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We initiate a study of the Euclidean distance degree in the context of sparse polynomials. Specifically, we consider a hypersurface [Formula omitted] defined by a polynomial f that is general given its support, such that the support contains the origin. We show that the Euclidean distance degree of [Formula omitted] equals the mixed volume of the Newton polytopes of the associated Lagrange multiplier equations. We discuss the implication of our result for computational complexity and give a formula for the Euclidean distance degree when the Newton polytope is a rectangular parallelepiped., Author(s): P. Breiding [sup.1] , F. Sottile [sup.2] , J. Woodcock [sup.2] Author Affiliations: (1) grid.419532.8, Max-Planck-Institute for Mathematics in the Sciences Leipzig, , Inselstr. 22, 04103, Leipzig, Germany (2) [...]

We consider the problem of positioning a cloud of points in the Euclidean space [R.sup.d], using noisy measurements of a subset of pairwise distances. This task has applications in various areas, such as sensor network localization and reconstruction of protein conformations from NMR measurements. It is also closely related to dimensionality reduction problems and manifold learning, where the goal is to learn the underlying global geometry of a data set using local (or partial) metric information. Here we propose a reconstruction algorithm based on semidefinite programming. For a random geometric graph model and uniformly bounded noise, we provide a precise characterization of the algorithm's performance: in the noiseless case, we find a radius [r.sub.0] beyond which the algorithm reconstructs the exact positions (up to rigid transformations). In the presence of noise, we obtain upper and lower bounds on the reconstruction error that match up to a factor that depends only on the dimension d, and the average degree of the nodes in the graph. Keywords Rigidity theory * Global rigidity * Stress matrix * Graph realization * Network localization * Manifold learning * Semidefinite programming, 1 Introduction 1.1 Problem Statement Given a set of n nodes in [R.sup.d], the localization problem requires one to reconstruct the positions of the nodes from a set of pairwise [...]

In this study, we present (i) a proof of the Menelaus theorem for quadrilaterals in hyperbolic geometry, (ii) and a proof for the transversal theorem for triangles, and (iii) the Menelaus's theorem for n-gons. Keywords Hyperbolic geometry, hyperbolic triangle, hyperbolic quadrilateral, Menelaus theorem, transversal theorem, gyrovector., §1. Introduction Hyperbolic Geometry appeared in the first half of the 19th century as an attempt to understand Euclid's axiomatic basis of Geometry. It is also known as a type [...]

A circular surface with a fixed radius can be swept out by moving a circle with its center following a curve, which acts as the spine curve. Based on a system of Euclidean invariants, the paper identifies those circular surfaces taking lines of curvature as generating circles and further explores the properties of the principal curvatures and Gaussian curvature of the tangent circular surfaces. The paper then applies the study to mechanism analysis by proving the necessary and sufficient condition for a circular surface to be generated by a serially connected C'R, HR, or RR mechanism, where C' joint can be visualized as a special H joint with a variable pitch of one degree of freedom. Following the analysis, this paper reveals for the first time the relationship between the invariants of a circular surface and the commonly used D-H parameters of C'R, HR, and RR mechanisms. [DOI: 10.1115/1.3212679] Keywords: circular surface, lines of curvature, Euclidean invariants, mechanisms, kinematics, differential geometry, robotics, workspace

The electric wind produced by corona discharge of a high-voltage electrode in air is employed for destabilizing a horizontal water film. In wire-to-plane geometry, the phenomenon is characterised by current versus voltage curves and visual observations of the onset of free-surface oscillations. The effect of the following parameters is examined for both positive and negative coronas: distance between the wire and the film (S), film thickness (h), wire diameter ([phi]) and composition, applied voltage (HV), and relative humidity (RH). The free-surface destabilisation is retarded by increasing d and [phi] and is insensitive to h in the tested range. The onset of corona discharge is predicted by Peek's law and compared with the experimentally observed threshold. In negative corona discharge, the current values are higher and the film is destabilised at lower HV than in positive polarity. Humidity tends to decrease the corona current at a given HV. Correlations are proposed for the current-voltage curves, in terms of the mean electric field in the inter-electrode gap and of RH, satisfactorily agreeing with the experimental data. Both positive and negative corona currents turn out to be stable for days of operation. The power loss by corona discharge is in any case lower than 12 W. Wave induction on the liquid-gas interface can effectively enhance heat and mass exchange between the two phases. Index Terms--Corona discharge, ionic wind, thin liquid films, free-surface waves, wire-to-plane geometry, current-voltage characteristics, humidity effects.

It is well known that an isothermal dielectric liquid can be driven by electroconvection. It has been demonstrated that two phenomena could generate space charge in isothermal dielectric liquids and then induce a fluid movement. These two electroconvective effects are bulk conduction and ion injection. In order to improve the performance of electroconvective devices, the velocity of the flows must be recorded as accurately and precisely as possible. In this paper, the Particle Image Velocimetry (PIV) method that was originally developed in the field of experimental fluid mechanics is adapted to electroconvective flow measurements. The choice and the size of seeding particles are discussed. The influence of the seeding particles density on the current is measured. In this work, experiments were investigated on a typical two-dimensional charged plume flow produced between a blade and a flat plate. Both negative and positive polarities are also investigated. Index Terms--Blade-plane geometry, charged plume, dielectric liquid, electrohydrodynamics, particle image velocimetry.

This paper demonstrates analytical relationship of influence coefficients between static-couple and multiplane methods on two-plane balancing as well as its application. For the static-couple approach, cross-effects are defined between static weights and couple response as well as between couple weights and static response, thus making it possible to offset both static and couple vibration vectors effectively with appropriate combination of static and couple weights. Relationship of influence coefficients between static/couple and individual probe due to static/couple weights is also given. Static, couple, or individual probe influence coefficients due to static or couple weights can be obtained directly without having to place static or couple trial weights if influence coefficients used in the multiplane approach are known. From static and couple influence data as well as cross-effects, influence data for the multiplane approach can be obtained directly as well without having to place any trial weights at either plane. The above findings and conversion equations are obtained analytically and verified by experimental results. Conversion of influence coefficients from multiplane to static-couple format can determine whether static or couple weights are more effective as well as running vibration modes, while conversion from static-couple to multiplane format can determine which balance plane is more effective. [DOI: 10.1115/1.2968866]

Quaternion operators have an important role as a screw and a rotation operator in Euclidean motion. We introduce a new operator similar to quaternion operator in Galilean motion. This operator is defined as a dual quaternion operator by using some information in [1]. Then, we have geneneralized this operator for n-dimensional Galilean space. Keywords Galilean transformation, dual complex numbers, Quaternion operators, lie group., [section]1. Introduction A unit real quaternion is a rotation operator on rigid body motion in Euclidean space. Unit dual quaternions are also used both as rotation and screw operators. Unit [...]

In this paper, we study minimum-time motion planning and routing problems for the Dubins vehicle, i.e., a nonholonomic vehicle that is constrained to move along planar paths of bounded curvature, without reversing direction. Motivated by autonomous aerial vehicle applications, we consider the Traveling Salesperson Problem for the Dubins vehicle (DTSP): given n points on a plane, what is the shortest Dubins tour through these points, and what is its length? First, we show that the worst-case length of such a tour grows linearly with n and we propose a novel algorithm with performance within a constant factor of the optimum for the worst-case point sets. In doing this, we also obtain an upper bound on the optimal length in the classical point-to-point problem. Second, we study a stochastic version of the DTSP where the n targets are randomly and independently sampled from a uniform distribution. We show that the expected length of such a tour is of order at least [n.sup.2/3] and we propose a novel algorithm yielding a solution with length of order [n.sup.2/3] with probability one. Third and finally, we study a dynamic version of the DTSP: given a stochastic process that generates target points, is there a policy that guarantees that the number of unvisited points does not diverge over time? If such stable policies exist, what is the minimum expected time that a newly generated target waits before being visited by the vehicle? We propose a novel stabilizing algorithm such that the expected wait time is provably within a constant factor from the optimum. Index Terms--Dubins traveling salesperson problem (DTSP), dynamic traveling repairman problem (DTRP), Euclidean traveling salesperson problem (ETSP), uninhabited aerial vehicles (UAVs).

Measuring flats in the horizontal posture with interferometers is analyzed in detail, taking into account the sag produced by gravity. A mathematical expression of the bending is provided for a plate supported at three unevenly spaced locations along the edge. It is shown that the azimuthal terms of the deformation can be recovered from a three-flat measuring procedure, while the pure radial terms can only be estimated. The effectiveness of the iterative algorithm for data processing is also demonstrated. Experimental comparison on a set of three flats in horizontal and upright posture is provided. OCIS codes: 120.0120, 120.3180, 120.3940, 120.4800, 120.6650.

In binary images, the Distance Transformation (DT) and the geometrical skeleton extraction are classic tools for shape analysis. In this paper, we present time optimal algorithms to solve the reverse Euclidean distance transformation and the reversible medial axis extraction problems for d-dimensional images. We also present a d-dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape. Index Terms--Shape representation, distance transformation, reverse Euclidean distance transformation, medial axis extraction, d-dimensional shapes.

Two algebraic methods for systematic construction of structured regular and irregular low-density parity-check (LDPC) codes with girth of at least six and good minimum distances are presented. These two methods are based on geometry decomposition and a masking technique. Numerical results show that the codes constructed by these methods perform close to the Shannon limit and as well as random-like LDPC codes. Furthermore, they have low error floors and their iterative decoding converges very fast. The masking technique greatly simplifies the random-like construction of irregular LDPC codes designed on the basis of the degree distributions of their code graphs. Index Terms--Degree distribution, Euclidean geometry, geometry decomposition, masking, permutation matrix.

Various kinds of distances of all negacyclic codes of length [2.sup.s] over [[??].sub.[2.sup.a]] are completely determined. Using our structure theorems of negacyclic codes of length [2.sup.s] over [[??].sub.[2.sup.a]], we first calculate the Hamming distances of all such negacyclic codes, which particularly lead to the Hamming weight distributions and Hamming weight enumerators of several codes. These Hamming distances are then used to obtain their homogeneous, Lee, and Euclidean distances. Our techniques are extendable to the more general class of constacyclic codes, namely, the [gamma]-constacyclic codes of length [2.sup.s] over [[??].sub.[2.sup.a]], where [gamma] is any unit of [[??].sub.[2.sup.a]] with the form 4k - 1. We establish the Hamming, homogeneous, Lee, and Euclidean distances of all such constacyclic codes. Index Terms--Binary codes, chain rings, codes over finite rings, constacyclic codes, cyclic codes, Euclidean distance, Hamming distance, homogeneous distance, Lee distance, negacyclic codes, quarternary codes, repeated-root codes.

We consider quantizations of observations represented by finite partitions of observation spaces. Partitions usually decrease the sensitivity of observations to their probability distributions. A sequence of quantizations is considered to be asymptotically sufficient for a statistical problem if the loss of sensitivity is asymptotically negligible. The sensitivity is measured by f-divergences of distributions or the closely related f-informations including the classical Shannon information. It is demonstrated that in some cases the maximization of f-divergences means the same as minimization of distortion of observations in the classical sense considered in mathematical statistics and information theory. The main result of the correspondence is a general sufficient condition for the asymptotic sufficiency of quantizations. Selected applications of this condition are studied leading to new simple criteria of asymptotic optimality for quantizations of vector-valued observations and observations on general Poisson processes. Index Terms--Abstract observation spaces, asymptotically sufficient partitions, asymptotically sufficient quantizations, Euclidean observation spaces, f-divergences, f-informations, general Poisson processes, optimal quantizations, sufficient statistics.

In recent years, the conversion of residue numbers to a binary integer has been intensively studied. The Chinese Remainder Theorem (CRT) is a solution to this conversion problem of a number to the Residue Number System with a general moduli set. This paper presents a new division-free conversion approach for the conversion of residue numbers to a binary integer. The algorithm differs from others employing a great number of division instructions by using shift instructions instead. These simple instructions keep the power consumption lower. This algorithm can also be implemented with a lookup table or upon a vector machine. Both make the conversion process efficient. This division-free algorithm employs the concept of Montgomery multiplication algorithm. There are two variations of Montgomery algorithm proposed, which are algorithms MMA and IMA. The algorithm MMA is to transform the input number into the output presentation of Montgomery algorithm. Algorithm IMA is therefore inverse the computation of Montgomery algorithm to obtain the multiplicand. These two algorithms are in the complexity of O(n), where n is [[log.sub.2][q.sub.i]]. [q.sub.i] is a modulus. The proposed algorithm for converting the residues to a binary integer therefore runs on O(n x log m) times on O(m) processors. There are O(log m) iterations of O(n) complexity. Compared with the traditional conversion algorithm, the advantages of this proposed algorithm are not only in employing simpler operations but also in performing fewer iterations. Keywords: Chinese Remainder Theorem; Residue Number System; Euclidean algorithm; Montgomery algorithm; Division-free

Abstract. The problem of homogenization is considered for the solutions of the Neumann problem for the Lame system of plane elasticity in two-dimensional domains with channels that have the form [...]

Byline: A. V. Tyutyuev (1), V. A. Shlyk (1) Abstract: Let D be a domain in the n-dimensional Euclidean space R.sup.n, n aY= 2, and let E be a compact in D. The paper presents conditions on the compact E under which any homeomorphic mapping f = D a E [right arrow] R.sup.n can be extended to a continuous mapping f = D [right arrow] RA- .sup.n = R.sup.n a {[infinity]}. These conditions define the class of NCS-compacts, which, for n = 2, coincides with the class of topologically removable compacts for conformal and quasiconformal mappings. Bibliography: 11 titles. Author Affiliation: (1) Far-Eastern State University, Vladivostock, Russia Article History: Registration Date: 01/02/2006 Received Date: 16/06/2004 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 213--220.

Byline: N. Manjavidze (1) Abstract: We present a survey of the theory of boundary-value problems in the plane with curvilinear cuts. The problem of linear conjugation, the Riemann-Hilbert problem, and the Riemann-Hilbert-Poincare problem are considered in detail both in the classical setting and for a cut plane, with an emphasis on problems with shifts. The main focus is on the solvability conditions and index formulas in various function classes. Author Affiliation: (1) Tbilisi Technical University, Georgia Article History: Registration Date: 11/01/2006 Article note: Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.

1. Definitions and Notations Throughout this paper, we shall work with probability measures on Euclidean space [R.sup.d] (d [greater than or equal to] 1). We introduce some notational conventions. For [...]

Byline: S. E. Pastukhova (1) Abstract: We study elasticity problems in the plane (space) which is reinforced with a periodic thin network (box structure). This highly contrasting medium depends on two small related parameters [epsilon] and h which control the size of a periodicity cell and thickness of the reinforcement. For combined structures, we prove the classical homogenization principle, which is the same for any relations between parameters [epsilon] and h this is in a contrast with the case of thin structures. We use the method of two-scale convergence with respect to a variable measure, which is natural for combined structures. Bibliography: 17 titles. Author Affiliation: (1) Moscow Institute for Radiotechnics, Electronics, and Automatics, Moscow, Russia Article History: Registration Date: 12/12/2005 Received Date: 23/09/2004 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 114--144.

The paper presents the formulation and illustrates the application of an asymptotic-numerical (semianalytical) method to analyze the geometrically nonlinear behavior of plane frames. The method adopts an 'internally constrained' beam model and involves two distinct procedures: (1) an asymptotic analysis, which employs a perturbation technique to establish a sequence of systems of equilibrium differential equations and boundary conditions, and (2) the successive numerical solution of such systems, by means of the finite element method. This method can be applied to investigate the behavior of frames with arbitrarily complex configurations (member number and orientation) and leads to the determination of analytical expressions which provide: (1) the initial postbuckling behavior of perfect frames and (2) the nonlinear equilibrium paths of frames containing small initial imperfections or acted by primary bending moments, including the influence of eventual buckling mode interaction phenomena. In order to validate and illustrate the application and potential of the proposed method, several numerical results are presented, concerning (1) four validation examples (Euler column and three simple frames--two or three members), for which there exist some (perfect frame) analytical and numerical asymptotic results reported in the literature; (2) a single-bay pitched-roof frame with partially restrained column bases; and (3) a three-bay frame with two leaning columns. These results comprise (1) the initial postbuckling behavior of perfect frames (individual and coupled buckling modes) and (2) geometrically nonlinear equilibrium paths describing the behavior of frames containing initial geometrical imperfections or primary bending moments. In the latter case, most of the semianalytical results are compared with fully numerical values, yielded by finite element analyses performed in the commercial code ABAQUS. CE Database subject headings: Postbuckling; Frames; Imperfections; Perturbation; Nonlinear response.

We investigate the projective properties of the feature consisting of two concentric circles. We demonstrate there exist geometric and algebraic constraints on its projection. We show how these constraints greatly simplify the recoveries of the affine and Euclidean structures of a 3D plane. As an application, we assess the performances of two camera calibration algorithms. Index Terms--Imaging geometry, concentric circles, projective plane, circular points, camera calibration.

Kernel methods are becoming increasingly popular for various kinds of machine learning tasks, the most famous being the support vector machine (SVM) for classification. The SVM is well understood when using conditionally positive definite (cpd) kernel functions. However, in practice, non-cpd kernels arise and demand application in SVMs. The procedure of "plugging" these indefinite kernels in SVMs often yields good empirical classification results. However, they are hard to interpret due to missing geometrical and theoretical understanding, in this paper, we provide a step toward the comprehension of SVM classifiers in these situations. We give a geometric interpretation of SVMs with indefinite kernel functions. We show that such SVMs are optimal hyperplane classifiers not by margin maximization, but by minimization of distances between convex hulls in pseudo-Euclidean spaces. By this, we obtain a sound framework and motivation for indefinite SVMs. This interpretation is the basis for further theoretical analysis, e.g., investigating uniqueness, and for the derivation of practical guidelines like characterizing the suitability of indefinite SVMs. Index Terms--Support vector machine, indefinite kernel, pseudo-Euclidean space, separation of convex hulls, pattern recognition.

In this paper, we present a method of kernel optimization by maximizing a measure of class separability in the empirical feature space, an Euclidean space in which the training data are embedded in such a way that the geometrical structure of the data in the feature space is preserved. Employing a data-dependent kernel, we derive an effective kernel optimization algorithm that maximizes the class separability of the data in the empirical feature space. It is shown that there exists a close relationship between the class separability measure introduced here and the alignment measure defined recently by Cristianini. Extensive simulations are carried out which show that the optimized kernel is more adaptive to the input data, and leads to a substantial, sometimes significant, improvement in the performance of various data classification algorithms. Index Terms--Class separability, data classification, empirical feature space, feature space, kernel machines, kernel optimization.

Abstract. A model of a cone can be constructed from a piece of paper by removing a wedge and taping the edges together. The paper models discussed here expand on [...]

Isometric surfaces share the same geometric structure, also known as the "first fundamental form." For example, all possible bendings of a given surface that includes all length preserving deformations without tearing or stretching the surface are considered to be isometric. We present a method to construct a bending invariant signature for such surfaces. This invariant representation is an embedding of the geometric structure of the surface in a small dimensional Euclidean space in which geodesic distances are approximated by Euclidean ones. The bending invariant representation is constructed by first measuring the intergeodesic distances between uniformly distributed points on the surface. Next, a multidimensional scaling (MDS) technique is applied to extract coordinates in a finite dimensional Euclidean space in which geodesic distances are replaced by Euclidean ones. Applying this transform to various surfaces with similar geodesic structures (first fundamental form) maps them into similar signature surfaces. We thereby translate the problem of matching nonrigid objects in various postures into a simpler problem of matching rigid objects. As an example, we show a simple surface classification method that uses our bending invariant signatures. Index Terms--MDS (Multi-Dimensional Scaling), FMTD (Fast Marching Method on Triangulate Domains), isometric signature, classification, geodesic distance.

A problem is presented concerning a tetrahedron spinning in space, focusing on finding the axis about which the rotational intertia is optimal. Topics include center of mass, critical vectors, and eigenvalues.

ABSTRACT It is considered the notion of absolute Geometry in its evolution, from the first Non-euclidean Geometry of Lobacewski, Bolyai and Gauss till that of Smarandache Anti-Geometry. Key words: Euclidean [...]

The geometric properties of 166 hexacubes are examined. Topics include background on the number of polycubes up to order of n = 12, and planar and nonplanar hexacubes.

Research was conducted to examine the electronic and geometrical structures of DR-DCNQI molecules using ab initio molecular orbital calculations at the HF/DZP level. CNQI monomer calculations revealed that the optimized structures were close to experimental measurements within errors of 0.04 angstrom in a six-membered ring. Results indicate that the transfer integrals correlate well with lattice parameters of the c axis.

Research was conducted to predict the binding energies and geometries of benzene-BX3 and ethene-BX3 complexes using quantum mechanical ab initio calculations. Single point measurements at a much higher correlation level and larger basis sets were also performed. Results based on binding energies indicate that all the complexes were weakly bound van der Waals complexes except for the C2H4-BH3 complex. There was also a strong evidence of an unusual increase in the nucleophilicity of one of the benzene carbons in the lowest energy conformer of systems involving benzene.

The paper presents a survey on recent results on the geometry of Riemann surfaces showing that the study of closed geodesics provides a link between different aspects of Riemann surface theory such as hyperbolic geometry, topology, spectral theory, and the theory of arithmetic Fuchsian groups. Of particular interest are the systoles, the shortest closed geodesics of a surface; their study leads to the hyperbolic geometry of numbers with close analogues to classical sphere packing problems.

Area-minimizing rectifiable currents are the nicest surfaces as they lack singularities. Rectifiable currents must be oriented unlike physical surfaces. Every surface can be decomposed as a rectifiable set and the boundary and topology of the set can be defined by viewing it as a current. Rectifiable sets give an approximate measure of the tangent plane at almost every point thus lending themselves to geometric manipulation. These sets have nice closure properties under limit operations.

A method has been developed for extracting solid geometry from a single surface and has been used to link three-dimensional topography simulation with three-dimensional interconnect analysis. It begins with the intersection of multiple triangulated surfaces to enclose the desired solid, followed by the removal of unwanted surfaces using the deloop algorithm and a new algorithm for the removal of thin triangles. The thin triangle removal algorithm runs in O(n) time and can remove a large number of unnecessary facets without significantly altering the surface topography. The solid extraction algorithm runs in O(n [multiplied by] lg n) time, limited by the performance of the deloop algorithm. The solid extraction capability enables more accurate three-dimensional interconnect analyses to be performed on rigorously simulated topography using tools such as FASTCAP from M.I.T. For comparison, FASTCAP analyses have been performed on polysilicon elbows generated by simple mask extrusion and a more rigorous SAMPLE-3D simulation. Results show differences up to 44% in capacitance values between the mask extrusion model and SAMPLE-3D topography.

The vector forms of Hamilton's formulas allow a direct computation of geometric constructions in solid geometry. The use of subroutines based on these formulas points to the degeneracy conditions in linear geometry problems and helps in the suitable designing of a graphics program. A generalized vector formula, for the area of a plane polygon, has been derived from the area formula of a triangle.

The depth-dependent fraction of a phase in a polycrystalline solid's surface layer is determined from a group of diffracted intensities in x-ray diffraction at different angles of incidence. A kinematic integrated intensity expression enables the solution for the depth-dependent fraction to be expressed as a polynomial. The determination of the fraction reveals that the fraction of the monoclinic phase of ground yttria-doped tetragonal zirconia polycrystals is large near the surface and the least at the deepest layer.

This paper presents a method to generate tetrahedral meshes in three dimensiona primitives for finite element computation. Input parameters define the global size of tetrahedra which can be increased or decreased locally. All the new nodes, which are not needed to describe the geometry, are generated automatically. The algorithm first discretizes all arises of primitives to edges, then all faces of primitives are split to triangles and finally the primitives are filled with tetrahedra. The algorithm tries to generate elements close to regular tetrahedra by maximizing locally the minimum solid angles associated to a set of a few neighbouring tetrahedra.

A method for testing the profiles of spherical surfaces is presented. It consists of measuring the transversal deflection of a reflected He-Ne laser beam when the surface is rotated around an axis located near its center of curvature. A set of formulas that enables us to calculate the shape of the profile as well as the decentering of the rotation axis is obtained. By using a simple experimental setup, we found the differences between the experimental profile with respect to the ideal one; the accuracy that was obtained is approximately 3 micrometers. The method may be improved and is useful for convex as well as for concave surfaces. With minor modifications it is possible to test large surfaces and weak aspherics.

The skeleton (medial-axis surface) of an object is the locus of all points in the object's interior that have equal minimum distance from at least two distinct parts of the boundary. The skeleton can be used in blending, motion planning, medical tomography, computer vision, and in mesh generation. We discuss some simple but frequently encountered shape elements of the skeleton of CSG objects, and investigate properties of trimming surfaces delimiting the faces of skeletons.

This paper deals with visual servoing for 6-degree-of-freedom robot manipulators, and considers the problem of establishing whether and how it is possible to reach the desired location while keeping all features in the field of view and following a straight line in the Euclidean space. A path-planning technique based on a parametrization of the camera path through polynomials is proposed, which overcomes existing methods dealing with this problem. The generated image trajectory can be tracked by using an image-based visual servoing controller. [DOI: 10.1115/1.2745882] Keywords: visual servoing, point correspondences, straight line, path-planning

Abstract: In the article, we propose a new algorithm for constructing a minimal-volume ellipsoid for a given finite collection of points in an m-dimensional Euclidean space. The algorithm is based on a construction of such ellipsoids for the most simple polyhedra. Article History: Registration Date: 23/10/2004

ABSTRACT Given a triangle in Euclidean geometry it is well known that there exist an infinity of triangles each of which is similar to the given one. In Section I [...]

Paul Erdos postulated that the sum of the distances from any point I inside a triangle ABC to the vertices is greater than or equal to twice the sum of distances from I to the sides of triangle ABC. Two proofs are given for this postulate: r + r raised to the negative one is greater than or equal to two for every r greater than zero, with r = 1, and Ptolemy's theorem which pertains to a convex quadrilateral inside a circle such that the sum of the products of opposite sides is equal to the product of diagonals.

The geometric effects in guide/antiguide intensity modulators were investigated. The results showed thatthe modulators with longer modulation active lengths had larger on/off ratios. The input light had difficulty in penetrating the antiguide regions when the gap size was too large. Optimum waveguide width decreased as the gap size increased. When the guide width was less than three micrometers, the confinement factor was small. However, when it was greater than eight micrometers, the waveguide exhibited multimode characteristics.

Butterfly theorem is one of the more appealing theorems in plane geometry and has attracted many lovers of mathematics, some of whom have produced simple and elegant proofs while others devised various generalizations. A proof of butterfly theorem for quadrilaterals is provided which depends primarily on the properties for areas of triangles.

A case is made for the importance of carrying on a dialogue with the past, both in the arts and in science. Examples from geometry show that some classical results have interesting generalizations to 'N' dimensions demonstrating the permanence of traditional theorems.