood vessel inflation. Implementation of this approach will facilitate the incorporation of novel hyperelastic material models for a soft tissue behavior into commercial FE software.

A photometric approach for estimating normals and tangents

Abstract: This paper presents a technique for acquiring the shape of real-world objects with complex isotropic and anisotropic reflectance. Our method estimates the local normal and tangent vectors at each pixel in a reference view from a sequence of images taken under varying point lighting. We show that for many real-world materials and a restricted set of light positions, the 2D slice of the BRDF obtained by fixing the local view direction is symmetric under reflections of the halfway vector across the normal-tangent and normal-binormal planes. Based on this analysis, we develop an optimization that estimates the local surface frame by identifying these planes of symmetry in the measured BRDF. As with other photometric methods, a key benefit of our approach is that the input is easy to acquire and is less sensitive to calibration errors than stereo or multi-view techniques. Unlike prior work, our approach allows estimating the surface tangent in the case of anisotropic reflectance. We confirm the accuracy and reliability of our approach with analytic and measured data, present several normal and tangent fields acquired with our technique, and demonstrate applications to appearance editing.

Does gaze influence steering around a bend

Abstract: M. F. Land and D. N. Lee (1994) suggested that steering around a bend is controlled through the estimation of curvature using the visual direction of a single road feature: the tangent point. The aim of this study was to evaluate, using a simulated environment, whether the high levels of tangent point fixation reported by some researchers are indeed related to steering control. In the first experiment, gaze patterns were examined when steering along roadways of varying widths and curvatures. Experiment 2 investigated the effects of enforced fixation on steering, when gaze was directed to the road ahead at a range of lateral eccentricities, including the tangent point. All participants completed both experiments. Overall, there was no evidence for extensive tangent point fixation in the free-gaze experiment and enforced tangent point fixation did not result in more accurate steering. The present results seem to suggest that participants tend to steer in the direction of their gaze; hence, looking at the tangent point causes the driver to steer toward it. These results provide some support for the R. M. Wilkie and J. P. Wann (2002) model of steering, which proposes that drivers will direct their gaze toward points they wish to pass through.

Strategy for selecting representative pointsviatangent spheres in the probability density evolution method

Abstract: A strategy of selecting efficient integration points via tangent spheres in the probability density evolution method (PDEM) for response analysis of non-linear stochastic structures is studied. The PDEM is capable of capturing instantaneous probability density function of the stochastic dynamic responses. The strategy of selecting representative points is of importance to the accuracy and efficiency of the PDEM. In the present paper, the centers of equivalent non-overlapping tangent spheres are used as the basis to construct a representative point set. An affine transformation is then conducted and a hypersphere sieving is imposed for spherically symmetric distributions. Construction procedures of centers of the tangent spheres are elaborated. The features of the point sets via tangent spheres, including the discrepancy and projection ratio, are observed and compared with some other typical point sets. The investigations show that the discrepancies of the point sets via tangent spheres are in the same order of magnitude as the point sets by the number theoretical method. In addition, it is observed that rotation transformation could greatly improve the projection ratios. Numerical examples show that the proposed method is accurate and efficient for situations involving up to four random variables. Copyright © 2007 John Wiley & Sons, Ltd.

End-Point Equations and Regularity of Sub-Riemannian Geodesics

Abstract: For a large class of equiregular sub-Riemannian manifolds, we show that length-minimizing curves have no corner-like singularities. Our first result is the reduction of the problem to the homogeneous, rank-2 case, by means of a nilpotent approximation. We also identify a suitable condition on the tangent Lie algebra implying existence of a horizontal basis of vector fields whose coefficients depend only on the first two coordinates x1, x2. Then, we cut the corner and lift the new curve to a horizontal one, obtaining a decrease of length as well as a perturbation of the end-point. In order to restore the end-point at a lower cost of length, we introduce a new iterative construction, which represents the main contribution of the paper. We also apply our results to some examples.

Dense and nearly jammed random packings of freely jointed chains of tangent hard spheres.

Abstract: Dense packings of freely jointed chains of tangent hard spheres are produced by a novel Monte Carlo method. Within statistical uncertainty, chains reach a maximally random jammed (MRJ) state at the same volume fraction as packings of single hard spheres. A structural analysis shows that as the MRJ state is approached (i) the radial distribution function for chains remains distinct from but approaches that of single hard sphere packings quite closely, (ii) chains undergo progressive collapse, and (iii) a small but increasing fraction of sites possess highly ordered first coordination shells.

Time-dependent fibre reorientation of transversely isotropic continua—Finite element formulation and consistent linearization

Abstract: Transverse isotropy is realized by one characteristic direction—for instance, the fibre direction in fibre-reinforced materials.Commonly, the characteristic direction is assumed to be constant, but in some cases—for instance, in the constitutive description of biological tissues, liquid crystals, grain orientations within polycrystalline materials or piezoelectric materials, as well as in optimization processes—it proves reasonable to consider reorienting fibre directions. Various fields can be assumed to be the driving forces for the reorientation process, for instance, mechanical, electric or magnetic fields. In this work, we restrict ourselves to reorientation processes in hyper-elastic materials driven by principal stretches. The main contribution of this paper is the algorithmic implementation of the reorientation process into a finite element framework. Therefore, an implicit exponential update of the characteristic direction is applied by using the Rodriguez formula to express the exponential term. The non-linear equations on the local and on the global level are solved by means of the Newton–Raphson scheme. Accordingly, the local update of the characteristic direction and the global update of the deformation field are consistently linearized, yielding the corresponding tangent moduli. Through implementation into a finite element code and some representative numerical simulations, the fundamental characteristics of the model are illustrated. Copyright © 2007 John Wiley & Sons, Ltd.

Approximation of an open polygonal curve with a minimum number of circular arcs and biarcs

Abstract: We present an algorithm for approximating a given open polygonal curve with a minimum number of circular arcs. In computer-aided manufacturing environments, the paths of cutting tools are usually described with circular arcs and straight line segments. Greedy algorithms for approximating a polygonal curve with curves of higher order can be found in the literature. Without theoretical bounds it is difficult to say anything about the quality of these algorithms. We present an algorithm which finds a series of circular arcs that approximate the polygonal curve while remaining within a given tolerance region. This series contains the minimum number of arcs of any such series. Our algorithm takes O(n^2logn) time for an original polygonal chain with n vertices. Using a similar approach, we design an algorithm with a runtime of O(n^2logn), for computing a tangent-continuous approximation with the minimum number of biarcs, for a sequence of points with given tangent directions.

Canard cycles in the presence of slow dynamics with singularities

Abstract: We study the cyclicity of limit periodic sets that occur in families of vector flelds of slow-fast type. The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point, the dynamics point strongly towards the curve of singularities, on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially-strong repulsion near this curve. All existing results deal with a non-zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities. In this paper we study what happens when the slow dynamics exhibit singularities. In particular our study includes the cyclicity of the slow-fast 2-saddle cycle, formed by a regular saddle-connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow divergence integral. This paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow-fast vector flelds on a 2-manifold M. We are interested in families of vector flelds X" (possibly depending on other parameters as well) where the unperturbed \fast" vector fleld X0 has a curve of singular points ∞, called a critical curve. We call a point p on ∞ normally attracting (resp. normally repelling) when DX0(p) has a strictly negative (resp. strictly positive) eigenvalue corresponding to an eigendirection not tangent to ∞. When ∞ has both normally attracting points and normally repelling points it may occur that X0 has orbits connecting two such points. Let F be such a fast orbit so that the !-limit and fi-limit lie on ∞. We study the limit periodic set LF formed by F and the piece of ∞ going from the !-limit of F to the fi-limit of F. Of course the nonzero eigenvalue of DX0(p) must bifurcate along this piece of ∞; assume that this happens in a unique point p⁄, called a turning point. (One also says normal hyperbolicity of X0 is lost in p⁄.) In such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits; such orbits are called canard cycles. ‡ @’

Making the Connection: Layers of Abstraction: Theory and Design for the Instruction of Limit Concepts

Abstract: Imagine asking a first-semester calculus student to explain the definition of the derivative using the epsilon-delta definition of a limit Given the difficulty of each of these concepts for students in such a course, you might not be surprised at the array of confused responses generated by a question requiring understanding of both Since the central ideas in calculus are defined in terms of limits, research on students' understanding of limits and the ways in which they can develop more powerful ways of reasoning about them has significant implications for instructional design Throughout this paper we will focus on calculus courses intended as an appropriate introduction for students who have never seen limits or derivatives and that are not intended to be a rigorous treatment of analysis The following typical response to the question relating the definitions of limit and the derivative illustrates the confusion that students exhibit when trying to make such connections This response was offered by an A-student, who we will call Bob, during a clinical interview late in a first-semester course: Your epsilon — this — the slope of this tangent line You want to pick a set of x 's, and that's here [ points at graph ] This x , it's barely changing such that it's equal to or less than this tangent line That would be your delta The slope — oh, OK The slope of this tangent line [ points at tangent ] — that's epsilon […]

Computation of medial axis and offset curves of curved boundaries in planar domain

Abstract: In this paper, we begin our research from the generating theory of the medial axis. The normal equidistant mapping relationships between two boundaries and its medial axis have been proposed based on the moving Frenet frames and Cesaro's approach of the differential geometry. Two pairs of adjoint curves have been formed and the geometrical model of the medial axis transform of the planar domains with curved boundaries has been established. The relations of position mapping, scale transform and differential invariants between the curved boundaries and the medial axis have been investigated. Based on this model, a tracing algorithm for the computation of the medial axis has been generated. In order to get the accurate medial axis and branch points, a Two_Tangent_Points_Circle algorithm and a Three_Tangent_Points_Circle algorithm have been generated, which use the results of the tracing algorithm as the initial values to make the iterative process effective. These algorithms can be used for the computation of the medial axis effectively and accurately. Based on the medial axis transform and the envelope theory, the trimmed offset curves of curved boundaries have been investigated. Several numerical examples are given at the end of the paper.

Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis

Abstract: The tangent line is a central concept in many mathematics and science courses. In this paper we describe a model of students’ thinking – concept images as well as ability in symbolic manipulation – about the tangent line of a curve as it has developed through students’ experiences in Euclidean Geometry and Analysis courses. Data was collected through a questionnaire administered to 196 Year 12 students. Through Latent Class Analysis, the participants were classified in three hierarchical groups representing the transition from a Geometrical Global perspective on the tangent line to an Analytical Local perspective. In the light of this classification, and through qualitative explanations of the students’ responses, we describe students’ thinking about tangents in terms of seven factors. We confirm the model constituted by these seven factors through Confirmatory Factor Analysis.

Possible adaptive control by tangent hyperbolic fixed point transformations used for controlling the - 6 -type van der pol oscillator

Abstract: In this paper a further step towards a novel approach to adaptive nonlinear control developed at Budapest Tech in the past few years is reported. Its main advantage in comparison with the complicated Lyapunov function based techniques is that its fundament is some simple geometric consideration allowing to formulate the control task as a Fixed Point Problem for the solution of which various Contractive Mappings can be created that generate Iterative Cauchy Sequences for Single Input - Single Output (SISO) systems. These sequences can converge to the fixed points that are the solutions of the control tasks. Recently alternative potential solutions were proposed and sketched by the use of special functions built up of the "response function" of the excited system under control. These functions have almost constant values apart from a finite region in which they have a "wrinkle" in the vicinity of the desired solution that is the "proper" fixed point of these functions. It was shown that at one of their sides these fixed points were repulsive, while at the opposite side they were attractive. It was shown, too, that at the repulsive side another, so called "false" fixed points were present that were globally attractive, with the exception of the basins of attraction of the "proper" ones. This structure seemed to be advantageous because no divergences could occur in the iterations, the convergence to the "false" values could easily be detected, and by using some ancillary tricks in the most of the cases the solutions could be kicked from the wrong fixed points into the basins of attraction of the "proper ones". It was expected that via adding simple rules to the application of these transformations good adaptive control can be developed. However, due to certain specialties of these functions practical problems arose. In the present paper novel transformations are presented that seem to evade these difficulties. Their applicability is illustrated via simulations in the ad

Constrained design of polynomial surfaces from geodesic curves

Abstract: In a recent work, Wang et al. [Wang G, Tang K, Tai CH. Parametric representation of a surface pencil with common spatial geodesic. Computer-Aided Design 2004;36(5): 447-59] discuss a constrained design problem appearing in the textile and shoe industry for garment design. Given a model and size, the characteristic curve called girth is usually fixed, and preferably should be a geodesic for manufacturing reasons. The designer must preserve this girth, being allowed to modify other areas according to aesthetic criteria. We present a practical method to construct polynomial surfaces from a polynomial geodesic or a family of geodesics, by prescribing tangent ribbons. Differently from previous procedures, we identify the existing degrees of freedom in terms of control points, and our method yields parametric polynomial surfaces that can be incorporated into commercial CAD programs. The extension to rational geodesics is also outlined.

Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods

Abstract: The paper deals with an optimal control problem with a scalar first-order state constraint and a scalar control. In presence of (nonessential) touch points, the arc structure of the trajectory is not stable. Under some reasonable assumptions, we show that boundary arcs are structurally stable, and that touch point can either remain so, vanish or be transformed into a single boundary arc. Assuming a weak second-order optimality condition (equivalent to uniform quadratic growth), stability and sensitivity results are given. The main tools are the study of a quadratic tangent problem and the notion of strong regularity. Those results enable us to design a new continuation algorithm, presented at the end of the paper, that handles automatically changes in the structure of the trajectory.

Analytical approach to initiation of propagating fronts.

Abstract: We consider the problem of initiation of a propagating wave in a one-dimensional bistable or excitable fiber. In the Zeldovich-Frank-Kamenetskii equation, also known as the Nagumo equation and Schlogl model, the key role is played by the "critical nucleus" solution whose stable manifold is the threshold surface separating initial conditions leading to the initiation of propagation and decay. An approximation of this manifold by its tangent linear space yields an analytical criterion of initiation which is in good agreement with direct numerical simulations.

A multi-body system approach for finite-element modelling of rail flexibility in railroad vehicle applications

Abstract: In this investigation, a non-linear finite-element formulation for modelling the rail structural flexibility in multi-body railroad vehicle systems is presented. Two different types of interpolations are used in the kinematic equations developed in this study; the geometry interpolation and the deformation interpolation. In the proposed formulation, the rails can have arbitrary geometry, which is described using the isoparametric geometric interpolation. The coordinates of the polynomials used in this interpolation represent constant position and gradient coordinates, which can be used to describe accurately the rail geometry. On the other hand, the rail deflections are described using the deformation interpolation and the non-linear finite-element floating frame of reference formulation. In the formulation proposed in this investigation, the rail tangent and normal vectors as well as other geometric parameters such as the curvature and torsion at the wheel/rail contact points are expressed in terms of th...

Necessary and sufficient conditions for viability for semilinear differential inclusions

Abstract: Given a set A" in a Banach space X, we define: the tangent set, and the quasi-tangent set to K atC G K, concepts more general than the one of tangent vector introduced by Bouligand (1930) and Severi (1931). Both notions prove very suitable in the study of viability problems referring to differential inclusions. Namely, we establish several new necessary, and even necessary and sufficient conditions for viability referring to both differential inclusions and semilinear evolution inclusions, conditions expressed in terms of the tangency concepts introduced.

Measuring the roughness of random paths by increment ratios

Abstract: A statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and multiplicative trends and applicability to infinite variance processes. The existence of the IR statistic limit (called the IR-roughness below) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. Firstly, for a diffusion process with smooth diffusion and drift coefficients, the IR-roughness coincides with the IR-roughness of a Brownian motion and its convergence rate is obtained. Secondly, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Thirdly, the IR-roughness of a Levy process with $\alpha-$stable tangent process is established and can be used to estimate the fractional parameter $\alpha \in (0,2)$ following a central limit theorem.

Sweeping solids on manifolds

Abstract: This work describes a new method for modeling of sweep solids on manifolds, considering various geometric and functional constrains. The proposed method is applied in semi-automatic computer aided design of ventilation tubes for hearing aid devices. The sweeping procedure begins with definition of a trajectory. Besides smoothness and minimal length, other requirements may be considered. Therefore, it is convenient to formulate the optimal trajectory problem as a geodesic computing over Riemannian manifold. The trajectory defined on the manifold is ofsetted, in order to make the sweep solid tangent to the manifold. The offset curve shape is iteratively smoothed while preserving minimal distance from the manifold. Then, a frame field is defined over the offset curve and the cross section contour is transformed according to this field. The major problem is how to construct the frame field such that the resulting sweep solid will be smooth and free of self-intersections. It is well known, that Frenet frame imposes restrictions on the trajectory and may create undesirable twist. In order to overcome these obstacles, an efficient procedure is proposed to compute the discrete minimal rotation frame. Finally, a new approach to the self-intersection problem of sweep solids is proposed. The key idea is to weaken the orthogonality requirement between the cross section plane and the trajectory curve, in order to avoid self-intersections.The described method was implemented and tested in real production environment, where it was proved robust and efficient. The proposed techniques can be utilized in many related applications where sweep surface modeling and manipulation is involved.

Free-Form Object Reconstruction from Silhouettes, Occluding Edges and Texture Edges: A Unified and Robust Operator Based on Duality

Abstract: In this paper, the duality in differential form is developed between a 3D primal surface and its dual manifold formed by the surface's tangent planes, that is, each tangent plane of the primal surface is represented as a four-dimensional vector that constitutes a point on the dual manifold. The iterated dual theorem shows that each tangent plane of the dual manifold corresponds to a point on the original 3D surface, that is, the "dual" of the "dual" goes back to the "primal." This theorem can be directly used to reconstruct 3D surface from image edges by estimating the dual manifold from these edges. In this paper, we further develop the work in our original conference papers resulting in the robust differential dual operator. We argue that the operator makes good use of the information available in the image data by using both points of intensity discontinuity and their edge directions; we provide a simple physical interpretation of what the abstract algorithm is actually estimating and why it makes sense in terms of estimation accuracy; our algorithm operates on all edges in the images, including silhouette edges, self occlusion edges, and texture edges, without distinguishing their types (thus, resulting in improved accuracy and handling locally concave surface estimation if texture edges are present); the algorithm automatically handles various degeneracies; and the algorithm incorporates new methodologies for implementing the required operations such as appropriately relating edges in pairs of images, evaluating and using the algorithm's sensitivity to noise to determine the accuracy of an estimated 3D point. Experiments with both synthetic and real images demonstrate that the operator is accurate, robust to degeneracies and noise, and general for reconstructing free-form objects from occluding edges and texture edges detected in calibrated images or video sequences.

A 9-node co-rotational quadrilateral shell element

Abstract: A new 9-node co-rotational curved quadrilateral shell element formulation is presented in this paper. Different from other existing co-rotational element formulations: (1) Additive rotational nodal variables are utilized in the present formulation, they are two well-chosen components of the mid-surface normal vector at each node, and are additive in an incremental solution procedure; (2) the internal force vector and the element tangent stiffness matrix are respectively the first derivative and the second derivative of the element strain energy with respect to the nodal variables, furthermore, all nodal variables are commutative in calculating the second derivatives, resulting in symmetric element tangent stiffness matrices in the local and global coordinate systems; (3) the element tangent stiffness matrix is updated using the total values of the nodal variables in an incremental solution procedure, making it advantageous for solving dynamic problems. Finally, several examples are solved to verify the reliability and computational efficiency of the proposed element formulation.

Stationary Configurations of a Tetrahedral Tethered Satellite Formation

Abstract: O RBITAL dynamics analysis of a connected multibody system is relevant for several missions, including tethered systems, space robots and manipulators, telescopes, space stations, etc. It is important to determine stationary motions of such structures due to their possible use as nominal motions in energy-saving mode. The study of the behavior of multibody systems in the orbital environment continues to grow since its beginning in the 1960s. Sarychev [1] and Wittenburg [2] investigate the equilibria of two connected rigid bodies in circular orbit with respect to the orbital reference frame. Cheng and Liu [3] consider the in-plane dynamics of a three-body systemwith two equal extreme bodies, examining the equilibrium orientations, bifurcations, and stability with respect to in-plane perturbations. Lavagna and Ercoli Finzi describe planar equilibrium configurations [4] and study their stability [5] for an extended rigid body with two pendulums attached either in a sequence or in a parallel scheme. Quite frequently, the orbital dynamics of multibody systems is studied for models that approximate connected satellites by an open chain of material points linked by weightless straight rods with spherical hinges. Misra and Modi [6] develop the general threedimensional formulation for an n-link chain. For twoand three-link systems, they determine the libration frequencies and study the control laws. In [7,8] this model is used to determine in-plane equilibrium configurations of a two-link chain and to study their stability. Sarychev [9] finds all spatial equilibrium orientations of a two-link chain in a circular orbit and describes their number as a function of problem parameters. Continued interest in such studies was confirmed recently by Correa and Gomez [10], who numerically study the equilibrium configurations of a three-link chain in the plane of the orbit and analyze their stability with respect to in-plane perturbations. An analytical study of chains with arbitrary number of links can be found in [11,12]. The center ofmass of the chain (CMC)moves along a circular orbit. All in-plane equilibrium configurations are described in [11]. All spatial equilibria are listed in [12], in which it is shown that each connecting rod can be oriented with respect to the tangent, normal and binormal to the CMC orbit in one of the following ways: 1) A rod is aligned with the tangent axis. 2) A rod belongs to an NBN group, that is, to a subchain of rods that lies in the plane parallel to normal and binormal so that its center of mass is on the tangent axis. 3) A rod links two NBN groups or an end of an NBN group with the tangent axis. Some of these equilibrium configurations are actually twodimensional, with all the system lying in one of the coordinate planes of the orbital reference frame. Yet, there exist essentially threedimensional configurations, for example, with masses placed at the vertices of a tetrahedron. Tetrahedral satellite formations are of significant interest due to numerous applications and many projects under development, including NASA research programs in formation flying [13–15]. Tetrahedral configuration has been successfully implemented in the Cluster mission to study three-dimensional structure of the Earth’s magnetosphere [16]. Being the simplest spatial configuration, a tetrahedral formation is a natural tool in experiments of this kind, because it enables one to execute simultaneous measurements at points of a large-span three-dimensional basis. In the present article, we apply the general results of [12] to study tetrahedral equilibrium configurations of a tethered satellite system. We identify the spectrum of spatial equilibrium configurations that can be achieved by varying the masses of the bodies and lengths of the rods.We determine the links that can be replacedwith tethers.We study the possibility of controlling such a system and of stabilizing its orientation.

Parabolic curves for diffeomorphisms in C2

Abstract: We give a simple proof of the existence of parabolic curves for diffeomorphisms in (C 2 , 0) tangent to the identity with isolated fixed point