On the circuits of a plane sextic curve

1936 ◽  
Vol 60 (1) ◽  
pp. 280-285 ◽  
Author(s):  
Harold Hilton
Keyword(s):  
Sextic Curve

scholarly journals Space sextic curves with six bitangents, and some geometry of the diagonal cubic surface

1997 ◽  
Vol 40 (1) ◽  
pp. 85-97
Author(s):  
R. H. Dye
Keyword(s):  
Finite Number ◽  
Space Curve ◽  
Rational Space ◽  
Cubic Surface ◽  
General Space ◽  
Cubic Surfaces ◽  
Related Pair ◽  
Sextic Curve

A general space curve has only a finite number of quadrisecants, and it is rare for these to be bitangents. We show that there are irreducible rational space sextics whose six quadrisecants are all bitangents. All such sextics are projectively equivalent, and they lie by pairs on diagonal cubic surfaces. The bitangents of such a related pair are the halves of the distinguished double-six of the diagonal cubic surface. No space sextic curve has more than six bitangents, and the only other types with six bitangents are certain (4,2) curves on quadrics. In the course of the argument we see that space sextics with at least six quadrisecants are either (4,2) or (5,1) quadric curves with infinitely many, or are curves which each lie on a unique, and non-singular, cubic surface and have one half of a double-six for quadrisecants.

The number of contact primes of the canonical curve of genus p

1930 ◽  
Vol 26 (4) ◽  
pp. 453-457
Author(s):  
W. G. Welchman
Keyword(s):  
General Formula ◽  
Theta Functions ◽  
Cubic Surface ◽  
General Curve ◽  
Projective Model ◽  
Quartic Curve ◽  
Canonical Curve ◽  
Riemann Theta Functions ◽  
Sextic Curve ◽  
Special Case

1. It is known, from the theory of the Riemann theta-functions, that the canonical series of a general curve of genus p has 2P−1 (2P − 1) sets which consist of p − 1. points each counted twice. Taking as projective model of the curve the canonical curve of order 2p − 2 in space of p− 1 dimensions, whose canonical series is given by the intersection of primes, we have the number of contact primes of the curve. The 28 bitangents of a plane quartic curve, the canonical curve of genus 3, have been studied in detail since the days of Plücker. The number, 120, of tritangent planes of the sextic curve of intersection of a quadric and a cubic surface, the canonical curve of genus 4, has been obtained directly by correspondence arguments by Enriques. Enriques also remarks that the general formula 2P−1 (2p −1) is a special case of the formula of de Jonquières, which was proved, by correspondence methods, by Torelli§.

scholarly journals The Coupler Surface of the RSRS Mechanism

2015 ◽  
Vol 8 (1) ◽  
Author(s):  
Nicolas Rojas ◽  
Aaron M. Dollar
Keyword(s):  
Rigid Body ◽  
Interested Reader ◽  
Implicit Equation ◽  
Spatial Mechanism ◽  
Full Equation ◽  
Sextic Curve ◽  
Spatial Linkages ◽  
Robotic Devices ◽  
Four Bar Linkage ◽  
Two Degree Of Freedom

Two degree-of-freedom (2-DOF) closed spatial linkages can be useful in the design of robotic devices for spatial rigid-body guidance or manipulation. One of the simplest linkages of this type, without any passive DOF on its links, is the revolute-spherical-revolute-spherical (RSRS) four-bar spatial linkage. Although the RSRS topology has been used in some robotics applications, the kinematics study of this basic linkage has unexpectedly received little attention in the literature over the years. Counteracting this historical tendency, this work presents the derivation of the general implicit equation of the surface generated by a point on the coupler link of the general RSRS spatial mechanism. Since the derived surface equation expresses the Cartesian coordinates of the coupler point as a function only of known geometric parameters of the linkage, the equation can be useful, for instance, in the process of synthesizing new devices. The steps for generating the coupler surface, which is computed from a distance-based parametrization of the mechanism and is algebraic of order twelve, are detailed and a web link where the interested reader can download the full equation for further study is provided. It is also shown how the celebrated sextic curve of the planar four-bar linkage is obtained from this RSRS dodecic.

III. A supplementary memoir on caustics

1867 ◽  
Vol 15 ◽  
pp. 268-268
Keyword(s):  
Orthogonal Trajectory ◽  
Sextic Curve ◽  
Arbitrary Line

It is near the conclusion of my “Memoir on Caustics,” Phil. Trans. vol. cxlvii. (1857), pp. 273, 312, remarked that for the case of parallel rays refracted at a circle, the ordinary construction for the secondary caustic cannot be made use of (the entire curve would in fact pass off to an infinite distance), and that the simplest course is to measure off the distance GQ from a line through the centre of the refracting circle perpendicular to the direction of the incident rays. The particular secondary caustic, or orthogonal trajectory of the refracted rays, obtained on the above supposition was shown to be a curve of the order 8; and it was further shown by consideration of the case (wherein the distance GQ is measured off from an arbitrary line perpendicular to the incident rays), that the general secondary caustic or orthogonal trajectory of the refracted rays was a curve of the same order 8. The last-mentioned curve in the case of reflexion, or for μ = — 1, degenerates into a curve of the order 6; and I propose in the present supplementary memoir to discuss this sextic curve; viz. the sextic curve which is the general secondary caustic or orthogonal trajectory of parallel rays reflected at a circle.

II. An inquiry into Newton’s rule for the discovery of ima­ginary roots

1864 ◽  
Vol 13 ◽  
pp. 179-183 ◽  
Keyword(s):  
Integral Function ◽  
Similar Process ◽  
Linear Transformations ◽  
Empirical Ground ◽  
Sextic Curve ◽  
Variable Point ◽  
Open Question ◽  
Rational Integral ◽  
Imaginary Roots ◽  
Second Degree

In the 'Arithmetica Universalis,' in the chapter "De Resolutione Equationum," Newton has laid down a rule, admirable for its simplicity and generality, for the discovery of imaginary roots in algebraical equations, and for assigning an inferior limit to their number. He has given no clue towards the ascertainment of the grounds upon which this rule is based, and has stated it in such terms as to leave it quite an open question whether or not he had obtained a demonstration of it. Maclaurin, Campbell, and others have made attempts at supplying a demonstration, but their efforts, so far as regards the more important part of the rule, that namely by which the limit to the number of imaginary roots is fixed, have completely failed in their object. Thus hitherto any opinion as to the truth of the rule rests on the purely empirical ground of its being found to lead to correct results in particular arithmetical instances. Persuaded of the insufficiency of such a mode of verification, the author has applied himself to obtain­ing a rigorous demonstration of the rule for equations of specified degrees. For the second degree no demonstration is necessary. For cubic equa­tions a proof is found without difficulty. For biquadratic equations the author proceeds as follows. He supposes the equation to be expressed homogeneously in x, y, and then, instituting a series of infinitesimal linear transformations obtained by writing x + hy for x , or y + hx for y , where h is an infinitesimal quantity, shows that the truth of Newton’s rule for this case depends on its being capable of being shown that the discriminant of the function (1, ± e , e 2 , ± e , 1)( x, y ) 4 is necessarily positive for all values of e greater than unity, which is easily proved. He then proceeds to consider the case of equations of the 5th degree, and, following a similar process, arrives at the conclusion that the truth of the rule depends on its being capable of being shown that the discriminant, say (D) of the function (1, ε, ε 2 , η 2 , η, 1,)( x , y ) 5 , which for facility of reference may be termed “ the (ε, η) function,” is necessarily positive when ε 4 — εη 2 and η 4 — ηε 2 are both positive. This discriminant is of the 12th degree in ε, η. But on writing x = εη, y =ε 5 + η 5 , it becomes a rational integral function of the 6th degree in x , and of the second degree in y , and such that, on making D = 0, the equation represents a sextic curve, of which x, y are the abscissa and ordinate, which will consist of a single close. It is then easily demonstrated that all values of ε, η which cause the variable point x , y to lie inside this curve, will cause D to he negative (in which case the function ε, η has only two imaginary factors), and that such values as cause the variable point to lie outside the curve, will make D positive, in which case the ε, η function has four imaginary factors. When the conditions concerning ε, η above stated are verified, it is proved that the variable point must be exterior to the curve, and thus the theorem is demonstrated for equations of the 5th degree.

Design, Construction, and Control of Curves and Surfaces via Deployable Mechanisms

2019 ◽  
Vol 11 (6) ◽  
Author(s):  
Vishal Ramadoss ◽  
Dimiter Zlatanov ◽  
Xilun Ding ◽  
Matteo Zoppi ◽  
Shengnan Lyu
Keyword(s):  
Computer Aided Design ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Approximation Technique ◽  
Polygonal Approximation ◽  
Curves And Surfaces ◽  
Cad Models ◽  
Sextic Curve ◽  
Aided Design ◽  
And Control

Abstract There has been an increasing interest in design and construction of deployable mechanisms (DMs) with multiple degrees of freedom (DOFs). This paper summarizes a family of deployable mechanisms that approximates a series of curves and surfaces using the polygonal approximation technique. These mechanisms are obtained by linking the two- and three-dimensional deployable units, which are constitutive of Sarrus and scissor linkages. Multiple unit mechanisms with varying sizes are assembled and alter their shape within a different family of parameterized curves and surfaces. A systematic methodology for polygonal approximation method is presented. Quadratic, semi-cubic, cubic, quartic and sextic curve boundaries, and quadric surfaces are approximated and controlled. Computer-aided design (CAD) models and kinematic simulations elucidate the mechanism’s ability to approximate a set of curves and surfaces.

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