The geometry of the secant caustic of a planar curve

2007

We find a geometrical method of analysing the singularities of a plane nodal curve. The main results will be used in a forthcoming paper on geometric Plucker formulas for such curves. Plane nodal curves, that is plane curves having at most nodes as singularities, form an important class of curves, as any projective algebraic curve is birational to a plane nodal curve.

In this paper, we have studied the striction curves of a normal ruled surface. We have shown that the evolute of a base curve is the striction curve of a normal ruled surface and the singularities of such surface lie on the evolute of the base curve. We have proved that the striction curves orthogonally cut the planar base curves. Also, we have proved that the surface area between a planar base curve and striction curve of a normal ruled surface is identical to the surface area between the evolutes of the base curve and striction curve. We have obtained different conditions for which the striction curve coincides with the base curve of some ruled surfaces. We have proved that the striction curve of a tangential Darboux developable of a space curve coincides with the base curve if and only if the space curve is a helix. We have deduced a beautiful form of surface area between the base curve and striction curve of a tangential Darboux developable.

2021

This paper is an elementary treatise on two-dimensional geometry. In this paper, a new form of dual curves with three inflection points is produced and graphically studied. The Laith's node is the locus of the dual vertexes of a right polygon, when a set of line segments from a fixed center point of an ellipse and a circle with radius equal to the minor axis, or by a drawing circle and 3 asymptotes, two of them tangent it while the third is passing through the center. Also, algebraic analyses and the construction method are geometrically designed and discussed. All obtained figures and geometrical aspects in this paper have been drawn and measured by AutoCAD program, (Version 2007), as a way to increase the correctness of results produced by these construction methods and equations presented in this paper.

2005

We discuss the behaviour of vertices and inflexions on one-parameter families of plane curves, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. This work is preliminary to an investigation of symmetry sets and medial axes for these families of curves, reported elsewhere.

Differential geometry of curves studies the properties of curves and higher-dimensional curved spaces using tools from calculus and linear algebra. This study has two aspects: the classical differential geometry which started with the beginnings of calculus and the global differential geometry which is the study of the influence of the local properties on the behavior of the entire curve. The local properties involves the properties which depend only on the behavior of the curve in the neighborhood of a point. The methods which have shown themselves to be adequate in the study of such properties are the methods of differential calculus. Due to this, the curves considered in differential geometry will be defined by functions which can be differentiated a certain number of times. The other aspect is the so-called global differential geometry which study the influence of the local properties on the behavior of the entire curve or surface. This paper aims to give an advanced introduction to the theory of curves, and those that are curved in general.

Differential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry. The theory of plane, curves and surfaces in the Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century. The core idea of both differential geometry and modern geometrical dynamics lies under the concept of manifold. A manifold is an abstract mathematical space, which locally resembles the spaces described by Euclidean geometry, but which globally may have a more complicated structure. The purpose of this paper is to give an elaborate introduction to the theory of curves, and those are, in general, curved. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by applying the concept of differential and integral calculus. The curves are represented in parametrized form and then their geometric properties and various quantities associated with them, such as curvature and arc length expressed via derivatives and integrals using the idea of vector calculus.

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

In this paper, we study some geometric invariants of closed plane curves, that can help us classify these curves. We focus on two invariants: the number of inection points and the number of vertex points. We intend to nd models of curves with a number of predened double points and with the smallest possible number of inection points and vertex points. Palavras-chave. Geometric modeling, plane curves, inections, vertices, graphs of stable maps.

Bulletin of the Brazilian Mathematical Society, New Series, 2019

In this paper we study singular points of the Wigner caustic and affine λ-equidistants of planar curves based on shapes of these curves. We generalize the Blaschke-Süss theorem on the existence of antipodal pairs of a convex curve.

2003

Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ 1117-PLANE CURVE SINGULARITIES AND CAROUSELS by LÊ DUNG TRÁNG Introduction. Let f : (C2, 0)-(C,0) be the germ of a complex analytic function. Since we are only interested in the topology of the germ of plane curve (r, 0) defined by f, we assume that f is reduced, i.e., if f-fi 1 ,. , is the decomposition of f into irreducible factors in the ring YI of convergent power series in two variables, then a1-= ak = 1. We still denote by f a representative of this germ defined on an open neighbourhood U of the origin 0 in E~2 and h = f-1 (o) n U. Since f is reduced at 0, the point 0 is an isolated singular point of r, i.e., there is a sufficiently small neighbourhood V of 0 in (~2, such that the space (h-{0}) n V is non-singular. In other words the ideal (~ 0 f /0X, 0 f /0Y) generated in C{X, YI by f and the partial derivatives of f is primary for the maximal ideal Jlil of It can be shown that this is equivalent to the fact that the Jacobian ideal (8 f/8X, 8 f/8Y) is M-primary, i.e., the quotient C-algebra: is a finite dimensional vector space over C. This dimension is called the Milnor number of f at 0. We shall denote it by p (f 0). It is known (see [M]) that, for E > 0 sufficiently small, the real 3sphere Sg(0) of C2 centered at 0 with radiusintersects r transversally.

Lecture Notes in Computer Science, 2005

In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of smooth plane curves (Bruce and Giblin, Giblin and Kimia, etc.) to the general case when the family of curves includes a singular member, as will happen if the curves are obtained by taking plane sections of a smooth surface, at the moment when the plane becomes tangent to the surface. Looking at those surface sections as isophote curves, of the pixel values of an image embedded in the real plane, this allows us to propose to combine object representation using a skeleton or symmetry set representation and the appearance modelling by representing image information as a collection of medial representations for the level-sets of an image.

Journal of Geometry, 1990

A brief review of the concepts of curvature, evolutes, and involutes is presented. This can help the student to understand the discussion on Huygens' pendulum in Marion and Thornton 5th edition Problem 3.8 uploaded to Academia.edu.

Journal of Evolution Equations, 2017

We discuss several examples of curvature flows for convex closed plane curves that preserve parallel curves and use this property to find singularities (curvature blow-up) of these flows. A precise curvature blow-up rate is also obtained.

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2021

It has been shown that the family of Poncelet N-gons in the confocal pair (elliptic billiard) conserves the sum of cosines of its internal angles. Curiously, this quantity is equal to the sum of cosines conserved by its affine image where the caustic is a circle. We show that furthermore, (i) when N=3, the cosine triples of both families sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). We also show that (ii) the family of triangles excentral to the confocal family conserves the same product of cosines as the one conserved by its affine image inscribed in a circle; and that (iii) cosine triples of both families sweep the same spherical curve. When the triple of log-cosines is considered, this curve becomes a planar, plectrum-shaped curve, rounder than the one swept by its parent confocal family.

Anais da Academia Brasileira de Ciências, 2002

In this paper is studied the relationship between quadratic tangencies of principal lines with the boundary of a surface and the Darbouxian umbilics of a smooth boundaryless surface which approximates it through the process of thickening and smoothing defined here.