Ruled invariants and associated ruled surfaces of a space curve

Based on space curve meshing theory, in this paper, we present a novel geometric design of a circular arc helical gear mechanism for parallel transmission with convex-concave circular arc profiles. The parameter equations describing the contact curves for both the driving gear and the driven gear were deduced from the space curve meshing equations, and parameter equations for calculating the convex-concave circular arc profiles were established both for internal meshing and external meshing. Furthermore, a formula for the contact ratio was deduced, and the impact factors influencing the contact ratio are discussed. Using the deduced equations, several numerical examples were considered to validate the contact ratio equation. The circular arc helical gear mechanism investigated in this study showed a high gear transmission performance when considering practical applications, such as a pure rolling process, a high contact ratio, and a large comprehensive strength.

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The surfaces whose prime-sections contain a

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100016583 ◽

1934 ◽

Vol 30 (2) ◽

pp. 170-177 ◽

Cited By ~ 2

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.

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FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

Nagoya Mathematical Journal ◽

10.1017/nmj.2016.22 ◽

2016 ◽

Vol 223 (1) ◽

pp. 1-20 ◽

Cited By ~ 1

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 $B$.

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