Elliptical Arcs Research Articles

Numerical ranges of weighted shift matrices with periodic weights

Let A be an n-by- n ( n ⩾ 2 ) matrix of the form 0 a 1 0 ⋱ ⋱ a n - 1 a n 0 . We show that if the a j ’s are nonzero and their moduli are periodic, then the boundary of its numerical range contains a line segment. We also prove that ∂ W ( A ) contains a noncircular elliptic arc if and only if the a j ’s are nonzero, n is even, | a 1 | = | a 3 | = ⋯ = | a n - 1 | , | a 2 | = | a 4 | = ⋯ = | a n | and | a 1 | ≠ | a 2 | . Finally, we give a criterion for A to be reducible and completely characterize the numerical ranges of such matrices.

Ellipse to Hyperbola: “With This String I Thee Wed”

SummaryWe introduce a string mechanism that traces both elliptic and hyperbolic arcs having the same foci. This suggests replacing each focus by a focal circle centered at that focus, a simple step that leads to new characteristic properties of central conics that also extend to the parabola.The classical description of an ellipse and hyperbola as the locus of a point whose sum or absolute difference of focal distances is constant, is generalized to a common bifocal property, in which the sum or absolute difference of the distances to the focal circles is constant. Surprisingly, each of the sum or difference can be constant on both the ellipse and hyperbola. When the radius of one focal circle is infinite, the bifocal property becomes a new property of the parabola.We also introduce special focal circles, called circular directrices, which provide equidistance properties for central conics analogous to the classical focus-directrix property of the parabola. Those familiar with paperfolding activities for constructing an ellipse or hyperbola using a circle as a guide, will be pleased to learn that the guiding circle is, in fact, a circular directrix.

Investigating the Performance of Alternate Regression Weights by Studying All Possible Criteria in Regression Models with a Fixed Set of Predictors

We describe methods for assessing all possible criteria (i.e., dependent variables) and subsets of criteria for regression models with a fixed set of predictors, x (where x is an n×1 vector of independent variables). Our methods build upon the geometry of regression coefficients (hereafter called regression weights) in n-dimensional space. For a full-rank predictor correlation matrix, Rxx, of order n, and for regression models with constant R2 (coefficient of determination), the OLS weight vectors for all possible criteria terminate on the surface of an n-dimensional ellipsoid. The population performance of alternate regression weights—such as equal weights, correlation weights, or rounded weights—can be modeled as a function of the Cartesian coordinates of the ellipsoid. These geometrical notions can be easily extended to assess the sampling performance of alternate regression weights in models with either fixed or random predictors and for models with any value of R2. To illustrate these ideas, we describe algorithms and R (R Development Core Team, 2009) code for: (1) generating points that are uniformly distributed on the surface of an n-dimensional ellipsoid, (2) populating the set of regression (weight) vectors that define an elliptical arc in ℝn, and (3) populating the set of regression vectors that have constant cosine with a target vector in ℝn. Each algorithm is illustrated with real data. The examples demonstrate the usefulness of studying all possible criteria when evaluating alternate regression weights in regression models with a fixed set of predictors.

Temperature dependence and fracture criterion of mixed mode I/II fracture toughness of phenolic resin for friction material

AbstractIn this study, the temperature dependence of the mixed‐mode fracture toughness of the phenolic resin for friction materials is investigated. For pure mode I, the fracture toughness decreases as the temperature increases, and it increases again after showing its minimum value. For pure mode II, the fracture toughness shows a similar trend but has its minimum value at a higher temperature. The temperature dependence of the mixed‐mode fracture toughness varies depending on the mode mixity, which is attributed to the different sensitivity to the relaxation phenomenon. At room temperature, as the fracture toughness for pure mode I and II are almost the same, the fracture locus shows a circular arc. At elevated temperatures, the locus becomes smaller and noncircular. At high temperature, the fracture locus shows an elliptical arc, where the fracture toughness for pure mode II is smaller than that for mode I. An empirical fracture criterion based on the time‐temperature dependence of the resin is proposed, and the proposed method successfully predicts the fracture toughness under various conditions of the temperature, time, and mode mixity. The crack initiation angles, on the other hand, are almost consistent regardless of the temperature, which agree with the maximum hoop stress theory. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011

Geometric properties for the design of unusual member cross-sections in bending

This short communication presents a series of general geometric properties of unusually-shaped member cross-sections. A skeleton derivation of the most important properties is presented and a method is illustrated for constructing more complex cross-sections. The focus is on the cross-sectional area, centroid and second moment of area, used in classical engineering design for bending. The shapes presented include circular and elliptical segments, sectors and arcs, semiellipses, superellipses, and combinations thereof. The superellipse allows a vast range of novel cross-sections to be described analytically and is a powerful mathematical tool for the analysis of unusual curved shapes. The equations are presented as composites of their geometric components so as to facilitate practising engineers in their implementation as programmed functions or subroutines.

Remarks on the Erroneous Dispersion Surfaces From a Pair of a Hyperbolic Branch and An Elliptical Arc of the Intersected Two Laue Spheres Based on the Usual Crude Approximation

In almost all previous works, the hyperbolic dispersion surfaces of the central proper quadrics have been crudely derived from the degree of reduction from the bi-quadratic equation by use of some roughly indefinable approximate relations. Moreover, neglecting the high symmetry of the hyperbola, both the branches have been approximated on the asymmetric surfaces composed of a pair of a branch of the hyperbola and a vertex of the ellipse without the presentation of reasonable evidence. Based upon the same dispersion surfaces equation, a new original gapless dispersion surfaces could be rigorously introduced without crude omission of even a term in the bi-quadratic equation based upon usual analogy with the extended band theory of solid as the close approximation to the truth.

An elliptical SPECT system with slit–slat collimation for cardiac imaging

Cardiac studies are a good candidate for SPECT (single photon emission computed tomography) because of the large clinical demand and the need for improved image quality. But SPECT imaging suffers from poor spatial resolution and high statistical noise. A new SPECT system with slit–slat collimation arranged on an elliptical arc for cardiac imaging is proposed in this paper. Simulated emission computed tomography data are generated along an elliptical moving orbit with system configuration parameters. The iterative reconstruction techniques are used to implement the cardiac imaging of the proposed SPECT system. Image reconstruction can be done using the OS-EM algorithm from the data collected. This system is developed to improve the reconstructed image if attenuation correction and depth-dependent correction are included in the reconstruction, and the body contour is used in the reconstruction for severely truncated data. The simulation results show that the present methods can provide significant improvement in the spatial resolution and the image quality of SPECT sets.

Reconstruction of individualized dress forms using parameterized silhouettes

PurposeThe purpose of this paper is to develop a fast parameterized modeling approach to generate individualized dress forms for realistic human bodies.Design/methodology/approachAn individualized dress form is created by deriving a new set of fitting functions from a number of key existing dressing parameters and pre‐defined templates. The fitting functions only contain simple shapes of circular and/or elliptical arcs, which can be modified computationally based on a few personal dressing data.FindingsThis paper reaffirms that individual body shape can be adequately described by a number of critical cross‐section silhouettes, and a personalized dress form can be constructed based on key dressing parameters and templates.Originality/valueThe fitting functions and relevant dressing data for specific cross‐sectional silhouettes are determined, permitting a user to create personalized dress forms only by inputting a simple set of dressing parameters.

Reconciling Circular and Elliptical Arcs

Abstract Most graphics packages allow users to create arcs from both circles and ellipses. Unfortunately, in some packages the elliptical arcs are drawn incorrectly, resulting in distorted arcs that do not begin and end at the specified angles. Based on the hypothesis that this is because the system is drawing elliptical arcs by non-uniformly scaling circular arcs, this paper presents the geometry for pre-distorting the desired angles so that the final image is correct.

Geometrical and wave-optical effects on the performance of a bent-crystal dispersive X-ray spectrometer

The X-ray focusing properties of a bent single crystal diffracting in Bragg geometry are discussed. First, it is assumed that a polychromatic point source is focused to a point image. The elliptical arc that the crystal must trace and the aberrations caused by bending the crystal cylindrically are derived from the ray paths. For a source of finite size, the magnification is found to vary over the crystal's length, so that rays of different wavelength produce images of different size. More realistic treatments of penetration and diffraction are performed with spherical monochromatic incident waves, using Takagi–Taupin calculations to create the diffracted wave and the Fresnel integral to trace the diffracted wave's evolution. Such “wave-optical” calculations on a symmetric Si (111) crystal with 7keV X-rays predict beam sizes different from those found in ray traces. Optimal sample and detector placement therefore requires wave effects to be considered.

Chaotic properties of the truncated elliptical billiards

Chaotic properties of symmetrical two-dimensional stadium-like billiards with piecewise flat and elliptical segments are studied numerically and analytically. Their sensitivity to small variations of the shape parameters can be usefully applied for optimal construction of the dielectric and polymer optical microresonators. For the two-parameter truncated elliptical billiards (TEB) the existence and linear stability of several periodic orbits are investigated in the full parameter space. Poincaré plots are computed and used for evaluation of the chaotic fraction of the phase space by means of the box-counting method. A highly chaotic behavior prevails in the region of elongated elliptical arcs, where most of the existing orbits are either neutral or unstable. In the parameter space, the upper limit of the fully chaotic behavior is reached when the relation between the two shape parameters becomes δ = 1 - γ 2 , corresponding to truncated circles. Above this limit, for flattened elliptical arcs, mixed dynamics with numerous stable elliptic islands is found. In both regions parabolic orbits are present, many of them identical to orbits within an ellipse. These properties of the TEB differ remarkably from the behavior in the elliptical stadium billiards (ESB), where the chaotic region in the parameter space was strictly bounded from both sides. In order to follow the transition from TEB to ESB, a generalisation to a novel three-parameter family (GTEB) of stadium-like boundary shapes with elliptical arcs is proposed.

Image matching with higher-order geometric features

We propose a geometric matching technique in which line segments and elliptical arcs are used as edge features. The use of these higher-order features renders feature representation efficient. We derive distance measures to evaluate the similarity between the features of the model and those of the image. The model transformation parameters are found by searching a 3-D transformation space using cell-decomposition. The performance of the proposed method is quite good when tested on a variety of images.

Light Energy Extraction From the Minor Surface Arc of an Electrically Pumped Elliptical Microcavity Laser

In this paper, the efficiency of extracting light energy from the minor surface arc of the elliptical microcavity laser is investigated for the first time using the dynamic thermal electron quantum medium finite-difference time-domain model. Light energy is extracted from the minor surface arc of the elliptical microcavity to a perpendicular waveguide. It is deduced from the field distribution that extraction efficiency of the TM polarization is higher due to wider transverse field profile. Optimum light energy is extracted when the waveguide is 0.342 ?m away from the edge of the elliptical minor arc. Higher extraction efficiency is also obtained when the waveguide index is lower than the microcavity index. It is believed that this method of extracting light energy allows for smaller device size with higher power output for building photonic integrated circuit.

New Algorithm for Great Elliptic Sailing (GES)

An analytical method and algorithm for great elliptic sailing (GES) calculations is presented. The method solves the complete GES problem calculating not only the great elliptic arc distance, but also other elements of the sailing such as the geodetic coordinates of intermediate points along the great elliptic arc. The proposed formulas provide extremely high accuracies and are straightforward to be exploited immediately in the development of navigational software, without the requirement to use advanced numerical methods. Their validity and effectiveness have been verified with numerical tests and comparisons to extremely accurate geodetic methods for the direct and inverse geodetic problem.

Eye synthesis using the eye curve model

Eyes are a critical part in exhibiting facial expressions. Because of the appearance diversity of eyes due to motion, it is difficult to synthesize eye with a particular facial expression. Traditional methods have failed to adequately catch motion-related appearance changes. In order to generate a photorealistic expression eye, we propose a two-step method. First, we propose a curve-based model to represent eyes. The model uses one circle and four skewed elliptical arcs to represent the shape of eyes, and divides the entire eye region into six sub-regions that correspond to different anatomical components of eyes. Then we propose a structure-based-similarity (SBS) framework to synthesize expression eyes using the eye curve model. The main contributions of this paper are: first of all, the proposed eye curve model can represent the diversity of eyes due to motion and structure, which is better than some traditional models. Second, the SBS framework is flexible. In multiple samples and multiple targets situation, the framework can synthesize eyes which exhibit same expression with different personal style. In single sample and single target situation, the framework can clone this expression successful. Experimental results show that synthesized eyes are realistic and expressive.

A generalized model for conic flexure hinges

Flexure hinges have been used to produce frictionless and backlashless transmission in a variety of precision instruments. Many kinds of flexure profile were proposed during the past decade. The present work brings elliptical arc, parabolic, and hyperbolic profiles together by proposing a generalized conic flexure hinge model. By utilizing the generalized equation for conic curves in polar coordinates, all the elements in the compliance and precision matrices for conic flexure hinges are deduced. These equations were verified by finite element analysis and experimentation. The analytical results are within 11% error compared to the finite element results and within 6% error compared to the experimental results.

Real-time elliptical head contour detection under arbitrary pose and wide distance range

In this paper, we propose a real time elliptical head contour detection method based on quadrant arcs, which is efficient and robust to arbitrary head pose and wide distance range. First, the moving object area is detected according to background model which is built on three color channels. Then, all the valid elliptical arcs are extracted out from connected edges, and classified into four kinds of quadrant arc sets according to their gradient information. Finally, the arcs lying in different sets are combined to fit out the elliptical head contour based on the least square method. Experimental results confirm the robustness and the accuracy of this method under arbitrary head pose and wider distance range, as well as the real time property, strong robustness to long or short hair and with or without hat.

The diffusion overpotential increase and appearance of overlapping arcs on the Nyquist plots in the low humidity temperature test conditions of polymer electrolyte fuel cell

We have investigated cell voltage characteristics and AC impedance characteristics of polymer electrolyte fuel cells (PEFCs) at various humidity temperatures for H 2/O 2 and H 2/air test conditions (current density: 200, 400, and 600 mA cm −2, cell temperature: 80 °C, humidity temperature at respective electrodes: 40, 50, 60, and 70 °C). The diffusion overpotential increases with decreasing humidity in the low humidity temperature region such as 40 and 50 °C and the Nyquist plots obtained from AC impedance measurements show a small arc superposed on an elliptic arc in the low frequency region. The diameter of this small arc increases with decreasing humidity temperature from 50 to 40 °C or with increasing current density. These results suggest that oxygen transport across the ionomer film in the catalyst layer is significantly reduced in the low humidity condition, which causes a decrease in cell voltage, increase in diffusion overpotential, the appearance of overlapping arcs (two separate arcs) in the lower frequency region on the Nyquist plots, and the increase of mass transport resistance from Nyquist plots.

A new generalized model for elliptical arc flexure hinges

Flexure hinges have been used in many engineering areas where high precision and sensitivity are required. Many kinds of flexure profiles were proposed during the past decade. Therefore, a general closed-form solution for flexure hinges of different profiles that incorporates the profile selection and parameter design will be of great benefit to the hinge design. The present work brings circular, right-circular, and elliptical profiles together by proposing a generalized flexure hinge model, which we call elliptical arc flexure hinges (whose maximum eccentric angle phi(m) ranges from 0 to pi/2) to distinguish from the existing elliptical flexure hinge (phi(m)=pi/2). Based on the theories of mechanics of materials, all the elements in the compliance matrix for elliptical arc flexure hinges are deduced by introducing the eccentric angle of ellipse as the integral variable. These compliance equations simply boil down to four integrals, thus simplifying the compliance calculation. These equations also apply to elliptical (phi(m)=pi/2), circular (a=b), and right-circular (phi(m)=pi/2 and a=b) hinges. These compliance equations were checked by comparing them with the results of finite element analysis, the existing equations, and experiment results. The comparison results show that these generalized equations are concise and adequate for most design purposes.

Algebraic computations and applications to geometry (abstract only)

Real algebraic numbers are the real numbers that are real roots of univariate polynomials with integer coefficients. We study exact algorithms, from a theoretical and an implementation point of view, based on integer arithmetic of arbitrary precision, for computations with real algebraic numbers and applications of these algorithms on problems and algorithms in non linear computational geometry. In order to construct a real algebraic number we must compute the real roots of a univariate polynomial with integer cofficients. We unify and simplify the theory behind the subdivision based algorithms for real root isolation and we improve the complexity of the algorithm that is based on the continued fraction expansion of the real numbers. The best known complexity bound up today is achieved using new techniques. Moreover, we prove that the bound holds for non square-free polynomials and that in the same complexity bound we can compute the multiplicities of the real roots. We prove a new bound for the expected complexity of the algorithm based on continued fractions. We generalize the real root isolation algorithms to bivariate polynomial systems. Our experimental analysis proves the effectiveness of our methods. The algorithms that we consider for computations with real algebraic numbers are construction, comparison, sign evaluation and quantifier elimination. If the degree of the polynomial is small, i.e. ≤ 4 in the univariate case and ≤ 2 in the bivariate case, we propose special purpose algorithms that have constant arithmetic complexity. For all the algorithms we present a C++ implementation and an experimental analysis. In computational geometry we study the predicates needed by the algorithms for the arrangement of elliptic arcs in the plane and the computation of the Voronoi diagram of ellipses, also in the plane. Finally, given a convex lattice polygon we study algorithms for decomposing it to two other convex lattice polygons, such that their Minkowski sum is the original polygon.