Showing papers on "Tangent published in 2002"

Graphic and numerical comparison between iterative methods

Abstract: generates a sequence {xn}n=0 that converges to ζ. In fact, Newton’s original ideas on the subject, around 1669, were considerably more complicated. A systematic study and a simplified version of the method are due to Raphson in 1690, so this iteration scheme is also known as the Newton-Raphson method. (Also as the tangent method, from its geometric interpretation.) In 1879, Cayley tried to use the method to find complex roots of complex functions f : C → C. If we take z0 ∈ C and we iterate

Hyperchaotic behaviour of two bi‐directionally coupled Chua's circuits

Abstract: In this paper, a non-linear bi-directional coupling of two Chua's circuits is presented. The coupling is obtained by using polynomial functions that are symmetric with respect to the state variables of the two Chua's circuits. Both a transverse and a tangent system are studied to ensure a global validity of the results in the state space. First, it is shown that the transverse system is an autonomous Chua's circuit, which directly allows the evaluation of the conditions on its chaotic behaviour, i.e. the absence of synchronization between the coupled circuits. Moreover, it is demonstrated that the tangent system is also a Chua's circuit, forced by the transverse system; therefore, its dynamics is ruled by a time-dependent equation. Thus, the calculus of conditional Lyapunov exponents is necessary in order to exclude antisynchronization along the tangent manifold. The properties of the transverse and tangent systems simplify the study of the coupled Chua's circuits and the determination of the conditions on their hyperchaotic behaviour. In particular, it is shown that hyperchaotic behaviour occurs for proper values of the coupling strength between the two Chua's circuits. Finally, numerical examples are given and discussed. Copyright © 2002 John Wiley & Sons, Ltd.

Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: Rigorous nonextensive solutions

Abstract: Using the Feigenbaum renormalization group (RG) transformation we work out exactly the dynamics and the sensitivity to initial conditions for unimodal maps of nonlinearity ζ > 1 at both their pitchfork and tangent bifurcations. These functions have the form of q-exponentials as proposed in Tsallis' generalization of statistical mechanics. We determine the q-indices that characterize these universality classes and perform for the first time the calculation of the q-generalized Lyapunov coefficient λq. The pitchfork and the left-hand side of the tangent bifurcations display weak insensitivity to initial conditions, while the right-hand side of the tangent bifurcations presents a super-strong (faster than exponential) sensitivity to initial conditions. We corroborate our analytical results with a priori numerical calculations.

Integrable systems in three-dimensional Riemannian geometry

Abstract: In this paper we introduce a new infinite-dimensional pencil of Hamiltonian structures. These Poisson tensors appear naturally as the ones governing the evolution of the curvatures of certain flows of curves in 3-dimensional Riemannian manifolds with constant curvature. The curves themselves are evolving following arclength-preserving geometric evolutions for which the variation of the curve is an invariant combination of the tangent, normal, and binormal vectors. Under very natural conditions, the evolution of the curvatures will be Hamiltonian and, in some instances, bi-Hamiltonian and completely integrable.

Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: rigorous nonextensive solutions

Abstract: Using the Feigenbaum renormalization group (RG) transformation we work out exactly the dynamics and the sensitivity to initial conditions for unimodal maps of nonlinearity $\zeta >1$ at both their pitchfork and tangent bifurcations. These functions have the form of $q$-exponentials as proposed in Tsallis' generalization of statistical mechanics. We determine the $q$-indices that characterize these universality classes and perform for the first time the calculation of the $q$-generalized Lyapunov coefficient $\lambda_{q} $. The pitchfork and the left-hand side of the tangent bifurcations display weak insensitivity to initial conditions, while the right-hand side of the tangent bifurcations presents a `super-strong' (faster than exponential) sensitivity to initial conditions. We corroborate our analytical results with {\em a priori} numerical calculations.

Tangent Fields and the Local Structure of Random Fields

Abstract: A tangent field of a random field X on ℝN at a point z is defined to be the limit of a sequence of scaled enlargements of X about z. This paper develops general properties of tangent fields, emphasising their rich structure and strong invariance properties which place considerable constraints on their form. The theory is illustrated by a variety of examples, both of a smooth and fractal nature.

Consistent tangent operators for constitutive rate equations

Abstract: A general approach for obtaining the consistent tangent operator for constitutive rate equations is presented. The rate equations can be solved numerically by the user's favourite time integrator. In order to obtain reliable results, the substepping in integration should be based on a control of the local error. The main ingredient of the consistent tangent operator, namely the derivative of the stress with respect to the strain increment must be computed simultaneously with the same integrator, applied to a numerical approximation of the variational equations. This information enables finite-element packages to assemble a consistent tangent operator and thus guarantees quadratic convergence of the equilibrium iterations. Several numerical examples with a hypoplastic constitutive law are given. As numerical integrator we used a second-order extrapolated Euler method. Quadratic convergence of the equilibrium iteration is shown. Copyright © 2002 John Wiley & Sons, Ltd.

The influence of limit cycle topology on the phase resetting curve

Abstract: Understanding the phenomenology of phase resetting is an essential step toward developing a formalism for the analysis of circuits composed of bursting neurons that receive multiple, and sometimes overlapping, inputs. If we are to use phase-resetting methods to analyze these circuits, we can either generate phase-resetting curves (PRCs) for all possible inputs and combinations of inputs, or we can develop an understanding of how to construct PRCs for arbitrary perturbations of a given neuron. The latter strategy is the goal of this study.We present a geometrical derivation of phase resetting of neural limit cycle oscillators in response to short current pulses. A geometrical phase is defined as the distance traveled along the limit cycle in the appropriate phase space. The perturbations in current are treated as displacements in the direction corresponding to membrane voltage. We show that for type I oscillators, the direction of a perturbation in current is nearly tangent to the limit cycle; hence, the projection of the displacement in voltage onto the limit cycle is sufficient to give the geometrical phase resetting. In order to obtain the phase resetting in terms of elapsed time or temporal phase, a mapping between geometrical and temporal phase is obtained empirically and used to make the conversion. This mapping is shown to be an invariant of the dynamics. Perturbations in current applied to type II oscillators produce significant normal displacements from the limit cycle, so the difference in angular velocity at displaced points compared to the angular velocity on the limit cycle must be taken into account. Empirical attempts to correct for differences in angular velocity (amplitude versus phase effects in terms of a circular coordinate system) during relaxation back to the limit cycle achieved some success in the construction of phase-resetting curves for type II model oscillators. The ultimate goal of this work is the extension of these techniques to biological circuits comprising type II neural oscillators, which appear frequently in identified central pattern-generating circuits.

Curvature Measures of 3D Vector Fields and their Applications

Abstract: Tangent curves are a powerful tool for analyzing and visualizing vector fields In this paper two of their most important properties are examined: their curvature and torsion Furthermore, the conc

Fair triangle mesh generation with discrete elastica

Abstract: Surface fairing, generating free-form surfaces satisfying aesthetic requirements, is important for many computer graphics and geometric modeling applications. A common approach for fair surface design consists of minimization of fairness measures penalizing large curvature values and curvature oscillations. The paper develops a numerical approach for fair surface modeling via curvature-driven evolutions of triangle meshes. Consider a smooth surface each point of which moves in the normal direction with speed equal to a function of curvature and curvature derivatives. Chosen the speed function properly, the evolving surface converges to a desired shape minimizing a given fairness measure. Smooth surface evolutions are approximated by evolutions of triangle meshes. A tangent speed component is used to improve the quality of the evolving mesh and to increase computational stability. Contributions of the paper include also art improved method for estimating the mean curvature.

Data Approximation Using Biarcs

Abstract: . An algorithm for data approximation with biarcs is presented. The method uses a specific formulation of biarcs appropriate for parametric curves in Bezier or NURBS formulation. A base curve is applied to obtain tangents and anchor points for the individual arcs joining in G 1 continuity. Data sampled from circular arcs or straight line segments is represented precisely by one biarc. The method is most useful in numerical control to drive the cutter along straight line or circular paths.

Solutions of fully nonlinear elliptic equations with patches of zero gradient: Existence, regularity and convexity of level curves

Abstract: In this paper, we first construct viscosity solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form F(D 2 u, x) = g(x, u) on {|⊇u| ¬= 0} In fact, viscosity solutions are surprisingly weak. Since candidates for solutions are just continuous, we only require that the test polynomials P (those tangent from above or below to the graph of u at a point x 0 ) satisfy the correct inequality only if |⊇P(x 0 )| ¬= 0. That is, we simply disregard those test polynomials for which |⊇P(x 0 )| = 0. Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for F = Δ, g = cu (c > 0) and constant boundary conditions on a convex domains, to prove that there is only one convex patch.

Fast reconstruction of curves with sharp corners

Abstract: We present an algorithm to reconstruct a collection of piecewise smooth simple closed curves in the plane from a set of n sample points in O(n log n) time We prove our algorithm correctly reconstructs the curves assuming certain sampling conditions which are based on the minimum angle made by tangents at any corner point but does not include any assumptions about the uniformity of the sampling.

Gamma-Lines: On the Geometry of Real and Complex Functions

Abstract: Preface. Tangent Variation Principle. Satellite Principles. Modification of Length-area Principle. Tangent Variation Principle. Estimates for collections of Gamma-Lines. Estimates of lengths of Gamma-Lines for angular-quasiconformal mappings. Remarks on applying of estimates of L (D, Gamma). Nevanlinna and Ahilfors Theories. Additions. Basic concepts and outcomes of Nevanlinna Value Distribution theory and Ahlfors theory of covering surfaces. Geometric deficient values. On some additions to L. Ahlfor's theory of covering surfaces. Bounds of some integrals. Gamma-Lines Approach in the Theory of Meromorphic Functions. Principle of closeness of sufficiently large sets of Alpha-points of meromorphic functions. Integrated Version of the Principle. Connections with known classes of functions. Distribution of Gamma-Lines for Functions Meromorphic in C. Applications. The main results on distribution of Gamma-Lines. "Wingdings" of Gamma-Lines. Average lengths of Gamma-Lines along concentric circles and the deficient values. Distribution of Gamma-Lines and value distribution of subclasses of modules and real parts of mermorphic functions. The number of Gamma-Lines crossing rings. Distribution of Gelfond points. Nevalinna's dream-description of transcendental ramification of Riemann surfaces. The proximity property of Alpha-points of meromorphic functions. A proof of the proximity property of Alpha-points based only on investigation of Gamma-Lines. Some Applied Problems. Gamma-Lines in Physics. On the cross road of value distribution, Gamma-Lines, free boundary theories and applied mathematics. "Pointmaps" of physical processes and Alpha-points of general classes of functions Principles. Nevanlinna and Ahilfors Theories. Additions. Gamma-Lines Approach in the Theory of Meromorphic Functions. Distribution of Gamma-Lines for Functions Mermorphic in C. Applications. Some Applied Problems.

Modelling of rc members under cyclic biaxial flexure and axial force

Abstract: A model is proposed for the incremental force-deformation behaviour of reinforced concrete sections and members, under generalised load or deformation histories in 3D, including cyclic loading, up to ultimate deformation. At the section level the model is of the Bounding Surface type and accounts for the coupling between the two directions of bending and between them and the axial direction. For the construction of the member tangent flexibility matrix on the basis of the section tangent flexibility matrix, a piecewise-linear variation along the member is assumed for the nine terms of the tangent section flexibility matrix. Model parameters are derived on the basis of available test results for: (a) the force-deformation response under cyclic biaxial bending with normal force; (b) the hysteretic energy dissipation; (c) the secant-to-yield member stiffness, and (d) the ultimate deformation of the member under cyclic biaxial load paths.

Lines tangent to 2-2 spheres in ℝⁿ

Abstract: We show that for n ≥ 3 there are 3.2 n-1 complex common tangent lines to 2n - 2 general spheres in R n and that there is a choice of spheres with all common tangents real.

Tangent height registration for the solar occultation satellite sensor ILAS: A new technique for Version 5.20 products

Abstract: [1] The Improved Limb Atmospheric Spectrometer (ILAS) was a solar occultation satellite sensor that was developed by the Environment Agency of Japan to monitor the stratospheric ozone layer. This paper describes methods of registering tangent heights for ILAS vertical profiles of gas mixing ratio and aerosol extinction coefficient. Accurate tangent height registration is crucial for the retrieval of accurate gas mixing ratios from atmospheric absorption spectra. Three methods for tangent height registration have been applied to retrieved ILAS data. The first method is the transmittance spectrum method (TS-M), which uses absorption spectra of oxygen molecules at around 760 nm (O2A band) measured by the ILAS visible channel and compares the average transmittance with that calculated theoretically from temperature and pressure using meteorological data. A tangent height is derived from these data. Version 3.10 ILAS data products use the TS-M. A second method is the Sun-edge sensor method (SES-M). This method for registering tangent heights was in mind when ILAS was originally designed. The SES-M geometrically determines the direction of the instantaneous field-of-view (IFOV) of the spectrometer from the angular difference between the top edge of the Sun determined with the SES and the spectrometer's IFOV. Information on the satellite's orbital position and solar position relative to the center of the Earth is used to register the tangent height. Version 4.20 ILAS data products use SES-M. The third method is a hybrid method (Hybrid-M) that was developed to correct for seasonal differences in tangent heights computed by the SES-M. The Hybrid-M assumes that the TS-M can correctly determine the tangent height at 30 km. Version 5.20 ILAS data products (the latest version) use Hybrid-M. Random and systematic errors in the Hybrid-M tangent height registration were estimated. The root-sum-square (RSS) total random error is 30 m, while the total systematic error is +300 ± 360 m in tangent height. Actual errors in tangent height registration in Version 5.20 algorithm are considered to be small judging from the results of comparisons with independent validation data sets. The Hybrid-M gives good estimates of tangent heights in Version 5.20 of the ILAS data processing algorithms.

Coupled Anisotropic Shear-wave Ray Tracing in Situations where Associated Slowness Sheets Are Almost Tangent

Abstract: Forshear waves in anisotropic media a frequency-dependent transport equation is derived, which involves coupling of both shear modes if the slowness vector points in a direction for which the associated slowness surfaces are (almost) tangent. This is achieved by averaging the Hamiltonians for both uncoupled shear modes. No assumption on weak anisotropy is made, and the shear polarisation is perpendicular to the P-wave polarisation for the actual slowness direction instead of an isotropically approximated P-wave polarisation. These are the main differences compared with the so-called zero-order quasi-isotropic approach.

Three-Dimensional Vibration Analysis of Paraboloidal Shells

Abstract: A method of analysis is presented for determining the free vibration frequencies and mode shapes of open paraboloidal shells of revolution having arbitrary thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell, as well as the inner and outer curved surfaces, may be free or may be subjected to any degree of constraint. The strain energy of deformation, as well as the kinetic energy of motion, are formulated in terms of three displacement components which are tangent or normal to the shell middle surface. The displacements are taken as periodic in the circumferential coordinate and in time, and as polynomials of arbitrary degree in the other two coordinates, and the Ritz method is used to formulate the eigenvalue problem. Convergence studies are presented, and frequencies are given for moderately thick and thick, moderately deep and deep, paraboloidal shells of uniform and variable thickness.

Method and apparatus for determining prices that maximize revenue

Abstract: A method and apparatus for determining one or more price that maximizes revenue for a given demand curve. For determining a single price that maximizes revenue, a geometric analysis of the demand curve is used to find a price that gives a first angle that is equal to a second angle. The first [103] and second [104] angles are calculated using a tangent line that is tangent to the demand curve at the price and reference lines that pass through the demand curve at the price. When the first angle is equal to the second angle [105], the determined price optimizes revenue for the product represented by the demand curve. For determining multiple prices that maximize revenue, a geometric calculation is performed using a first price [504] so as to determine additional prices. The geometric error that is associated with the first price and the additional prices is determined [505] and the first price is changed until a first price and additional prices are found that minimize the geometric error [506]. In another embodiment, a method and apparatus are disclosed for determining prices that maximize revenue for multiple products.

Application of the ψ-s curve to road geometry extraction and modeling

Abstract: A method for determining the line segments, circular arcs, and clothoidal arcs that form a complex curve along a length thereof is disclosed. A ψ-s curve of the complex curve is determined, which is a plot of tangent angles, wherein the angle is made with a fixed line by a tangent to the complex curve along the length thereof. The straight line portions and parabolic portions of the plot of the ψ-s curve are determined and used to determine the corresponding circular arcs and straight lines that form the complex curve and clothoidal arcs that form the complex curve, respectively. The ψ-s curve can be used to identify the curves and straight lines that define the geometry of roads and therefore can be used to store data that indicate the geometry of roads in a geographic database that contains data representing the roads.

Disk drive with head supporting device

Abstract: A disk drive comprises a recording medium, spinning means, a head supporting device having a support arm disposed in a rotatable manner about a bearing unit in a radial direction as well as a perpendicular direction to a writing surface of the recording medium, and rotating means for rotating the support arm in the radial direction of the recording medium. A shape of the housing is defined based on a first straight line tangent to both an outermost rim of the rotating means and an outer periphery of the recording medium, and according to a quadrilateral comprised of the first straight line, a second straight line tangent to the outer periphery of the recording medium and perpendicular to the first straight line, a third straight line tangent to the outermost rim of the rotating means and perpendicular to the first straight line, and a fourth straight line tangent to the outer periphery of the recording medium and parallel with the first straight line, thereby realizing the disk drive that is compatible in external size with memory cards.

Curves, circles, and spheres

Abstract: The standard radius of curvature at a point q(s) on a smooth curve can be defined as the limiting radius of circles through three points that all coalesce to q(s). In the study of ideal knot shapes it has recently proven useful to consider a global radius of curvature of the curve at q(s) defined as the smallest possible radius amongst all circles passing through this point and any two other points on the curve, coalescent or not. In particular, the minimum value of the global radius of curvature gives a convenient measure of curve thickness. Given the utility of the construction inherent to global curvature, it is also natural to consider variants of global radii of curvature defined in related ways. For example multi-point radius functions can be introduced as the radius of a sphere through four points on the curve, circles that are tangent at one point of the curve and intersect at another, etc. Then single argument, global radius of curvature functions can be constructed by minimizing over all but one argument. In this article we describe the interrelations between all possible global radius of curvature functions of this type, and show that there are two of particular interest. Properties of the divers global radius of curvature functions are illustrated with the simple examples of ellipses and helices, including certain critical helices that arise in the optimal shapes of compact filaments, in �-helical proteins, and in B-form DNA.

The Improvement Made by a Modified TLM in 4DVAR with a Geophysical Boundary Layer Model

Abstract: The strong nonlinearity of boundary layer parameterizations in atmospheric and oceanic models can cause difficulty for tangent linear models in approximating nonlinear perturbations when the time integration grows longer. Consequently, the related 4—D variational data assimilation problems could be difficult to solve. A modified tangent linear model is built on the Mellor-Yamada turbulent closure (level 2.5) for 4-D variational data assimilation. For oceanic mixed layer model settings, the modified tangent linear model produces better finite amplitude, nonlinear perturbation than the full and simplified tangent linear models when the integration time is longer than one day. The corresponding variational data assimilation performances based on the adjoint of the modified tangent linear model are also improved compared with those adjoints of the full and simplified tangent linear models.

On Pencils of Tangent Planes and the Recognition of Smooth 3D Shapes from Silhouettes

Abstract: This paper presents a geometric approach to recognizing smooth objects from their outlines We define a signature function that associates feature vectors with objects and baselines connecting pairs of possible viewpoints Feature vectors, which can be projective, affine, or Euclidean, are computed using the planes that pass through a fixed baseline and are also tangent to the object's surface In the proposed framework, matching a test outline to a set of training outlines is equivalent to finding intersections in feature space between the images of the training and the test signature functions The paper presents experimental results for the case of internally calibrated perspective cameras, where the feature vectors are angles between epipolar tangent planes

Is the world full of circles

Abstract: The statistical arrangement of oriented segments in natural scenes was recently proposed to be indicative of a cocircularity rule. In particular, the probability density function of the relative position of two oriented segments was found to be maximal along fixed angles on the plane, consistent with the two segments being tangent to two points of a circle. Does this observation point to a prevalence of circles in natural scenes? Here we demonstrate that similar statistics can be obtained even when circles are not very common in visual scenes. The reason is that circles or near circular objects can heavily skew the distribution in favor of the cocircularity rule.

The Application of Chain Code Sum in the Edge form Analysis

Abstract: This dissertation expands on a new algorithm, which has the function of analyzing edge form with the chain code. By the introduction of the concepts of relative chain code and absolute chain code, we proposed a simple and direct algorithm to compute chain code sum(average chain code). We found that tangent direction(slope) of edge point can be figured by the absolute chain code sum of three sequential points and curvature of edge can be figured by the difference of chain code sum between the three sequential points ingoing and outgoing. Furthermore, we provided the criterion distinguishing edge inflexion and sleek curve section and the method computing inaccurately curvature radius and approximate perimeter. In the end of the paper, we introduced the use method with cell edge hollow repairing and overlap or conglutination cell segmenting for example. This algorithm has proved high speed and has a good effect of cell segmention on more than twenty groups of conglutinate and absent cells which are gathered from three kinds of cells.