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Tortuosity is among the first alterations in the retinal vessel network to appear in many retinopathies, such as those due to hypertension. An automatic evaluation of retinal vessel tortuosity would help the early detection of such retinopathies. Quite a few techniques for tortuosity measurement and classification have been proposed, but they do not always match the clinical concept of tortuosity. This justifies the need for a new definition, able to express in mathematical terms the tortuosity as perceived by ophthalmologists. We propose here a new algorithm for the evaluation of tortuosity in vessels recognized in digital fundus images. It is based on partitioning each vessel in segments of constant-sign curvature and then combining together each evaluation of such segments and their number. The algorithm has been compared with other available tortuosity measures on a set of 30 arteries and one of 30 veins from 60 different images. These vessels had been preliminarily ordered by a retina specialist by increasing perceived tortuosity. The proposed algorithm proved to be the best one in matching the clinically perceived vessel tortuosity. Index Terms--Curvature, fundus imaging, hypertensive retinopathy, retina, tortuosity, vessels.

The comparison of curvature of three-dimensional objects within and between the modalities of vision and touch using a dual-task paradigm to test for some of the processes is presented.

A new law of the wall accounting for curvature effects in swirling axial flows is derived. The influence of the curvature on the turbulence mixing lengths in both axial and tangential directions is examined theoretically using the Reynolds stress transport equations. For equilibrium flows with weak curvature, identical mixing lengths are derived for the axial and tangential directions. Additionally, the effect of finite local curvature and shear stress ratio on the near-wall velocities is systematically explored. It is found that the curvature effect in swirling axial flows is suppressed by a factor of 1/(1 + [[sigma].sup.2.sub.w]) compared to that in curved channel flows, where [[sigma].sub.w] is the ratio of the axial to swirl shear stress. For a given curvature radius, the maximum velocity deviation occurs when the axial-to-swirl shear stress ratio is zero. Finally, the performance of the new curvature law is evaluated by implementing it as a wall function in a well-established CFD code. The new wall function provides improved agreement for swirl velocity distributions inside labyrinth cavities in comparison with existing experimental laser Doppler anemometry measurements. [DOI: 10.1115/1.2409360] Keywords: dynamic modeling, curvature, law of the wail swirling axial flow

An improved shear deformable curved beam theory to overcome the drawback of currently available beam theories is newly proposed for the spatially coupled stability analysis of thin-walled curved beams with nonsymmetric cross sections. For this, the displacement field is introduced considering the second order terms of semitangential rotations. Next the elastic strain energy is newly derived by using transformation equations of displacement parameters and stress resultants and considering shear deformation effects due to shear forces and restrained warping torsion. Then the potential energy due to initial stress resultants is consistently derived with accurate calculation of the Wagner effect. Finally, equilibrium equations and force-deformation relations are obtained using a stationary condition of total potential energy. The closed-form solutions for in-plane and out-of-plane buckling of curved beams subjected to uniform compression and pure bending are newly derived. Additionally, finite-element procedures are developed by using curved beam elements with arbitrary thin-walled sections. In order to illustrate the accuracy and the practical usefulness of this study, closed-form and numerical solutions for spatial buckling are compared with results by available references and ABAQUS' shell elements. CE Database subject headings: Spatial analysis; Beams; Curvature; Deformation; Warpage; Thin wall sections.

Functional and structural maps, such as a curvature, cortical thickness, and functional magnetic resonance imaging (MRI) maps, indexed over the local coordinates of the cortical manifold play an important role in neuropsychiatric studies. Due to the highly convoluted nature of the cerebral cortex and image quality, these functions are generally uninterpretable without proper methods of association and smoothness onto the local coordinate system. In this paper, we generalized the spline smoothing problem (Wahba, 1990) from a sphere to any arbitrary two-dimensional (2-D) manifold with boundaries. We first seek a numerical solution to orthonormal basis functions of the Laplace-Beltrami (LB) operator with Neumann boundary conditions for a 2-D manifold M then solve the spline smoothing problem in a reproducing kernel Hilbert space (r.k.h.s.) of real-valued functions on manifold M with kernel constructed from the basis functions. The explicit discrete LB representation is derived using the finite element method calculated directly on the manifold coordinates so that finding discrete LB orthonormal basis functions is equivalent to solving an algebraic eigenvalue problem. And then smoothed functions in r.k.h.s can be represented as a linear combination of the basis functions. We demonstrate numerical solutions of spherical harmonics on a unit sphere and brain orthonormal basis functions on a planum temporale manifold. Then synthetic data is used to quantify the goodness of the smoothness compared with the ground truth and discuss how many basis functions should be incorporated in the smoothing. We present applications of our approach to smoothing sulcal mean curvature, cortical thickness, and functional statistical maps on submanifolds of the neocortex. Index Terms--Cortical thickness, curvature, Laplace-Beltrami (LB) operator, Neumann boundary conditions, reproducing kernel Hilbert space, spline smoothing.

A method is developed for two-dimensional (2-D) skeleton extraction that circumvents the problem of threshold parameter dependence by eliminating the curvature contribution to the error. It is done by taking into account variations of density due to boundary curvature and yields a skeletionization algorithm that gives both better localization and less susceptibility to boundary noise and parameter choice than the Hamilton-Jacobi method.

Byline: P. D. Andreev (1) Abstract: Let (x,d) be a locally compact geodesically complete CAT(0)-space of topological dimension greater than 1. It is proved that if each geodesic segment in X is uniquely extended to a complete geodesic, then the metric d is determined by the diagonal tube V a X x X corresponding to an arbitrary positive r. This partly generalizes V. N. Berestovskii's results on A. D. Aleksandrov spaces of negative curvature. Bibliography: 8 titles. Author Affiliation: (1) Pomor State University, Arkhangelsk, Russia Article History: Registration Date: 05/10/2005 Received Date: 25/09/2003 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 5--29.

The estimated parameters of output distance functions frequently violate the monotonicity, quasi-convexity and convexity constraints implied by economic theory, leading to estimated elasticities and shadow prices that are incorrectly signed, and ultimately to perverse conclusions concerning the effects of input and output changes on productivity growth and relative efficiency levels. We show how a Bayesian approach can be used to impose these constraints on the parameters of a translog output distance function, Implementing the approach involves the use of a Gibbs sampler with data augmentation. A Metropolis-Hastings algorithm is also used within the Gibbs to simulate observations from truncated pdfs. Our methods are developed for the case where panel data is available and technical inefficiency effects are assumed to be time-invariant. Two models--a fixed effects model and a random effects model are developed and applied to panel data on 17 European railways. We observe significant changes in estimated elasticities and shadow price ratios when regularity restrictions are imposed. JEL Classification: C11; C15; D24 Keywords: Markov chain Monte Carlo: Inequality constraints: Output distance function; European railways

6. Historical Essay Before starting the presentation of this section on the construction of 'non-Euclidean geometry' (the term was introduced by Gauss, a German mathematician), we can cite P. Beyl, [...]

ABSTRACT. The technique of topological analysis of integrable problems developed by A. T. Fomenko is applied for studying certain problems of celestial mechanics. Introduction The problem of integrability and nonintegrability [...]

Abstract: In Riemannian (differential) geometry, the differences between Euclidean geometry, elliptic geometry, and hyperbolic geometry are understood in terms of curvature. I think Gauss and Riemann captured the essence of [...]

The conditions under which an infinite three-dimensional graph manifold M carries a metric of nonpositive bounded curvature (an NPC-metric) having finite volume are studied. A complete list of all such [...]

Separation control on NACA 0012 and NACA 0015 airfoils was compared under incompressible conditions, using leading-edge periodic excitation, in order to assess the effect of leading-edge curvature. Both lift and moment coefficients were considered to compare and analyse control effectiveness. In contrast to the relatively mild NACA 0015 trailing-edge stall, NACA 0012 stall was dominated by a leading-edge bubble-bursting mechanism that gave rise to alternating intervals of partial attachment and separation, but with no regular frequency. Low-amplitude excitation downstream of the bubble enhanced poststall lift and significantly attenuated the associated unsteadiness. In general, larger momentum coefficients were required for NACA 0012 separation control due to the largecentrifugal acceleration of the flow around the leading edge. Because of the different stalling characteristics, relatively high- and low-excitation frequencies were effective for the NACA 0012 and NACA 0015 airfoils, respectively. However, the combination of high-excitation amplitudes with relatively low frequencies was effective on the NACA 0012, and this was believed to be associated with the large harmonic content of the evolving perturbations.

The application of composites in the aircraft industry has increased significantly over the past few decades. With traditional composite laminate shaping, each layer is made to conform to the mold surface by hand before subsequent layers are added. This is a very labor- and time-intensive process. There is a great deal of interest in developing an automated process for forming composite parts with compound curvatures. The proposed composite forming process utilizes a computer-controlled, reconfigurable discrete element mold to incrementally form a compound curvature part shape from a flat lay-up, thereby facilitating process automation. An elastomeric interpolating layer, called an interpolator, is placed on top of the hemispherical forming ends of the die elements to prevent dimpling of the composite lay-up. The process employs vacuum to pull a single diaphragm (top), composite, and interpolator into contact with the mold surface. Through an experimental investigation, this new composites forming process with 'active' tooling has been successfully demonstrated. Heating of the composite is accomplished by uncontained, forced convection using a matrix of heated air jets mounted above the composite. However, low-powered conduction is shown to be the best heating method in terms of both composite heating time and minimization of through-thickness temperature. Using vacuum to conform both the composite and the interpolator to the mold, and choosing sufficiently stiff diaphragm and interpolator materials, undimpled and wrinkle-free composite parts have been formed in an incremental fashion. [DOI: 10.1115/1.1561456]

Byline: Paul J.F. Gandy (1), Jacek Klinowski (1) Keywords: equipotential surfaces; nodal surfaces; triply periodic minimal surfaces; mean curvature; Gaussian curvature Abstract: Models of electrostatic surfaces in atomic crystals rely on equations involving the Jacobi theta functions. Numerical integration of these is prohibitively time consuming, making it difficult to examine the properties of the fields which give rise to the surfaces. We give simple expressions for the key electrostatic surfaces using Fourier expansions in basis sets of nodal surfaces. Any surface may be computed in seconds in a form ammenable to further analysis. The distribution of the mean and Gaussian curvatures over each surface has been visualised by assigning colours so that the range from minimum to maximum value spans blue to red. We similarly explore the mean and Gaussian scalar fields over a range of triply periodic surfaces of the same morphology. Author Affiliation: (1) Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, UK Article History: Registration Date: 09/10/2004

This paper presents a curvature-based bound on the number of frictionless fingers or fixtures required to immobilize three-dimensional (3-D) objects. A recently developed second-order mobility theory has shown that in addition to first-order geometrical effects, second-order or curvature effects play an important role in the kinematics of contact. Based only on first-order geometrical effects, it was previously thought that seven contacts are required to immobilize generic 3-D objects. We show that when second-order effects are included, four convex fingers or fixtures with sufficiently flat curvature can immobilize any generic smooth or polyhedral 3-D object. The derivation of the improved bound proceeds by first constructing a suitable equilibrium grasp of the given object, called a preimmobilizing grasp. Depending on the object's geometry, a preimmobilizing grasp may have two, three, or four contacts. The conversion of a preimmobilizing grasp to a four-finger immobilizing grasp depends on the object's curvature at the contacts. This curvature can be convex, concave, or saddle-like. Since a preimmobilizing grasp can have k = 2, 3, 4 contacts, there are 31 cases to consider (see text). We construct immobilizing grasps for all of these cases, and present simulation results showing the immobilization of selected object types. Index Terms--Curvature, fixturing, grasping, mobility bounds.

A curvature law of the wall, which determines the mean velocity profile, is analytically derived for near-wall turbulent flows to include strong curved-channel wall curvature effects through a perturbation analysis. The new law allows improved analysis of such flows, and it provides the basis for improved wall function boundary conditions for their computation (CFD), even for strong curvature cases. The improved law is based on the algebraic eddy viscosity and curvature-corrected mixing length concepts, the latter of which is a linear function of the gradient Richardson number. To include the complete Richardson number effects, the local streamline curvature effects in the gradient Richardson number are kept. To overcome the mathematical difficulty of keeping all of these local streamline curvature terms, an innovative nonconstant-parameter perturbation solution technique is successfully applied.

Supercritical flow in a bend manhole for combined sewer systems is considered for a typical relative curvature and deflection angles of 45 degrees and 90 degrees. The typical features of bend flow in a U-shaped channel are determined, including the surface profiles along the inner and the outer bend walls, impact flow at the manhole outlet into the downstream sewer pipe, air entrainment characteristics, and wave development in the downstream sewer. Also, typical velocity distributions along the manhole are presented, and the capacity of the conventional bend structure is determined. To increase the discharge capacity across a bend manhole, the so-called bend cover is introduced. This cover element finished in steel can be removed for maintenance and inspection purposes. It limits the wave height close to the bend outlet to 90% of the sewer diameter and allows air entrainment into the downstream sewer. The performance and capacity of bend flow may significantly be increased, and the novel element may easily be added to existing manholes.

In this article we survey some of the developments in Riemannian geometry. We place special emphasis on explaining the relationship between curvature and topology for Riemannian manifolds with lower curvature bounds.

This paper describes a novel method for image corner detection based on the curvature scale-space (CSS) representation. The first step is to extract edges from the original image using a Canny detector. The corner points of an image are defined as points where image edges have their maxima of absolute curvature. The corner points are detected at a high scale of the CSS and tracked through multiple lower scales to improve localization. This method is very robust to noise, and we believe that it performs better than the existing corner detectors. An improvement to Canny edge detector's response to 45 [degrees] and 135 [degrees] edges is also proposed. Furthermore, the CSS detector can provide additional point features (curvature zero-crossings of image edge contours) in addition to the traditional corners. Index Terms - Low-level processing, feature extraction, corner detection, multiscale analysis, curvature scale space, Canny edge-detector.

Some of the views of Ker-Chau Li regarding the method of principal Hessian directions for estimating plotting directions that detect curvature in regression functions invite comment. The method, which was based on normally distributed predictors, can be improved wherein the conditions of methodological points are cumulative. Moreover, the use of distributed predictors in constant variance conditions lead to the interpretation of the average hessian matrix of the residual regression function., 1. INTRODUCTION Empirical evidence accumulated during the past few years indicates that the method of principal Hessian directions (pHd), as proposed by Li (1992), can be quite effective for detecting [...]

Although the straightness of hand paths is a widely accepted feature of human multijoint reaching movement, detailed examinations have revealed slight curvatures in some regions of the workspace. This observation raises the question of whether planned trajectories are straight or curved. If they are straight, 3 possible factors can explain the observed curvatures: (a) imperfect control, (b) visual distortion, or (c) interaction between straight virtual trajectories and the dynamics of the arm. Participants instructed to generate straight movement paths produced movements much straighter than those generated spontaneously. Participants generated spontaneously curved trajectories in the frontoparallel plane, where visual distortion is not expected. Electromyograms suggested that participants generated straighter paths without an increase in arm stiffness. These findings argue against the 3 factors. It follows that planned trajectories are likely to be curved.

A turbulent boundary layer subjected to multiple, additional strain rates, namely convex curvature coupled with streamwise pressure gradients (zero and favorable, ZPG and FPG) was investigated experimentally using laser Doppler velocimetry. The inapplicability of the universal flat-plate log-law to curved flows is discussed. However, a logarithmic region is found in the curved and accelerated turbulent boundary layer examined here. Similarity of the mean velocity and Reynolds stress profiles was achieved by 45 deg of curvature even in the presence of the strongest FPG investigated (k = 1.01 X [10.sup.-6]). The Reynolds stresses were suppressed (with respect to fiat plate values) due primarily to the effects of strong convex curvature ([[Delta].sub.o]/R = 0.10). In curved boundary layers subjected to different favorable pressure gradients, the mean velocity and normal Reynolds stress profiles collapsed in the inner region, but deviated in the outer region ([y.sup.+] [greater than or equal to] 100). Thus, inner scaling accounted for the impact of the extra strain rates on these profiles in the near-wall region. Combined with curvature, the FPG reduced the strength of the wake component, resulted in a greater suppression of the fluctuating velocity components and a reduction of the primary Reynolds shear stress throughout almost the entire boundary layer relative to the ZPG curved case.

The history of the Gauss-Bonnet theorem and its generalization depicts how mathematics advances by the work of a large number of unknown mathematicians in the face of challenges. The theorems which are well-known at present were once the most difficult to understand and dominated mathematical discussions. Mathematicians fail to name key concepts and forget important theorems and simple ideas, confusing efforts to track the history of mathematical theorems.

This paper focuses on the experimental evaluation of a range image segmentation system which partitions range data into homogeneous surface patches using estimates of the sign of the mean and Gaussian curvatures. We report the results of an extensive testing program aimed at investigating the behavior of important experimental parameters such as the probability of correct classification and the accuracy of curvature estimates, measured over variations of significant segmentation variables. Evaluation methods in computer vision are often unstructured and subjective: this paper contributes a useful example of extensive experimental assessment of surface-based range segmentation. Index Terms - Machine vision, range image segmentation, curvature-based methods, surfaces, image representation, system evaluation.

Cosmological horizons play an essential role in determining the causal structure of spacetime and are of central importance in the inflationary universe scenario. We review the topic of horizons in simple language, pointing out a number of widespread misconceptions. The use of spacetime diagrams plotted in terms of proper time and proper distance coordinates helps sort out some of these difficulties. They complement the widely used conformal diagrams, which show causal relations clearly but severely distort proper distances.

The topic of Hermitian symmetric spaces is discussed in the context of its relations to the curvature of the natural Kahler metrics. A Kahler manifold is considered Hermitian symmetric if there is an involute holomorphic isometry for every point in the system. A simpler characterization of Grasmann manifolds in terms of curvature, which is applicable to all classes of irreducible Hermetian spaces, is yet to be formulated.

The caustic test is capable of a high degree of accuracy in the measurement of aspheric surfaces. The essence of the test is to measure the locations of the centers of curvature for off-axis zones of an aspheric surface with a modified Foucault test apparatus. We present equations that extend the usefulness of the caustic test to general aspheric surfaces. Equations are also given for calculating derivatives for several caustic test parameters that are useful for tolerancing optical systems or in derivative increment calculations for computer lens design applications. Guidelines are given for the correct use of the caustic test in actual testing setups. Key words: Caustic test, optical testing, Foucault test, knife-edge, zonal testing, zone, paraboloid, conic, asphere, center of curvature, mirror, optical tolerancing, derivative increment, caustic.

The Sun model developed from the lower bound approach of plasticity is extended by introducing a family of dilatant plasticity theories. This kinematic hardening version model is analyzed with three others including, the isotropic hardening version of Sun, the Gurson model and the Mear and Hutchinson model. Results show that local necking predictions are affected by yield surface curvature and that the kinematic hardening model is more accurate for local necking analysis.

A technique for improving the numerical predictions of turbulent flows with the effect of streamline curvature is developed. Separated flows and the flow in a curved duct are examples of flow fields where streamline curvature plays a dominant role. New algebraic formulations for the eddy viscosity [mu.sub.t], incorporating the k - [epsilon] turbulence model are proposed to account for various effects of streamline curvature. The loci of flow reversal (where axial velocities change signs) of the separated flows over various backward-facing steps are employed to test the capability of the proposed turbulence model in capturing the effect of local curvature. The inclusion of the effect of longitudinal curvature in the proposed turbulence model is validated by predicting the distributions of the longitudinal velocity and the static pressure in an S-bend duct and in 180 deg turn-around ducts. The numerical predictions of different curvature effects by the proposed turbulence models are also reported.

A simple mechanical device is introduced, the parallelometer, that can be used to measure the curvatures of surfaces. The device can be used as a practical illustration of parallel transport of a vector. Its connection to the Foucault pendulum is discussed. PACS Nos: 01.50.My, 01.50.Pa, 01.55.+b, 02.40.Ma, 02.40.Yy, 03.65.Vf Nous presentons un dispositif mecanique simple, le parallelometre, qui peut etre utilise pour mesurer la courbure des surfaces. Le dispositif illustre de facon pratique le transport parallele d'un vecteur. Nous analysons sa relation avec le pendule de Foucault. [Traduit par la Redaction], 1. Introduction It is intuitively obvious how to transport a vector in a plane in such a way that it remains unchanged: just pick a set of Cartesian coordinates and [...]

Diagnostics are investigated for assessing the agreement between likelihood and standard large-sample confidence regions for parameters from an exponential family nonlinear model (Cordeiro and Paula 1989). The development connects the contour methods proposed by Hodges (1985. 1987) with the curvature measures for normal, nonlinear regression proposed by Bates and Watts (1980) and Jennings's (1982. 1986) diagnostics for logistic regression. The proposed methodology is illustrated with several examples. KEY WORDS: Exponential family nonlinear models; Logistic regression; Nonlinear regression; Parameter-effects curvature., 1. INTRODUCTION Large-sample, normal approximations pervade statistical theory and practice as a mechanism for producing easily computed and interpreted confidence regions. It is widely recognized, however, that such approximations can [...]

Salvadori and Luccio demonstrated that, for the purpose of psychophysical analysis, the curvature of a point on a continuous single-valued function can be quantified as the inverse ray of the circle osculating the function at the point of interest. While this method is mathematically sound, it does not allow for estimation of curvature in complex shapes containing discontinuities and irregularities such as sharp corners, closed contours, or line intersections--features that commonly occur in normal visual experience. In this paper, a simple modification of this algorithm is presented which overcomes these limitations by using discrete rather than continuous contour sampling, thereby allowing estimation of curvature for a wider variety of shape contours.

An answer is provided to the problem on when a periodic function is the curvature of a closed plane curve. It is also shown that the associative curve of any periodic function can be forced to close up by scaling or by addition of a constant, and some applications of the results are provided.

Two-component vesicles with coherent domains of the components, where the domains are separated by an interface are studied. Results reveal that the interface width is narrow when the difference between the spontaneous curvatures is small, and it grows instantly when the difference becomes large enough.

The flow drag and heat transfer reduction phenomena of nonionic aqueous surfactant solution in helically coiled pipes are examined and compared with those of straight pipes. Observations reveal that the friction factors and heat transfer coefficients of the surfactant solution in helical pipes are higher than in a straight pipe.

An elegant symmetry of normal lines measure non-linearity and tangent lines measure non-circularity is obtained. The classical version of aberrancy is discussed and some related exercises that are suitable as undergraduate projects are provided.

An asymptotic formula relating the mean curvature of a hypersurface at a given point and the volume of small balls centered at this point is proved.

The historical development of curvature, the subject comes under calculus, is discussed. The radius calculation of curvature and the existence of curvature in elastic force are described with some experimental processes.

The impact of the clubhead of a driver with a golf ball is modeled. The effect of the convex clubface of a driver on the flight of the golf ball is considered and the dependence of the optimum curvature of the clubface on the volume, mass, and impact speed of the clubhead is determined.

We announce the classification of complete almost embedded surfaces of constant mean curvature, with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.