Curves on ruled surfaces under infinitesimal bending

Najdanović, Marija S.; Maksimović, Miroslav; Velimirović, Ljubica S.

doi:10.5937/bnsr11-32015

Bull Nat Sci Research

2021

DOI: 10.5937/bnsr11-32015

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Curves on ruled surfaces under infinitesimal bending

Marija S. Najdanović¹,

Miroslav Maksimović²,

Ljubica S. Velimirović³

Abstract: Infinitesimal bending of curves lying with a given precision on ruled surfaces in 3-dimensional Euclidean space is studied. In particular, the bending of curves on the cylinder, the hyperbolic paraboloid and the helicoid are considered and appropriate bending fields are found. Some examples are graphically presented.

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Cited by 4 publications

References 15 publications

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Ruled surfaces obtained by infinitesimal bendings of curves in Minkowski 3-space

Han,

Shin,

Jang

2024

Int. J. Geom. Methods Mod. Phys.

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In this paper, we define a ruled surface obtained by an infinitesimal bending of a curve in Minkowski [Formula: see text]-space. We also characterize such ruled surfaces according to causal characters of the curve. Furthermore, we find appropriate conditions of ruled surfaces obtained by infinitesimal bendings of curves to be (lightlike) developable, in particular, cylindrical or conical in Minkowski [Formula: see text]-space, with examples presented.

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Ruled surfaces obtained by infinitesimal bendings of curves in Minkowski 3-space

Han,

Shin,

Jang

2024

Int. J. Geom. Methods Mod. Phys.

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Helicoid and curvature based functional variations

Cvetkovic,

Velimirovic

2023

Filomat

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In view of the meaning of ruled surfaces in aesthetics, statics, scale and manufacturing technologies, we point out the possibility of a mathematical analysis in the case of infinitesimal deformations by considering the rigidity of surfaces. In case of bendable surfaces it is useful to discuss the variation of magnitudes such as the shape operator. The shape operator is a good way to measure how a regular surface S bends in R3 by valuation how the surface normal ? changes from point to point. In this work we consider variations of the shape operator, the normal curvature and the principal curvatures of helicoid under infinitesimal bending of the surface.

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Surfaces defined by bending of knots

Rancic,

Najdanovic,

Velimirovic

2023

Filomat

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We consider the definition of the infinitesimal bending of a curve as a vector parametric equation of a surface defined by two free variables: one of them is free variable u which define curve and another is bending parameter ?. In this way, while being bent curve is deformed and moved through the space forming a surface. If infinitesimal bending field is of constant intensity, deformed curves form a ruled surface that represents a ribbon. In particular, we consider surfaces obtain by bending of knots both analytically and graphically. We pay attention to the torus knot and possibility of its infinitesimal bending so that the surface determined by bending is a part of the initial torus.

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Curves on ruled surfaces under infinitesimal bending

Cited by 4 publications

References 15 publications

Ruled surfaces obtained by infinitesimal bendings of curves in Minkowski 3-space

Ruled surfaces obtained by infinitesimal bendings of curves in Minkowski 3-space

Helicoid and curvature based functional variations

Surfaces defined by bending of knots

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