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The authors would like to acknowledge King Fahd University of Petroleum & Minerals (KFUPM) for supporting this work. This paper proposes a new concept for the evaluation of fatigue damage parameters. It is suggested that evaluation of fatigue damage may not necessarily be confined to a single plane, i.e., a critical plane. Rather, it is suggested here that the culmination of fatigue crack on a particular plane(s) is due to the contributions of damage on all planes. Mathematically, the polar damage sum concept is calculated as the area enclosed by the polar curve of the damage parameter. The proposed concept is examined using multiaxial experimental data for six different alloys. A total of 241 samples are included in this study. Using the proposed concept, fatigue life estimations are improved compared to those obtained from the original critical plane based on Fatemi-Socie and Smith-Watson-Topper models.

We renormalize, using suitable lenses, small domains of a singular holomorphic line field of ${\mathbb C}^2$ or plane field of ${\mathbb C}^3$ where the curvature of a plane-field is concentrated. At a proper scale the field is almost invariant by translations. When the field is integrable, the leaves are locally almost translates of a surface that we will call {\it profile}. When the singular rays of the tangent cone (a generalization to a plane-field of the tangent cone of a singular surface is defined) are isolated, we obtain more precise results. We also generalize a result of Merle (\cite{Me}) concerning the contact order of generic polar curves with the singular level $f=0$ when $\omega = df$. On the way we obtain some classical results (Lê's carousels) on the knot $K = (\{f=0\} \cap B_{\epsilon}(0,0,0))$ in dimension $2$ an a maybe less classical ones in dimension $3$ .

Recently we have shown that the distance trisector curve is a transcendental curve. Since from the computational point of view this implies that no closed expression to describe the curve in algebraic terms can be found, it is still of interest to know how to approximate it efficiently by means of polynomial or rational functions. We discuss here some of the remarkable properties of this curve, that among other things lead to very good approximations.

The paper deals with geometric invariants of an algebraic curve such as the minimal number of crucial values of rational functions and the minimal transcendence degree of definition fields. The main question is if the difference of these two invariants is always equal to 3 for any curve with genus g > 0. For curves defined over an algebraic number field, a positive answer is given by Belyi’s theorem. In the paper, the main question is answered in the affirmative for some other cases.

The goal of this paper is to measure the non-convexity of compact and smooth connected components of real algebraic plane curves. We study these curves first in a general setting and then in an asymptotic one. In particular, we consider sufficiently small levels of a real bivariate polynomial in a small enough neighbourhood of a strict local minimum at the origin of the real affine plane. We introduce and describe a new combinatorial object, called the Poincare-Reeb graph, whose role is to encode the shape of such curves and to allow us to quantify their non-convexity. Moreover, we prove that in this setting the Poincare-Reeb graph is a plane tree and can be used as a tool to study the asymptotic behaviour of level curves near a strict local minimum. Finally, using the real polar curve, we show that locally the shape of the levels stabilises and that no spiralling phenomena occur near the origin., 32 pages, 34 figures

We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex, as was shown by Coste.The aim of the present thesis is twofold. Firstly, to construct examples of non-Morse strict local minima whose sufficiently small level curves are far from being convex. And secondly, to study a combinatorial object measuring this non-convexity, namely the Poincaré-Reeb tree of the restriction of the first coordinate to the region bounded by a given level curve. These planar trees are rooted and their vertices roughly speaking correspond to points on the curve with vertical tangent lines.The main objective of our study is to characterise all possible topological types of Poincaré-Reeb trees. To this end, we construct a family of examples realising a large class of such trees. As a preliminary step, we restrict our attention to the univariate case, using a tool inspired by Ghys’ work. One of our main results gives a new and constructive proof of the existence of Morse polynomials whose associated permutation (the so-called “Arnold’s snake”) is separable.; Nous considérons une fonction polynomiale de deux variables réelles qui s’annule à l’origine et qui a un minimum local strict en ce point. Nous nous plaçons dans un voisinage de l’origine dans lequel les lignes de niveau non nulles de cette fonction sont des courbes de Jordan lisses. Chaque fois que l’origine est un point critique de Morse, les niveaux suffisamment petits deviennent des bords de disques convexes. Si l’origine n’est pas de Morse, ces courbes de niveau peuvent ne pas être convexes, comme l’a montré Coste.Le but de cette thèse est double. Tout d'abord, nous nous intéressons à la construction d’exemples de minimums locaux stricts et non-Morse dont les lignes de niveau suffisamment petites sont loin d’être convexes. Et deuxièmement, nous étudions un objet combinatoire mesurant cette non-convexité : l’arbre de Poincaré-Reeb de la restriction de la première coordonnée à la région délimitée par une ligne de niveau donnée. Ces arbres planaires sont enracinés et leurs sommets correspondent en gros aux points de la courbe où les tangentes sont verticales.L’objectif principal de notre étude est de caractériser tous les types topologiques possibles d’arbres de Poincaré-Reeb. À cette fin, nous construisons une famille d’exemples réalisant une grande classe de tels arbres. Dans un premier temps, nous concentrons notre attention sur le cas des polynômes d’une variable, en utilisant un outil inspiré du travail de Ghys. L’un de nos résultats principaux donne une preuve nouvelle et constructive de l’existence de polynômes de Morse dont la permutation associée (appelée «le serpent d’Arnold») est séparable.

We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex, as was shown by Coste.The aim of the present thesis is twofold. Firstly, to construct examples of non-Morse strict local minima whose sufficiently small level curves are far from being convex. And secondly, to study a combinatorial object measuring this non-convexity, namely the Poincaré-Reeb tree of the restriction of the first coordinate to the region bounded by a given level curve. These planar trees are rooted and their vertices roughly speaking correspond to points on the curve with vertical tangent lines.The main objective of our study is to characterise all possible topological types of Poincaré-Reeb trees. To this end, we construct a family of examples realising a large class of such trees. As a preliminary step, we restrict our attention to the univariate case, using a tool inspired by Ghys’ work. One of our main results gives a new and constructive proof of the existence of Morse polynomials whose associated permutation (the so-called “Arnold’s snake”) is separable.; Nous considérons une fonction polynomiale de deux variables réelles qui s’annule à l’origine et qui a un minimum local strict en ce point. Nous nous plaçons dans un voisinage de l’origine dans lequel les lignes de niveau non nulles de cette fonction sont des courbes de Jordan lisses. Chaque fois que l’origine est un point critique de Morse, les niveaux suffisamment petits deviennent des bords de disques convexes. Si l’origine n’est pas de Morse, ces courbes de niveau peuvent ne pas être convexes, comme l’a montré Coste.Le but de cette thèse est double. Tout d'abord, nous nous intéressons à la construction d’exemples de minimums locaux stricts et non-Morse dont les lignes de niveau suffisamment petites sont loin d’être convexes. Et deuxièmement, nous étudions un objet combinatoire mesurant cette non-convexité : l’arbre de Poincaré-Reeb de la restriction de la première coordonnée à la région délimitée par une ligne de niveau donnée. Ces arbres planaires sont enracinés et leurs sommets correspondent en gros aux points de la courbe où les tangentes sont verticales.L’objectif principal de notre étude est de caractériser tous les types topologiques possibles d’arbres de Poincaré-Reeb. À cette fin, nous construisons une famille d’exemples réalisant une grande classe de tels arbres. Dans un premier temps, nous concentrons notre attention sur le cas des polynômes d’une variable, en utilisant un outil inspiré du travail de Ghys. L’un de nos résultats principaux donne une preuve nouvelle et constructive de l’existence de polynômes de Morse dont la permutation associée (appelée «le serpent d’Arnold») est séparable.

Given any light position S ϵ P2 and any algebraic curve C of P2 (with any kind of singularities), we consider the incident lines coming from S (i.e. the lines containing S) and their reflected lines after reflection on the mirror curve C. The caustic by reflection ΣS(C) is the Zariski closure of the envelope of these reflected lines. We introduce the notion of reflected polar curve and express the class of ΣS(C) in terms of intersection numbers of C with the reflected polar curve, thanks to a fundamental lemma established in [16]. This approach enables us to state an explicit formula for the class of ΣS(C) in every case in terms of intersection numbers of the initial curve C.

In this paper, the procedure for manually drawing the Nyquist plot of a generic transfer function is revised and, in particular, two novel parameters ( Δ τ , Δ p ), which allow to simplify this process, are presented. Thanks to these parameters, the analysis of the frequency response at low and high frequencies is considerably enhanced, with a very little effort. These parameters allow to predict initial and final directions of the polar curve in the vicinity of initial and final points and consequently the sectors of the complex plane where the plot starts/ends. In many cases it is possible to obtain a qualitative Nyquist plot, able to correctly predict the stability properties of the closed-loop system, by simply joining the initial and final tracts found with the proposed procedure. Moreover, the analysis based on these parameters can aid to correctly interpret the plots obtained with computer programs which often, in particular when poles at the origin are present, hide the behavior of the frequency response in the area close to the origin of the complex plane.

In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated 2-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the 2-Hessian. In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the 2-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve., Updated version to be published in Mathematica Scandinavica. Substantially rewritten, but with essential results unchanged. Contains results from first author's master's thesis. 21 pages, 2 figures

We consider a finite analytic morphism $\phi = (f,g) : (X,p)\to (\C^2,0)$ where $(X,p)$ is a complex analytic normal surface germ and $f$ and $g$ are complex analytic function germs. Let $\pi : (Y,E_{Y})\to (X,p)$ be a good resolution of $\phi$ with exceptional divisor $E_{Y}=\pi ^{-1}(p)$. We denote $G(Y)$ the dual graph of the resolution $\pi $. We study the behaviour of the Hironaka quotients of $(f,g)$ associated to the vertices of $G(Y)$. We show that there exists maximal oriented arcs in $G(Y)$ along which the Hironaka quotients of $(f,g)$ strictly increase and they are constant on the connected components of the closure of the complement of the union of the maximal oriented arcs.

In this paper, an algorithm is given for determining the topology of an algebraic space curve and to compute a certified G^1 rational parametric approximation of the algebraic space curve. The algorithm works by extending to dimension one the local generic position method for solving zero-dimensional polynomial equation systems. Here, certified means that the approximation curve and the original curve have the same topology and their Hausdorff distance is smaller than a given precision. The main advantage of the algorithm, inherited from the local generic position method, is that the topology computation and approximation for a space curve are directly reduced to the same tasks for two plane curves. In particular, an error bound of the approximation space curve is deduced explicitly from the error bounds of the approximation plane curves. The complexity of the algorithm is also analyzed. Its effectivity is shown on some non-trivial examples.

Usually, the tangent line is considered to be a calculus notion. However, it is also a graphical and an algebraic notion. The graphical frame, where our primary conceptions are conceived, could give rise to algebraic methods to obtain the tangent line to a curve. In this pre-calculus perspective, two methods are described and discussed according to their potential for secondary students and teacher training.

It is well known that the equisingularity class, which is the same as the topological type, of the general polar curve of a plane branch is not the same for all branches in a given equisingularity class, but depends upon the analytic type of the branch. It was shown in [1] that, for sufficiently general branches in a given equisingularity class, the topology of the general polar is constant. The aim of this paper is to go beyond generality and show how one could describe the topology of the general polars of all branches in a given equisingularity class, making use of the analytic classification of branches as described in [5]. We will show how this works in some particular equisingularity classes for which one has the complete explicit analytic classification, and in particular for all branches of multiplicity less or equal than four, based on the classification given in [4].

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. This paper gives an upper bound of the Bezout number, the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite, over arbitrary triangulations.

We consider closed curves C≃C∗ in the affine plane S≃C2 that admit a good or very good asymptote. By this we mean a curve L≃C in S that in suitable coordinates for S is linear and tangent to C at infinity, and meets C once or not at all at finite distance. We classify these curves up to automorphism of S. Relying on the theory of open algebraic surfaces we first determine the possibilities for the singularities of C at infinity and then proceed to give explicit equations.

This paper presents the dual bases for Wang-Bezier curves with a position parameter L, which include Bezier curve, Wang-Ball curve and some intermediate curves. The Marsden identity and the transformation formulas from Bezier curve to Wang-Bezier curve are also given. These results are useful for the application of Wang-Bezier curve and their popularization in Computer Aided Geometric Design.

We consider the global differential geometry of polar closed convex curves in the spherical resp. hyperbolic plane (in the last case after restriction to horocyclic convex curves) by means of a curve representation by a suitable support function. In a certain sense the results are simpler than in the euclidean case. There exists a connection between the polar curve and the parallel curve at distance \({\frac{{\pi}}{{2}}}\,{\rm resp.}\,{\frac{{\pi}}{{2}}\,·\,i}\). The evolute of a convex curve may also be represented by its support function in a simple manner.

We prove that, given a smooth projective curve C of genus g≥2, the forgetful morphism $\mathcal {M}_{\mathbf {O}_r} \longrightarrow \mathcal {M}_{\mathbf {GL}_r}$ (respectively $\mathcal M_{\mathbf {Sp}_{2r}}\longrightarrow \mathcal M_{\mathbf {GL}_{2r}}$) from the moduli space of orthogonal (respectively symplectic) bundles to the moduli space of all vector bundles over C is an embedding. Our proof relies on an explicit description of a set of generators for the polynomial invariants on the representation space of a quiver under the action of a product of classical groups.

Let C be an algebraic curve of genus g ≥ 2. We prove an analogue of Clifford’s theorem for coherent systems on C and some refinements using results of Re and Mercat.

We consider the polar curves PS,0 arising from generic projections of a germ (S, 0) of complex surface singularity onto C. Taking (S, 0) to be a minimal singularity of normal surface (i.e. a rational singularity with reduced tangent cone), we give the δ-invariant of these polar curves, as well as the equisingularity-type of their generic plane projections, which are also the discriminants of generic projections of (S, 0). These two (equisingularity)-data for PS,0 are described in term, on the one side of the geometry of the tangent cone of (S, 0) and on the other side of the limit-trees introduced by T. de Jong and D. van Straten for the deformation theory of these minimal singularities. These trees give a combinatorial device for the description of the polar curve which makes it much clearer than in our previous Note on the subject. This previous work mainly relied on a result of M. Spivakovsky. Here we give a geometrical proof via deformations (on the tangent cone, and what we call Scott deformations) and blow-ups, although we need Spivakovsky’s result at some point, extracting some other consequences of it along the way.

In this paper we introduce the notion of local shape to describe the behavior of a real place of an algebraic curve around its center. We analyze how the local shape is affected by the offsetting process, and we relate this phenomenon to the curvature of the curve. Furthermore, we characterize the situations when the offsetting process behaves locally well, so that the local shape is preserved.

In this paper, based on modifications to the well-known subdivision scheme in combination with modified affine arithmetic method to guide the subdivision, we propose four robust and reliable algorithms for plotting polar algebraic curves, space algebraic curves and offsets of planar algebraic curves.

The number of rational points of a plane non-singular algebraic curve X defined over a finite field is computed, provided that the generic point of X is not an inflexion and that X is Frobenius non-classical with respect to conics., Latex2e, 8 pages