scholarly journals I. On skew surfaces, otherwise scrolls

1863 ◽  
Vol 12 ◽  
pp. 446-448
Keyword(s):  
Ruled Surfaces ◽  
Nodal Curve

It may be convenient to mention at the outset that in the paper “On the Theory of Skew Surfaces,” Camb. and Dubl. Math. Journ. vol. vi. pp. 171-173 (1852), I pointed out that upon any skew surface of the order n there is a singular (or nodal) curve meeting each generating line in ( n — 2) points, and that the class of the circumscribed cone, or what is the same thing, the class of the surface, is equal to the order n of the surface. In the paper “On a Class of Ruled Surfaces,” Camb. and Dubl. Math. Journ. vol. viii. pp. 45, 46 (1853), Dr. Salmon considered the surface generated by a line which meets three curves of the orders m, n, p respectively.

scholarly journals XXI. On skew surfaces, otherwise scrolls

1863 ◽  
Vol 153 ◽  
pp. 453-483 ◽  
Keyword(s):  
Three Dimensions ◽  
Linear Functions ◽  
Ruled Surfaces ◽  
Nodal Curves ◽  
Nodal Curve ◽  
Double Points ◽  
The Subject ◽  
Line Meeting

It may be convenient to mention at the outset that, in the paper “On the Theory of Skew Surfaces”, I pointed out that upon any skew surface of the order n there is a singular (or nodal) curve meeting each generating line in ( n -2) points, and that the class of the circumscribed cone (or, what is the same thing, the class of the surface) is equal to the order n of the surface. In the paper “On a Class of Ruled Surfaces”, Dr. Salmon considered the surface generated by a line which meets three curves of the orders m , n , p respectively : such surface is there shown to be of the order =2 mnp ; and it is noticed that there are upon it a certain number of double right lines (nodal gene­rators); to determine the number of these, it was necessary to consider the skew surface generated by a line meeting a given right line and a given curve of the order m twice; and the order of such surface is found to be =½ m ( m —1)+ h , where h is the number of apparent double points of the curve. The theory is somewhat further developed in Dr. Salmon’s memoir “On the Degree of a Surface reciprocal to a given one”, where certain minor limits are given for the orders of the nodal curves on the skew surface generated by a line meeting a given right line and two curves of the orders m and n and respectively, and on that generated by a line meeting a given right line and a curve of the order m twice. And in the same memoir the author considers the skew surface generated by a line the equations whereof are ( a , ..)( t , 1) m =0 ( a' , ..)( t , 1) n =0, where a , .. a' , .. are any linear functions of the coordinates, and t is an arbitrary para­meter. And the same theories are reproduced in the ‘Treatise on the Analytic Geo­metry of Three Dimensions’ §. I will also, though it is less closely connected with the subject of the present memoir, refer to a paper by M. Chasles, “Description des Courbes à double courbure de tous les ordres sur les surfaces réglées du troisiѐme et du quatriѐme ordre”||.

The surfaces whose prime-sections contain a

1934 ◽  
Vol 30 (2) ◽  
pp. 170-177 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Hyperelliptic Curves ◽  
Ruled Surfaces ◽  
Minimum Order ◽  
Rational Surfaces ◽  
Pencil Of Conics

The surfaces whose prime-sections are hyperelliptic curves of genus p have been classified by G. Castelnuovo. If p > 1, they are the surfaces which contain a (rational) pencil of conics, which traces the on the prime-sections. Thus, if we exclude ruled surfaces, they are rational surfaces. The supernormal surfaces are of order 4p + 4 and lie in space [3p + 5]. The minimum directrix curve to the pencil of conics—that is, the curve of minimum order which meets each conic in one point—may be of any order k, where 0 ≤ k ≤ p + 1. The prime-sections of these surfaces are conveniently represented on the normal rational ruled surfaces, either by quadric sections, or by quadric sections residual to a generator, according as k is even or odd.

scholarly journals FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

2016 ◽  
Vol 223 (1) ◽  
pp. 1-20 ◽  
Author(s):  
ADRIEN DUBOULOZ ◽  
TAKASHI KISHIMOTO
Keyword(s):  
Minimal Model ◽  
Finite Extension ◽  
Smooth Curve ◽  
Ruled Surfaces ◽  
Model Program ◽  
Minimal Model Program ◽  
Generic Fiber ◽  
Projective Model ◽  
The Family

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.

Polar multiplicities and Euler obstruction for ruled surfaces

2012 ◽  
Vol 43 (3) ◽  
pp. 443-451 ◽  
Author(s):  
Nivaldo G. Grulha ◽  
Marcelo E. Hernandes ◽  
Rodrigo Martins
Keyword(s):  
Ruled Surfaces ◽  
Euler Obstruction

scholarly journals On ruled surfaces of genus 1

1969 ◽  
Vol 21 (2) ◽  
pp. 291-311 ◽  
Author(s):  
Tatsuo SUWA
Keyword(s):  
Ruled Surfaces

scholarly journals Characteristics of the double-cycled motion-ruled surface of the Schatz linkage based on differential geometry

2014 ◽  
Vol 229 (5) ◽  
pp. 957-964 ◽  
Author(s):  
Lei Cui ◽  
Jian S Dai ◽  
Chung-Ching Lee
Keyword(s):  
Differential Geometry ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Full Rotation ◽  
Kinematic Properties

This paper applies Euclidean invariants from differential geometry to kinematic properties of the ruled surfaces generated by the coupler link and the constraint-screw axes. Starting from investigating the assembly configuration, the work reveals two cycle phases of the coupler link when the input link finishes a full rotation. This leads to analysis of the motion ruled surface generated by the directrix along the coupler link, where Euclidean invariants are obtained and singularities are identified. This work further presents the constraint ruled surface that is generated by the constraint screw axes and unveils its intrinsic characteristics.

scholarly journals EXTENSIONS AND RANK-2 VECTOR BUNDLES ON IRREDUCIBLE NODAL CURVES

2005 ◽  
Vol 16 (10) ◽  
pp. 1081-1118
Author(s):  
D. ARCARA
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Rational Map ◽  
Nodal Curves ◽  
Nodal Curve ◽  
Stable Vector Bundles ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use the natural rational map [Formula: see text] defined by [Formula: see text] to study a compactification [Formula: see text] of the moduli space [Formula: see text] of semi-stable vector bundles of rank 2 and determinant L on C. In particular, we resolve the indeterminancy of ϕL in the case deg L = 3,4 via a sequence of three blow-ups with smooth centers.

scholarly journals On Motion of Robot End-Effector Using the Curvature Theory of Timelike Ruled Surfaces with Timelike Rulings

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
Cumali Ekici ◽  
Yasin Ünlütürk ◽  
Mustafa Dede ◽  
B. S. Ryuh
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
End Effector ◽  
Timelike Ruled Surface ◽  
Curvature Theory ◽  
Differential Properties

The trajectory of a robot end-effector is described by a ruled surface and a spin angle about the ruling of the ruled surface. In this way, the differential properties of motion of the end-effector are obtained from the well-known curvature theory of a ruled surface. The curvature theory of a ruled surface generated by a line fixed in the end-effector referred to as the tool line is used for more accurate motion of a robot end-effector. In the present paper, we first defined tool trihedron in which tool line is contained for timelike ruled surface with timelike ruling, and transition relations among surface trihedron: tool trihedron, generator trihedron, natural trihedron, and Darboux vectors for each trihedron, were found. Then differential properties of robot end-effector's motion were obtained by using the curvature theory of timelike ruled surfaces with timelike ruling.

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