Family Of Ellipses Research Articles

The cusphere

A cusphere, portmanteau of cube and sphere, is the constant magnitude surface in imaginary scator algebra. The computer renderings exhibit a fascinating geometry with great aesthetic value. The scator space in 1 + 2 dimensions, is endowed with a scalar (real) and two hyperimaginary components that can be represented in the three orthogonal axes of Euclidean space. A myriad of plane curves are obtained on the cusphere surface: Families of ellipses, circles and lemniscatae are three of the familiar ones. There are also less conventional ones, like four pointed stars and squircles. Implicit as well as parametric equations of these curves are derived. The three dimensional geometrical object is explored from different perspectives.

VISROC 2.0: Updated Software for the Visualization of the significance of Receiver Operating Characteristics based on confidence ellipses

The Receiver Operating Characteristics (ROC) method is used to evaluate the diagnostic accuracy of binary quantitative tests in a broad spectrum of disciplines, including medicine, physics of complex systems, geophysics, meteorology, etc. The estimation of the significance of the examined prediction method is of high importance and it's usually approximated by Monte Carlo calculations. To simplify this problem, a FORTRAN code called VISROC was submitted to the CPC Program Library in 2014. VISROC evaluates the significance of binary diagnostic and prognostic tools for a family of k-ellipses which are based on confidence ellipses and cover the whole ROC space. Since that time, the code has been significantly improved and several new capabilities have been added. Most importantly, a Graphical User Interface (GUI) has been implemented, which can be invoked using either the R shiny web application or the Python application available for Windows, Mac, and Linux operating systems, both of which are described here. New version program summaryProgram Title: VISROC 2.0CPC Library link to program files:https://doi.org/10.17632/dktybrydvf.1Licensing provisions: AGPLv3Programming language: R and PythonJournal reference of previous version: N.V. Sarlis, S.-R. G. Christopoulos, Comput. Phys. Commun. 185 (2014) 1172.Does the new version supersede the previous version?: YesReasons for the new version: Overcome previous version's limitations and implement GUI.Summary of revisions: R and Python applications, GUI, ability to input files containing ROC data.Nature of problem: The Receiver Operating Characteristics (ROC) [1] is a method used to evaluate the diagnostic ability of binary tests and prediction methods in a broad spectrum of disciplines. Apart from the sensitivity (or True Positive rate, TPr) and specificity (which is complementary to False Positive rate, FPr, i.e., specificity=1-FPr) measures, the estimation of the statistical significance of the examined method is of high importance. VISROC evaluates the significance of binary diagnostic and prognostic tools for a family of k-ellipses which are based on confidence ellipses and cover the whole ROC space.Solution method: Using the statistics of random binary predictions for a given value of the predictor threshold ϵt, one can construct the corresponding confidence ellipses. The envelope of these confidence ellipses is estimated by varying ϵt in the interval [0,1] and one obtains a new family of ellipses, called k-ellipses [2]. They cover the whole ROC space and lead to a well defined Area Under the Curve (AUC). Mason and Graham [3] have shown that AUC follows the Mann-Whitney U-statistics [4] which can be used [5] to estimate the statistical significance of each k-ellipse. As the transformation is invertible, any point on the ROC plane corresponds to a unique value of k, hence it belongs to a unique k-ellipse that allows the estimation of the probability (p-value) to obtain this point by chance. The present GUI applications provide the p-value on the ROC plane as well as the k-ellipses corresponding to the (p=)10%, 5% and 1% significance levels using as input the number of the positive (P) and negative (Q) cases to be predicted.

Mathematical Models and Nonlinear Optimization in Continuous Maximum Coverage Location Problem

This paper considers the maximum coverage location problem (MCLP) in a continuous formulation. It is assumed that the coverage domain and the family of geometric objects of arbitrary shape are specified. It is necessary to find such a location of geometric objects to cover the greatest possible amount of the domain. A mathematical model of MCLP is proposed in the form of an unconstrained nonlinear optimization problem. Python computational geometry packages were used to calculate the area of partial coverage domain. Many experiments were carried out which made it possible to describe the statistical dependence of the area calculation time of coverage domain on the number of covering objects. To obtain a local solution, the BFGS method with first-order differences was used. An approach to the numerical estimation of the objective function gradient is proposed, which significantly reduces computational costs, which is confirmed experimentally. The proposed approach is shown to solve the maximum coverage problem of a rectangular area by a family of ellipses.

Inellipses of Convex Polygons, Brianchon’s Theorem, and Bradley’s Theorem

Three problems are discussed: the classification theorem for families of ellipses inscribed in convex polygons, Brianchon’s theorem, and Bradley’s theorem. Each is presented with an elementary synthetic proof.

Twin Composites, Strange Continued Fractions, and a Transformation that Euler Missed (Twice)

We introduce a polynomial E(d, t, x) in three variables that comes from the intersections of a family of ellipses described by Euler. For fixed odd integers $$t\ge 3$$, the sequence of E(d, t, x) with d running through the integers produces, conjecturally, sequences of “twin composites” analogous to the twin primes of the integers. This polynomial and its lower degree relative R(d, t, x) have strikingly simple discriminants and resolvents. Moreover, the roots of R for certain values of d have continued fractions with at least two large partial quotients, the second of which mysteriously involves the 12th cyclotomic polynomial. Various related polynomials whose roots also have conjecturally strange continued fractions are also examined.

The Focus Locus Problem and Toric Sections

SummaryWe investigate curves (foci loci) traced by the foci of one-parameter families of ellipses. The elliptical families are shadows cast by a circle. The focal paths turn out to be cross sections of tori.

Some Exact Analytical Solutions for Two-Dimensional Flow of an Incompressible Second Grade Fluid

This investigation deals with some exact analytical solutions of the incompressible second grade fluid by using the method based on the separation of variables. In many cases, this method can derive exact analytical solutions easier than other methods. A family of solutions is derived in this paper, which governs scientific and engineering experimentations. The derived solutions represent the flows having streamlines as a family of ellipses, parabolas, concentric circles, and rectangular hyperbolas. From practical point of view, these flows have applications in many manufacturing processes in industry. Some physical features of the derived solutions are also illustrated by their contour plots.

Some Exact Solutions to Equations of Motion of an Incompressible Second Grade Fluid

In this paper, we derive some exact solutions of the equations governing the steady plane motions of an incompressible second grade fluid. For this purpose, the vorticity and stream functions both are expressed in terms of complex variables and complex functions. The derived solutions represent the flows having streamlines as a family of ellipses, parabolas, concentric circles, and rectangular hyperbolas. Some physical features of the derived solutions are also illustrated by their contour plots.

A study on unsteady seepage field of fractured horizontal well in tight gas reservoir

Based on geologic and development characteristics of tight gas reservoir, a tight gas reservoir/fractured horizontal wellbore coupling model at unsteady state is derived by Green functions and Newman product principle, and the analytical solutions of dimensionless velocity and dimensionless pressure are obtained as well. Practical calculation example shows that main seepage of a fractured horizontal well surrounds the artificial fractures. When more fractures are fractured, the shapes of isobars from inner to outer are: families of conjugate ellipses, bone-shaped curves and a family of ellipses. According to the seepage law, effective technological measures of drilling and producing are given in the paper.

PRESSURE BEHAVIOR OF FRACTURED GAS WELLS IN LOW-PERMEABILITY RESERVOIRS WITH THRESHOLD PRESSURE GRADIENT

This paper presents analytical solutions to the pressure transient equations of a uniform-flux hydraulically fractured gas well in low-permeability tight formation with threshold pressure gradient. A method to determine the location of the moving boundary front is also presented. This paper concludes that the pseudopressure equipotential lines for a fully penetrating vertical uniform-flux hydraulically fractured gas well are a family of ellipses, whose focii are the two end points of the hydraulic fracture. This paper also concludes that both pressure transient distance in the direction perpendicular to the fracture and pseudopressure drop at the wellbore are approximately linear funcitons of the square root of producing time. Finally, the solution procedure proposed in this paper is a fast tool to evaluate hydraulically fractured well perfomance in a tight gas formation.

Generic Ellipses as Envelopes

SummaryThe astroid is well known as an envelope both of a family of line segments and a family of ellipses. The relationship between these two families is investigated by asking when two generic ellipses are tangent to each other, where a generic ellipse is described by |x/a|m + |y/b|m = 1 for some a, b, m > 0. Results are illustrated with several examples of envelopes of families of generic ellipses.

PRESSURE BEHAVIOR OF UNIFORM-FLUX HYDRAULICALLY FRACTURED WELLS IN LOW-PERMEABILITY RESERVOIRS WITH THRESHOLD PRESSURE GRADIENT

The threshold pressure gradient, which is associated with non-Darcy flow in low-permeability reservoirs, is defined as the level of pressure gradient that must be attained to enable the fluid to overcome the viscous forces and start to flow. With low-velocity and non-Darcy flow, the fluid flow boundary is controlled by the threshold pressure gradient and can extend outward continuously, while the fluid beyond this boundary cannot flow. This paper presents analytical solutions to the pressure transient equations of uniform-flux hydraulically fractured wells in low-permeability reservoirs with a threshold pressure gradient. These solutions are obtained by using the Green’s functions method with numerical approximations. A method to determine the location of the moving boundary front is also presented. This paper concludes that the equipotential surfaces of a fully penetrating vertical uniform-flux hydraulically fractured well are a family of ellipses, whose focuses are the two end points of the hydraulic fracture. The paper also concludes that both the pressure transient equivalent radius and pressure drop at the wellbore are linear functions of the square root of producing time. It is known that the radius of investigation of conventional reservoirs is independent of flow rate, but this study concludes that the pressure transient equivalent radius is a function of flow rate. When threshold pressure gradient is equal to zero, the proposed formula to calculate the pressure transient equivalent radius almost reduces to the formula to calculate radius of investigation of conventional reservoirs in the literature. Finally, the solution procedure proposed in this paper is a fast tool to evaluate hydraulically fractured well performance in low-permeability reservoirs.

Productivity Formulae of an Infinite‐Conductivity Hydraulically Fractured Well Producing at Constant Wellbore Pressure Based on Numerical Solutions of a Weakly Singular Integral Equation of the First Kind

In order to increase productivity, it is important to study the performance of a hydraulically fractured well producing at constant wellbore pressure. This paper constructs a new productivity formula, which is obtained by solving a weakly singular integral equation of the first kind, for an infinite‐conductivity hydraulically fractured well producing at constant pressure. And the two key components of this paper are a weakly singular integral equation of the first kind and a steady‐state productivity formula. A new midrectangle algorithm and a Galerkin method are presented in order to solve the weakly singular integral equation. The numerical results of these two methods are in accordance with each other. And then the solutions of the weakly singular integral equation are utilized for the productivity formula of hydraulic fractured wells producing at constant pressure, which provide fast analytical tools to evaluate production performance of infinite‐conductivity fractured wells. The paper also shows equipotential threads, which are generated from the numerical results, with different fluid potential values. These threads can be approximately taken as a family of ellipses whose focuses are the two endpoints of the fracture, which is in accordance with the regular assumption in Kuchuk and Brigham, 1979.

Visualizing the extra symmetry of the Kepler problem

The extra symmetry of the Kepler problem corresponding to the conservation of the Runge–Lenz vector can be visualized by studying how entire Kepler ellipses evolve under symmetry transformations. After discussing the general case, we show that the evolution generated by a component of the Runge–Lenz vector in the plane of the orbit yields the same family of ellipses in the plane as the projection onto the plane of a rotation of a certain circle in three dimensions.

Modeling of the Sagittal Cervical Spine as a Method to Discriminate Hypolordosis

Computer analysis of digitized vertebral body corners on lateral cervical radiographs. Using elliptical and circular modeling, the geometric shape of the path of the posterior bodies of C2-C7 was sought in normal, acute pain, and chronic pain subjects. To determine the least squares error per point for paths of geometric shapes, minor axis to major axis elliptical ratios (b/a), Cobb angles, sagittal balance of C2 above C7, and posterior tangent segmental and global angles. When restricted to cervical lordotic configurations, normal, acute pain, and chronic pain subjects have not been compared for similarities or differences of these parameters. Conventional Cobb angles provide only a comparison of the endplates of the distal vertebrae, while geometric modeling provides the shape of the entire sagittal curves, the orientation of the spine, and segmental angles. Radiographs of 72 normal subjects, 52 acute neck pain subjects, and 70 chronic neck pain subjects were digitized. For normal subjects, the inclusion criteria were no kyphotic cervical segments, no cranial-cervical symptoms, and less than +/- 10 mm horizontal displacement of C2 above C7. In pain subjects, inclusion criteria were no kyphotic cervical segments and less than 25 mm of horizontal displacement of C2 above C7. Measurements included segmental angles, global angles of lordosis (C1-C7 and C2-C7), height-to-length ratios, anterior weight bearing, and from modeling, circular center, and radius of curvature. In the normal group, a family of ellipses was found to closely approximate the posterior body margins of C2-C7 with a least squares error of less than 1 mm per vertebral body point. The only ellipse/circle found to include T1, with a least squares error of less than 1 mm, was a circle. Compared with the normal group, the pain group's mean radiographic angles were reduced and the radius of curvature was larger. For normal, acute, and chronic pain groups, the mean angles between posterior tangents on C2-C7 were 34.5 degrees, 28.6 degrees, and 22.0 degrees, C2-C7 Cobb angles were 26.8 degrees, 16.5 degrees, and 12.7 degrees, and radius of curvature were r = 132.8 mm, r = 179 mm, and r = 245.4 mm, respectively. The mean cervical lordosis for all groups could be closely modeled with a circle. Pain groups had hypolordosis and larger radiuses of curvature compared with the normal group. Circular modeling may be a valuable tool in the discrimination between normal lordosis and hypolordosis in normal and pain subjects.

Calculational technique for an iris with an elliptical aperture

This paper proposes an analytical calculational technique for a uniform iris with an aperture of elliptical shape. The purpose of the device is to regulate the depth of sharply imaged space. Anamorphic objectives are not suitable for this, because the depth of field is different in the vertical and horizontal directions. The technique is based on an original theory of envelopes, which in this case considers the working face of the leaf of the iris that coincides with a curve that envelops a family of ellipses that open out to a certain angle relative to the fixed pin of the iris leaf.

Stress trajectories in the presence of friction

We study the stress field around an oblique closed defect in polymethylmethacrylate (PMMA) plates under uniaxial compression in the presence of friction. Every specimen (PMMA plate) with a slot may be compared to an infinite plate with an elliptical cavity where the long axis is equal to the slot length. We have calculated the stress tensor for an elliptical oblique closed slot using the system of curvilinear co-ordinates defined by two families of ellipses and homofocal hyperbolas and we have plotted the isovalues and the stress trajectories in terms of the friction intensity.

Failure Prediction for Cross-Rolled Beryllium Sheet Material

A failure model for cross-rolled beryllium SR-200 sheets is developed for material loaded in a complex state of stress. Coefficients of the Tsai-Wu criterion are determined from a series of special laboratory experiments. Tests include circular plates loaded by a concentric ring, as well as in-plane compression and off-axis plate specimens. Complex states of stress lead to brittle failure of the anisotropic material. Failure surfaces obtained from the criterion form a family of ellipses when plotted in standard Cartesian coordinates. The criterion is incorporated into a general purpose finite element analysis code. Numerical simulation incrementally applies loads to a structural component that is being designed and checks each nodal point in the model for exceedance of the failure criterion. To demonstrate applicability of the predictive capability of the criterion, a 2.54-mm thick beryllium plate is placed under clamped edge conditions and loaded to failure by a central transverse point load. A numerical model of the structure predicts the failure load to within three percent.

Ion separation in imperfect fields of the quadrupole mass analyzer Part II. Ion beam dynamics in the phase-space of the fringing field

The theoretical aspects of ion separation taking account of fringing fields in the quadrupole mass analyzer are analyzed by applying beam dynamics in phase-space. The mathematical simulation of fringing fields is developed by the principles of statistical mechanics. Numerical solutions of differential equations (the Hill equation) for ion motion in the fringing field are considered using matrix conversions with Galerkin-Slezkin's integral averaging in the iteration procedure. Analysis of beam dynamics in the phase-space where the law of phase volume conservation (Liouville's theorem) is valid indicates that, for stable ion motion, beam boundaries are described by a family of ellipses. The parameters of the ellipses are determined by the elements of the state transition matrix for an arbitrary set of initial conditions. Theoretical analysis indicates that beam divergence has a major influence on transmission. Transmission is an exponential function of relative transit time. The optimum time of transit through the fringing field has exponential dependences on the initial values of radial and angular envelopes of a beam with an asymptotic value of 1.77π.

Diffusive void bifurcation in stressed solid

Interconnects are susceptible to solid diffusion under residual stress, electric current, and elevated temperature. As atoms diffuse, voids nucleate, drift, and enlarge. At some point, the voids of rounded shape can collapse to narrow slits and sever the interconnects. The fatal slits are often found to be transgranular, i.e., each slit cuts across a single grain. They have raised many concerns, but the underlying mechanism has remained unclear. It is proposed that a void changes shape due to surface diffusion under the combined action of surface energy, elastic energy, and electric current. The void will be rounded if surface energy prevails, but will collapse to a slit if the elastic energy or the electric current prevails. A cylindrical void in an infinite crystal under biaxial stresses, but under no electric current, is analyzed. Four things are done, as follows: (1) A suitable thermodynamic potential is minimized and maximized to select, among a family of ellipses, equilibrium void shapes. The bifurcation diagram consists of a subcritical pitchfork and two Griffith cracks. (2) A void under biased stresses is analyzed to illustrate the effect of imperfections. (3) Exact initial bifurcation modes are determined. The critical loads for the successive modes are closely separated, indicating that the shape evolution will be sensitive to initial imperfections. (4) A variational principle for shape evolution under stress, current and surface energy is identified. Stress-induced evolution time is estimated by using this variational principle.