Class Of Planar Curves Research Articles

A New Class of Plane Curves with Arc Length Parametrization and Its Application to Linear Analysis of Curved Beams

The objective of this paper is to define one class of plane curves with arc-length parametrization. To accomplish this, we constructed a novel class of special polynomials and special functions. These functions form a basis of L2(R) space and some of their interesting properties are discussed. The developed curves are used for the linear static analysis of curved Bernoulli–Euler beam. Due to the parametrization with arc length, the exact analytical solution can be obtained. These closed-form solutions serve as the benchmark results for the development of numerical procedures. One such example is provided in this paper.

A Note on Superspirals of Confluent Type

Superspirals include a very broad family of monotonic curvature curves, whose radius of curvature is defined by a completely monotonic Gauss hypergeometric function. They are generalizations of log-aesthetic curves, and other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. In this work, we study superspirals of confluent type via similarity geometry. Through a detailed investigation of the similarity curvatures of superspirals of confluent type, we find a new class of planar curves with monotone curvature in terms of Tricomi confluent hypergeometric function. Moreover, the proposed ideas will be our guide to expanding superspirals.

Uniform approximation of curvature for smooth classes of plane curves

We consider problems of approximating the curvature of plane curves from smooth classes by the curvature of elements of smooth finite-dimensional function spaces (trigonometric polynomials, splines with equidistant knots) in the uniform norm.

Construction of a class of quintic curves with rational offsets

In this paper, a method for constructing a class of quintic curves with rational offsets is presented. Although this class of planar curves does not include PH curves, such curves are widely applied in CAD because they have rational offsets. A complex variate model is employed to deduce the proposed method. The quintic OR curves are first classified into two classes according to the different factorization of their hodographs, and are then discussed. With the given $C^1$ Hermite data, the curves are determined by specifying a real parameter. This real parameter can be used to adjust the shape of the constructed curves, and affects the parameter values of the cusps.

Intrinsically Defined Curves and Special Functions

SummaryWe study two classes of plane curves with prescribed curvature. First, we investigate spirals whose curvature is a power function, and express coordinates of the spirals’ centers in terms of the gamma function. For curves in the second family, the curvature is a multiple of the sine function. We show that this family contains infinitely many closed curves and provide their characterization in terms of the zeroth-order Bessel function.

Geometry of plane curves

In this paper we obtain, for a class of plane curves, extensions of the well-known relation of inflection points, double points and bitangencies established by Fabricius-Bjerre for closed curves.

Pedal Curves Part I: Homogeneous Differential Equation

We study the class of plane curves with positive curvature κ and spherical parametrization s. t. that the curves and their derived curves like evolute, caustic, pedal and co-pedal curve, resp., have the same shape. We characterize these properties by a homogeneous linear differential equation of first order for the support function ρ and the radii of curvature κ -1 and we give a complete local classification of this class.

Partial Localization, Lipid Bilayers, and the Elastica Functional

Partial localization is the phenomenon of self-aggregation of mass into highdensity structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of two-dimensional spatial mass densities, exhibits partial localization and displays the “solid-like” behaviour of cell membranes. Specifically, we show that density fields of moderate energy are partially localized, that is, resemble thin structures. Deviation from a specific uniform thickness, creation of “ends,” and the bending of such structures all carry an energy penalty, of different orders in terms of the thickness of the structure. These findings are made precise in a Gamma-convergence result. We prove that a rescaled version of the energy functional converges in the zero-thickness limit to a functional that is defined on a class of planar curves. Finiteness of the limit enforces both optimal thickness and non-fracture; if these conditions are met, then the limit value is given by the classical elastica (bending) energy of the curve.

Cell membranes, Lipid Bilayers, and the Elastica Functional

AbstractWe study an energy functional that arises in a simplified two‐dimensional model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of spatial mass densities, favours concentrations on ‘thin structures’. Stretching, fracture and bending of such structures all carry an energy penalty. In this sense we show that the models captures essential features of lipid bilayers, namely partial localisation and a solid‐like behaviour.Our findings are made precise in a Gamma‐convergence result. We prove that a rescaled version of the energy functional converges in the ‘zero thickness limit’ to a functional that is defined on a class of planar curves. Finiteness of the limit value enforces both optimal thickness and non‐fracture; if these conditions are met, then the value of this functional is given by the classical Elastica (bending) energy. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Finding the Convex Hull of Discs in Parallel

We present a parallel method for finding the convex hull of planar discs in the EREW PRAM model. We show that the convex hull of n discs in the plane can be computed in O( log 1+ε n) time using O(n/ log ε n) processors or in O( log n log log n) time using O(n log 1+ε n) processors for any positive constant ε. The first result achieves cost optimal and the second one runs faster. We also show that the convex hull of planar discs can be constructed in O( log n) time using O(n) processors when the number of different kinds of radii is restricted to 2O( log α n) for any positive constant α with 0 < α < 1. Finally, we show that our method can be generalized to computing the convex hull of a large class of planar curves. We use a technique called multi-level divide-and-conquer in our algorithm.

Generalized Bezout's theorem and its applications in coding theory

This paper presents a generalized Bezout theorem which can be used to determine a tighter lower bound of the number of distinct points of intersection of two or more plane curves. A new approach to determine a lower bound on the minimum distance for algebraic-geometric codes defined from a class of plane curves is introduced, based on the generalized Bezout theorem. Examples of more efficient linear codes are constructed using the generalized Bezout theorem and the new approach. For d=4, the linear codes constructed by the new construction are better than or equal to the known linear codes. For d/spl ges/5, these new codes are better than the known AG codes defined from whole spaces. The Klein codes [22, 16, 5] and [22, 15, 6] over GF(2/sup 3/), and the improved Hermitian code [64, 56, 6] over GF(2/sup 4/) are also constructed.

Null curves in diffraction patterns

We show that there is a large class of plane curves which never appear in partial form, as curves along which destructive interference takes place. These curves have an analytic property which, when combined with an analytic property of the free light field, gives rise to patterns of interference that are commonly observed.

On a Class of Plane Curves

Proceedings of the London Mathematical SocietyVolume s1-33, Issue 1 p. 193-196 Articles On a Class of Plane Curves J. H. Grace, J. H. GraceSearch for more papers by this author J. H. Grace, J. H. GraceSearch for more papers by this author First published: November 1900 https://doi.org/10.1112/plms/s1-33.1.193Citations: 3AboutPDF 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 onFacebookTwitterLinked InRedditWechat Citing Literature Volumes1-33, Issue1November 1900Pages 193-196 RelatedInformation