Limacon Of Pascal Research Articles

Some results for the family of univalent functions related with Limaçon domain

The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limacon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which $ \frac{\varsigma f^{\prime}(\varsigma)}{f(\varsigma)} $ and $ \frac{(\varsigma f^{\prime}(\varsigma))^{\prime}}{f^{\prime}(\varsigma)} $ map open unit disc onto region bounded by limacon are studied. Coefficients bounds, Fekete Szeg $ \ddot{ \rm{o}} $ inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.

Automated calibration of an optomechanical derotator using 6-axes parallel kinematics and industrial image processing algorithms

A concept for an optomechanical derotator is presented. The derotator allows optical measurements of rotating objects during operation. To guarantee a stationary optical image, the optical axis of the derotator needs to be coaxially aligned to the rotational axis of the measured object. The correlation between the movement of the optical image and a miscalibration is explained by a mathematical model. The movement of the optical image is tracked with a high-speed camera. Approximating the tracked path as a Limacon of Pascal, a parameter corresponding to the magnitude of the miscalibration is identified. This parameter is minimized by modifying the position and the orientation of the derotator with a hexapod. The limitations of this procedure are analyzed and further approaches are discussed.

Localization of the eigenvalues of a pencil of positive definite matrices

Let A and B be real square positive definite matrices close to each other. A domain S on the complex plane that contains all the eigenvalues λ of the problem Az = λBz is constructed analytically. The boundary ∂S of S is a curve known as the limacon of Pascal. Using the standard conformal mapping of the exterior of this curve (or of the exterior of an enveloping circular lune) onto the exterior of the unit disc, new analytical bounds are obtained for the convergence rate of the minimal residual method (GMRES) as applied to solving the linear system Ax = b with the preconditioner B.

Refraction holodiagrams and Snell's law

Summary The refraction holodiagram RHD is analyzed here with respect to the law of refraction. Particularly, we study the surface that exactly conjugates by refraction a virtual point source with a real image or conversely. By using the total optical path as a parameter we build a diagram that consists in a family of Descartes ovals of the apple type that contains the Pascal's limacon as a particular extreme case and the spherical surface with the Weierstrass points as another. These representations permit the straightforward application of Fermat's principle in the case of arbitrary refracting surfaces and show the shape of generalized Fresnel's zones in the intersections with any surface. Snell's law is applied to rays incident on the apple type surfaces to find out the conditions for exact conjugation. Sensitivity to optical path variations is also discussed. The RHD curves family can be represented in a Cartesian way where the ovals appear as equally spaced straight lines.

The stressed state in a neighborhood of a plane crack whose border has points of negative curvature

Using the example of a plane crack bounded by the limacon of Pascal we study the influence of the points of negative curvature on the stressed state of a body in a neighborhood of border of the crack. It is shown that under tensile loads the damage of the body begins in a neighborhood of such points. In the case of shear loads prescribed on the edges of the crack damage of the body begins in a neighborhood of points of the border farthest from the points of negative curvature.

Fundamentals of optical design in rotating mirror cameras

The elementary optics of rotating-mirror high-speed cameras are analysed geometrically. The images in an off-axis scanning mirror are shown to lie on Pascal's limacon. The Miller principle by which shuttering is accomplished is described. A simple estimate of motion blurring is compared with the Rayleigh criterion.

Shape of spherulitic boundaries

AbstractA quantitative description of shape of spherulitic boundaries is given in the present publication for the case of two‐dimensional spherulitic crystallization. Shape of the spherulitic boundaries depending on time of formation of nuclei and relative growth rate of spherulites corresponds to one of the following: circular, straight line, hyperbolic line, limacon of Pascal or a curve describable by a fourth degree equation. The deduced formulae are illustrated by experimental data obtained in spherulitic crystallization of α‐ and β‐modifications of polypropylene.

Design synthesis of a gyrogrinder using direct search optimization

A novel grinding mechanism operating on the principle of gyroscopic tracking of arbitrary contours is presented. The mathematical model describing the mechanism of gyrogrinding and the conditions under which the grinding wheel remains in contact with the periphery of template are established. The design parameters that greatly influence the performance of gyrogrinding are identified as the radius of the grinding wheel, the length of the spin axis, rotational speed of grinding wheel, and the mass polar moment of inertia of rotor and inner gimbal. In this paper, the design synthesis of a gyrogrinder is formulated as an optimization problem so that the optimal values of the design variables are calculated using a direct search multiparameter optimization method for continuous grinding with no dynamic shattering of grinding wheel. Three typical templates: elliptical, Limacon of Pascal and Four Leaved Rose, which represent possible shapes in cam production were considered and the optimal values of design parameters for each cam configurations were found using the optimization strategy. Results show that optimal design parameters lie within a small range for the three cam configurations. The mean optimal values of the design parameters were calculated and used in the dimensioning of the rotor and inner gimbal. A prototype was then built based on this investigation and when tested functioned effectively grinding the template. This paper illustrates the use of computer-aided optimization in the design synthesis of a gyrogrinder.

�ber Drehzykliden vierter Ordnung

The algebraic surfaces of revolution considered in this paper are generated either (a) by a circle or a lemniscate, or (b) by a Cartesian oval or especially a limacon of Pascal. The absolute circle at infinity is a double curve for the cyclides of type (a), in particular a cuspidal curve for the cyclides of type (b). Continuing earlier investigations on general cyclides there are deduced several remarkable properties of the spherical quartics situated on the mentioned cyclides of revolution.

Mass spectrum of dynamite effluent: J R Hobbs et al, J Vac Sci Technol, 11 (3), 1974, 557–560

The elementary optics of rotating-mirror high-speed cameras are analysed geometrically. The images in an off-axis scanning mirror are shown to lie on Pascal's limacon. The Miller principle by which shuttering is accomplished is described. A simple estimate of motion blurring is compared with the Rayleigh criterion.

A method for improving the estimation of membrane frequencies

A method for obtaining approximations of the natural frequencies of membranes is developed. An approximate expression for the radius of the bounding curve is first written as a truncated Fourier series. The deflection, which is written as a superposition of the modes of the circular membrane, is forced to vanish (approximately) on the approximated boundary. This generates a system of linear homogeneous equations, the unknowns in which are the amplitudes of the modes of the circle. Equating the determinant of coefficients to zero yields an equation from which the approximate frequencies may be found. It is shown that the first-order approximation obtained through this procedure is identical to a method given by Rayleigh. Approximate frequencies of the first several modes of membranes in the shapes of a square, an ellipse, and the limacon of Pascal are then determined as demonstrations of the new second-order approximation. The approximations of the first three natural frequencies of the ellipse were found to be in error by less than 5% for eccentricities of 0·8 or less, and the approximations of the first four frequencies of the square were found to be in error by less than 3%.

The Limacon of Pascal as a Basis for Computed and Graphic Methods of Determining Astronomic Positions

NAVIGATIONVolume 6, Issue 5 p. 310-316 The Limacon of Pascal as a Basis for Computed and Graphic Methods of Determining Astronomic Positions Alfred H. Kerrick, Alfred H. Kerrick Mr. Kerrick, a resident of Philadelphia, is a retired Navigation Specialist (Marine), formerly with the United States Navy Hydrographic Office.Search for more papers by this author Alfred H. Kerrick, Alfred H. Kerrick Mr. Kerrick, a resident of Philadelphia, is a retired Navigation Specialist (Marine), formerly with the United States Navy Hydrographic Office.Search for more papers by this author First published: Spring 1959 https://doi.org/10.1002/j.2161-4296.1959.tb01959.xCitations: 3 AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume6, Issue5Spring 1959Pages 310-316 RelatedInformation