Showing papers on "Tangent published in 1998"

Transformation Invariance in Pattern Recognition-Tangent Distance and Tangent Propagation

Abstract: In pattern recognition, statistical modeling, or regression, the amount of data is a critical factor affecting the performance. If the amount of data and computational resources are unlimited, even trivial algorithms will converge to the optimal solution. However, in the practical case, given limited data and other resources, satisfactory performance requires sophisticated methods to regularize the problem by introducing a priori knowledge. Invariance of the output with respect to certain transformations of the input is a typical example of such a priori knowledge. In this chapter, we introduce the concept of tangent vectors, which compactly represent the essence of these transformation invariances, and two classes of algorithms, “tangent distance” and “tangent propagation”, which make use of these invariances to improve performance.

Recursively generated B-spline surfaces on arbitrary topological meshes

Abstract: This paper describes a method for recursively generating surfaces that approximate points lying-on a mesh of arbitrary topology. The method is presented as a generalization of a recursive bicubic B-spline patch subdivision algorithm. For rectangular control-point meshes, the method generates a standard B-spline surface. For non-rectangular meshes, it generates surfaces that are shown to reduce to a standard B-spline surface except at a small number of points, called extraordinary points. Therefore, everywhere except at these points the surface is continuous in tangent and curvature. At the extraordinary points, the pictures of the surface indicate that the surface is at least continuous in tangent, but no proof of continuity is given. A similar algorithm for biquadratic B-splines is also presented.

Slopes of a Receiver Operating Characteristic Curve and Likelihood Ratios for a Diagnostic Test

Abstract: This paper clarifies two important concepts in clinical epidemiology: the slope of a receiver operating characteristic (ROC) curve and the likelihood ratio. It points out that there are three types of slopes in an ROC curve--the tangent at a point on the curve, the slope between the origin and a point on the curve, and the slope between two points on the curve. It also points out that there are three types of likelihood ratios that can be defined for a diagnostic test that produces results on a continuous scale--the likelihood ratio for a particular single test value, the likelihood ratio for a positive test result, and the likelihood ratio for a test result in a particular level or category. It further illustrates mathematically and empirically the following three relations between these various definitions of slopes and likelihood ratios: 1) the tangent at a point on the ROC curve corresponds to the likelihood ratio for a single test value represented by that point; 2) the slope between the origin and a point on the curve corresponds to the positive likelihood ratio using the point as a criterion for positivity; and 3) the slope between two points on the curve corresponds to the likelihood ratio for a test result in a defined level bounded by the two points. The likelihood ratio for a single test value is considered an important parameter for evaluating diagnostic tests, but it is not easily estimable directly from laboratory data because of limited sample size. However, by using ROC analysis, the likelihood ratio for a single test value can be easily measured from the tangent. It is suggested that existing ROC analysis software be revised to provide estimates for tangents at various points on the ROC curve.

Analysis of rotating crack model

Abstract: This paper extends the standard rotating crack (RC) model to a formulation with multiple orthogonal cracks. The corresponding stress evaluation algorithm is described, and a derivation of the tangent stiffness matrix is presented. The derivation is extended to the case of equal principal strains, in which the classical formula for the tangent shear modulus fails. A condition excluding snapback of the stress-strain diagram for an arbitrary loading path is derived. Attention then shifts to stress locking, meaning here spurious stress transfer across widely opening cracks. The problem is illustrated by numerical examples. The mechanism of stress transfer is thoroughly analyzed, and the source of locking is detected. A remedy and extension of the model to a nonlocal formulation is described in a separate paper.

Connections on non-parametric statistical manifolds by Orlicz space geometry

Abstract: The non-parametric version of Information Geometry has been developed in recent years. The first basic result was the construction of the manifold structure on ℳμ, the maximal statistical models associated to an arbitrary measure μ (see Ref. 48). Using this construction we first show in this paper that the pretangent and the tangent bundles on ℳμ are the natural domains for the mixture connection and for its dual, the exponential connection. Second we show how to define a generalized Amari embedding AΦ:ℳμ→SΦ from the Exponential Statistical Manifold (ESM) ℳμ to the unit sphere SΦ of an arbitrary Orlicz space LΦ. Finally we show that, in the non-parametric case, the α-connections ∇α(α∈(-1,1)) must be defined on a suitable α-bundle ℱα over ℳμ and that the bundle-connection pair (ℱα, ∇α) is simply (isomorphic to) the pull-back of the Amari embedding Aα: ℳμ→S2/1-α were the unit sphere S2/1-αcL2/1-α is equipped with the natural connection.

Decaying Singular Vectors and Their Impact on Analysis and Forecast Correction

Abstract: The full set of kinetic energy singular values and singular vectors for the forward tangent propagator of a quasigeostrophic potential vorticity model is examined. In contrast to the fastest growing singular vectors, the fastest decaying vectors exhibit a downward and downscale transfer of energy and an eastward tilt with height. The near-neutral singular vectors resemble small-scale noise with no localized structure or coherence between levels. Post-time forecast and analysis correction techniques are examined as a function of the number of singular vectors included in the representation of the inverse of the forward tangent propagator. It is found that for the case when the forecast error is known exactly, the best corrections are obtained when using the full inverse, which includes all of the singular vectors. It is also found that the erroneous projection of the analysis uncertainty onto the fastest decaying singular vectors has a significant detrimental effect on the estimation of analysis error. Therefore, for the more realistic case where the forecast error is known imperfectly, use of the full inverse will result in an inaccurate estimate of analysis errors, and the best corrections are obtained when using an inverse composed only of the growing singular vectors. Running the tangent equations with a negative time step is a very good approximation to using the full inverse of the forward tangent propagator.

Cohomological degrees and hilbert functions of graded modules

Abstract: Making use of the recent construction of cohomological degrees functions, we give several estimates on the relationship between number of generators and degrees of ideals and modules with applications to Hilbert functions. They extend results heretofore known from generalized Cohen-Macaulay local rings to nearly arbitrary local rings. The rules of computation these functions satisfy enables comparison with Castelnuovo-Mumford's regularity in the graded case. As application, we derive sharp improvements on predicting the outcome of effecting Noether normalizations in tangent cones.

The Non-Linear Laplacian for Finsler Manifolds

Abstract: For a Finsler manifold (M,F), there is a canonical energy function E defined on the Sobolev space. The variation of E gives rises to a non-linear Laplacian. Although this Laplacian is non-linear, it has a close relationship with curvatures and other geometric quantities. There are two curvatures involved. The first one is the Ricci curvature, which is a Riemannian quantity, and the second one is the mean tangent curvature defined in [S2]. The mean tangent curvature is a non-Riemannian quantity. In this report, we shall briefly describe the recent developments in the study of this non-linear Laplacian.

Characterization of corners of curvilinear segment

Abstract: A system determines discontinuities on a drawn curve. Local tangent values for a curvilinear segment are generated by determining the most likely tangent value for all points on the segment and by acting on a sampling of points around the current point whose tangent is to be determined. A robust statistical estimator is then applied to all angle values generated for the sample set to determine the likely intended tangent direction of the current focal point. The system then selects points with significant changes in tangent values to determine curve discontinuities such that the corners of the segment can be derived.

Parametrically deformed free-form surfaces as part of a variational model

Abstract: A new approach is described which provides deformation methods for multi-patch tensor based free-form surfaces. The surface deformation generated is controlled by global geometric constraints. For example, the objective can be to deform a free-form surface until it becomes tangent to a pre-defined plane at a given point. This point can be fixed or free to slide on the surface. The parametric deformation of surfaces is dedicated to modifications of free-form surfaces within CAD software and to the design of objects submitted to aesthetic requirements. It is an alternative to previous approaches and it works with multiple surfaces through a simple mechanical model. The deformation method uses an analogy between the control polyhedron of each surface (based on a B-Spline model) and the mechanical equilibrium of a rigid bar network. The user can localize the surface deformation into an arbitrary shaped area through the selection of control polyhedron vertices spread over the entire surface. These vertices are used to automatically construct the associated bar network. The bar network equilibrium parameters are set up to achieve isotropic or anisotropic deformation as required by the designer. The surface deformation is then automatically carried through an optimization process which modifies mechanical parameters to agree with the global geometric constraint set up. The G1 continuity across the different during the deformation process using a set of geometric constraints in addition to mechanical ones. Parametric free-form surface deformation can be subjected to non-linear geometric constraints such as the tangency of a surface to a pre-defined plane. The resolution of such a problem uses an optimization process which minimizes the variation of the parameters governing the equilibrium of the bar network, namely the external forces applied to the nodes of the network. Several examples illustrate basic deformation types with various sets of constraints.

Specified–Precision Computation of Curve/Curve Bisectors

Abstract: The bisector of two plane curve segments (other than lines and circles) has, in general, no simple — i.e., rational — parameterization, and must therefore be approximated by the interpolation of discrete data. A procedure for computing ordered sequences of point/tangent/curvature data along the bisectors of polynomial or rational plane curves is described, with special emphasis on (i) the identification of singularities (tangent–discontinuities) of the bisector; (ii) capturing the exact rational form of those portions of the bisector with a terminal footpoint on one curve; and (iii) geometrical criteria the characterize extrema of the distance error for interpolants to the discretely–sample data. G1 piecewise– parabolic and G2 piecewise–cubic approximations (with O(h4) and O(h6) convergence) are described which, used in adaptive schemes governed by the exact error measure, can be made to satisfy any prescribed geometrical tolerance.

On stagnation points and streamline topology in vortex flows

Abstract: The problem of locating stagnation points in the flow produced by a system of N interacting point vortices in two dimensions is considered. The general solution follows from an 1864 theorem by Siebeck, that the stagnation points are the foci of a certain plane curve of class N−1 that has all lines connecting vortices pairwise as tangents. The case N=3, for which Siebeck's curve is a conic, is considered in some detail. It is shown that the classification of the type of conic coincides with the known classification of regimes of motion for the three vortices. A similarity result for the triangular coordinates of the stagnation point in a flow produced by three vortices with sum of strengths zero is found. Using topological arguments the distinct streamline patterns for flow about three vortices are also determined. Partial results are given for two special sets of vortex strengths on the changes between these patterns as the motion evolves. The analysis requires a number of unfamiliar mathematical tools which are explained.

Low frequency behavior of solutions to electromagnetic scattering problems in chiral media

Abstract: This paper deals with the low frequency scattering problem by a conducting object in a chiral medium. Using boundary integral equation methods, we obtain the limit of the scattered field solution of the Drude--Born--Fedorov equations as frequency tends to zero. Making use of a Hodge decomposition of the tangent fields shows that the asymptotics strongly depend on the topological properties of the domains under consideration. The asymptotic expansion's dependence on the chirality parameter is shown completely.

On an -manifold in ⁿ near an elliptic complex tangent

Abstract: In this paper, we will be concerned with the local biholomorphic properties of a real n-manifold M in C At a generic point, such a manifold basically has the nature of the standard R in C Near a complex tangent, however, the consideration can be much more complicated and the manifold may acquire a nontrivial local hull of holomorphy and many other biholomorphic invariants The study of such a problem was first carried out in a celebrated paper of E Bishop [BIS] where, for each sufficiently non-degenerate complex tangent, he attached a biholomorphic invariant λ, called the Bishop invariant When the complex tangent is elliptic, ie, when 0 ≤ λ < 12 (for a more precise definition, see §2), he showed the existence of families of complex analytic disks with boundary on M that shrink down to the locus of points in M with complex tangents In particular, using the well-known continuity principle, one sees that the image M of such families is contained in the holomorphic hull of the manifold At the time, he asked whether M gives precisely the local holomorphic hull of M , as well as certain uniqueness properties of the attached disks He also proposed the problem of determining the fine structure of M near such complex tangents Later, there appeared a sequence of papers concerning the smooth character of M in case M ⊂ C Here we would like to mention, in particular, the famous theorem proved by Kenig-Webster in their deep work [KW1] which states that the local hull of holomorphy M near an elliptic complex tangent is a smooth Levi flat hypersurface with M ⊂ C as part of its smooth boundary In another important paper of Moser-Webster [MW], a systematic normal form theory was employed for the understanding of the local biholomorphic invariants of M in case M is real analytic When the Bishop invariant λ 6= 0, their method works in any dimension and even for some hyperbolic complex tangents; but it breaks down for complex tangents with λ = 0 Among other things, they showed that M can be biholomorphically mapped into the affine space

Tangent measure distributions of fractal measures

Abstract: Tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in Euclidean spaces. The idea is, loosely speaking, to attach to every point of the set a family of random measures, called the \(\alpha\)-dimensional tangent measure distributions at the point, which describe asymptotically the \(\alpha\)-dimensional scenery seen by an observer zooming down towards this point. This tool has been used by Bandt [BA] and Graf [G] to study the regularity of the local geometry of self similar sets, but in this paper we show that its scope goes much beyond this situation and, in fact, it may be used to describe a strong regularity property possessed by every measure: We show that, for every measure \(\mu\) on a Euclidean space and any dimension \(\alpha\), at \(\mu\)-almost every point, all \(\alpha\)-dimensional tangent measure distributions are Palm measures. This means that the local geometry of every dimension of general measures can be described – like the local geometry of self similar sets – by means of a family of statistically self similar random measures. We believe that this result reveals a wealth of new and unexpected information about the structure of such general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations between ordinary and average densities.

Constrained optimal framings of curves and surfaces using quaternion Gauss maps

Abstract: We propose a general paradigm for computing optimal coordinate frame fields that may be exploited to visualize curves and surfaces. Parallel transport framings, which work well for open curves, generally fail to have desirable properties for cyclic curves and for surfaces. We suggest that minimal quaternion measure provides an appropriate heuristic generalization of parallel transport. Our approach differs from minimal tangential acceleration approaches due to the addition of "sliding ring" constraints that fix one frame axis, but allow an axial rotational freedom whose value is varied in the optimization process. Our fundamental tool is the quaternion Gauss map, a generalization to quaternion space of the tangent map for curves and of the Gauss map for surfaces. The quaternion Gauss map takes 3D coordinate frame fields for curves and surfaces into corresponding curves and surfaces constrained to the space of possible orientations in quaternion space. Standard optimization tools provide application specific means of choosing optimal, e.g., length- or area-minimizing, quaternion frame fields in this constrained space.

Explicit Elastic Curves

Abstract: The equilibria of thin rods are given by curves which are critical points of the modified total squared curvature. The critical curves are known as elastic curves. It is shown how all the elastic curves are given explicitly in terms of elliptic functions as soon as a certain set of three parameters is known. Every regular curve can be parametrized to have a constant speed but the parametrization is rarely known explicitly. Remarkably, all the elastic curves are here explicitly parametrized to have a constant speed. Curves with fixed distinct endpoints as well as closed curves are admitted. The tangent direction may be constrained at one, both, or neither of the endpoints. There are three major strands of formulas corresponding to: fixed length L, variable length without tension, and variable length with tension (let ν > 0 and add a term νL to the total squared curvature). In the most complicated cases the three parameters are given as solutions to a non-linear system of three equations. In the least complicated case everything is given explicitly in terms of elliptic functions. If the length is variable and there is no tension, at least one of the parameters is completely determined (the elliptic modulus m = 1/2). Using the same set of parameters explicit formulas are given for: the length when it is variable, the total squared curvature, and the tangent angle along the elastic curve. A number of examples are presented which illustrate the full range of constraints.

Slow-Fast Autonomous Dynamical Systems

Abstract: In this paper, we consider a class of slow-fast autonomous dynamical systems, i.e. systems having a small parameter ∊ multiplying a component of velocity. At first, the singular perturbation method (∊ = 0+) is recalled. Then we consider the case ∊ ≠ 0. Starting from a working hypothesis and particularly in the case of a singular approximation, our purpose is to show that there exists slow manifolds which can be defined as the slow manifolds of a so-called tangent linear system. The method allowed us to plot the slow manifold and to go further into the qualitative study and the geometric characterization of attractors. As an example, we give the explicit slow manifold equation of the van der Pol limit cycle. The value of the parameter corresponding to bifurcations is computed. Other third order systems are also treated. The method is extended to dynamical systems with no small parameter, and, therefore, which have no singular approximations, but have at least one real and negative eigenvalue in a large domain. It is numerically shown from the Lorenz model and from a laser model that there exists slow manifolds which can be defined as the slow manifods of a so-called tangent linear system, as in the previous cases. The implicit equation of these slow manifolds has been calculated too.

Curvatures of the quadratic rational Bézier curves

Abstract: We find necessary and sufficient conditions for the curvature of a quadratic rational Bezier curve to be monotone in [0, 1], to have a unique local minimum, to have a unique local maximum, and to have both extrema in (0, 1), and we also visualize them in figures. As an application, we present a necessary and sufficient condition for the offset curve to be regular and to have the same tangent direction with the given quadratic rational Bezier curve, and give a simple algorithm to find it.

Fundamentals of Flow

Abstract: There are two methods for studying the movement of flow. One method follows any arbitrary particle with its kaleidoscopic changes in velocity and acceleration. This is called the Lagrangian method. The other one is a method by which, rather than following any particular fluid particle, changes in velocity and pressure are studied at fixed positions in space x, y, z and at time t . This method is called the Eulerian method. A curve formed by the velocity vectors of each fluid particle at a certain time is called a streamline. In other words, the curve where the tangent at each point indicates the direction of fluid at that point is a streamline. Floating aluminum powder on the surface of flowing water and then taking a photograph, gives the flow trace of the powder. Whenever streamlines around a body are observed, they vary according to the relative relationship between the observer and the body. By moving both a cylinder and a camera placed in a water tank at the same time, it is possible to observe relative streamlines. Furthermore, a flow whose flow state; expressed by velocity, pressure, and density at any position: does not change with time, is called a steady flow. On the other hand, a flow whose flow state does change with time is called an unsteady flow.

Planar G2 curve design with spiral segments

Abstract: Spiral segments are useful in the design of fair curves. Recent work demonstrated the composition of G2 curves from planar cubic and Pythagorean hodograph quintic spiral segments. Practical cases that arise in the use of such spiral segments for computer-aided design are now explored. This paper describes an extension to additional cases of the technique for drawing with Bezier spiral segments that match the position, tangent and curvature of the end of another segment, to additional cases. The advantage of this technique is its control of the curvature and inflection points of a designed curve. The benefit of using such curves in the design of surfaces, in particular surfaces of revolution and swept surfaces, is the control of unwanted flat spots and undulations.

Cubic parametric curves of given tangent and curvature

Abstract: We propose a constructive solution to the problem of finding a cubic parametric curve in a plane if the tangent vectors (derivatives with respect to the parameter) and signed curvatures are given at its end-points but the end-points themselves are unknown. We also show how these curves can be applied to construct blending curves subject to curvature, arc length, inflection and area constraints.

An Implicit integration scheme for generalized viscoplasticity with dynamic recovery

Abstract: This paper is concerned with the development of a general implicit time-stepping integrator for the flow and evolution equations in a recent representative class of generalized viscoplastic models, involving both hardening and dynamic recovery mechanisms. To this end, the computational framework is developed on the basis of the unconditionally stable, backward Euler difference scheme. Its mathematical structure is of sufficient generality to allow a systematic treatment of several internal variables of the tensorial and scalar types. The matrix forms developed are directly applicable in general (three-dimensional) situations as well as subspace applications (i.e., plane stress/strain, axisymmetric, generalized plane stress in shells). The closed-form expressions for residual vectors and the algorithmic, (consistent) material tangent stiffness array are given explicitly, with the maximum matrix sizes “optimized” to depend only on the number of independent stress components, but not the number of internal state variables involved. Several numerical simulations are given to assess the performance of the developed schemes.

Optimum balancing designs of the drag-link drive of mechanical presses for precision cutting

Abstract: Proposing optimum designs by adding disk counterweights to reduce the shaking force and shaking moment of the drag-link drive of mechanical presses is the main subject of this study. The two-phase optimization technique is proposed for the multi-objective optimization. The effects of implementing the designs on the flywheel and brake of the machine are also evaluated. Regarding the two-phase optimization technique proposed, it starts with an initial estimate but generates a few feasible designs. It is found that the results obtained with the technique are usually better than those starting with the criterion value as the objective function. The optimum designs with or without the tangent constraints, which insure the contours of the added counterweights on the links with fixed pivots that are tangent to the pivot centers, are investigated. It is found that the results with the tangent constraints are more cost-effective. Subject to the very strict space constraints, the shaking effects can be reduced by almost a half, using the proposed design.

Irregularity of Optimal Trajectories in a Control Problem for a Car-like Robot

Abstract: We study the problem to find (a) shortest plane curve(s) joining two given points with given tangent angles and curvatures. The tangent angle and the curvature of the path are continuous and the derivative of the curvature is bounded by $2$. At a regular (i.e. of the class $C^3$) point such a curve must be locally a piece of a clothoid or a line segment (up to isometry a clothoid is given by Fresnel's integrals $x(t)=\int _0^t\cos \tau ^2d\tau $, $y(t)=\int _0^t\sin \tau ^2d\tau $). We prove that if the distance between the initial and final points is greater than $320\sqrt{\pi }$, then a generic shortest curve contains infinitely many switching points.

Study of the yield point of the thread

Abstract: In this work the study of yield point of thread and stress in flow point is shown. The determination of this point is done with the well‐known Meredith’s and Coplan’s construction, which has the basis in tangent method and numerical method, which is developed from sufficient smooth approximation of the curve. The approximation polynomial expression of the ninth order was chosen for the construction of the mean curve σ (e) which, as the sufficiently smooth curve, fits the measured values.

The interfacial thickness of symmetric diblock copolymers : theory and experiment

Abstract: A recent application of density functional theory to the structure and thermodynamics of the ordering of symmetric, tangent hard site, diblock copolymers [S. K. Nath et al., J. Chem. Phys. 106, 1950 (1997)] predicted an interfacial thickness larger than would be expected from previous self-consistent-field studies of thread chains. Here we compare the theoretical predictions with the few experimental measurements of interfacial thickness in symmetric diblocks. It is observed that predictions of the thickness of the interface are sensitive to the details of the monomer structure included in the underlying model, and that the range of the experimental measurements is spanned by the two theoretical models.