Generalized osculating‐type ruled surfaces of singular curves

Yazıcı, Bahar Doğan; İşbilir, Zehra; Tosun, Murat

doi:10.1002/mma.8997

Cited by 3 publications

(1 citation statement)

References 23 publications

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“…They defined a smooth surface with a moving frame as the framed surface and gave criteria for singular points of the framed surface. Using this powerful tool, some researchers have studied the singular properties of different singular surfaces in recent years [12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space

Liu,

Li,

Pei

2023

Mathematics

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In this paper, we study the surfaces of osculating circles, which are the sets of all osculating circles at all points of regular curves. Since the surfaces of osculating circles may be singular, it is necessary to investigate the singular points of these surfaces. However, traditional methods and tools for analyzing singular properties have certain limitations. To solve this problem, we define the framed surfaces of osculating circles in the Euclidean 3-space. Then, we discuss the types of singular points using the theory of framed surfaces and show that generic singular points of the surfaces consist of cuspidal edges and cuspidal cross-caps.

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Section: Introductionmentioning

confidence: 99%

Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space

Liu,

Li,

Pei

2023

Mathematics

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Framed Curve Families Induced by Real and Complex Coupled Dispersionless-Type Equations

Popović¹,

Eren

Savić³

et al. 2023

Mathematics

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In this study, we investigate coupled real and complex dispersionless equations for curve families, even if they have singular points. Even though the connections with the differential equations and regular curves were considered in various ways in the past, since each curve does not need to be regular, we establish the connections for framed base curves, which generalize regular curves with linear independent conditions. Also, we give the Lax pairs of the real and complex coupled dispersionless equations from the motions of any framed curve. These give us significant conditions based on the framed curvatures and associated curvatures of the framed curves for integrability since it is well known that the Lax pair provides the integrability of differential equations.

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Special non-lightlike ruled surfaces in Minkowski 3-space

Sun,

Yao,

Pei

2023

MATH

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<abstract><p>In this paper, we give the existence and uniqueness theorems for non-lightlike framed surfaces and define a special non-lightlike ruled surface in Minkowski 3-space. It may have singularities. We give the conditions for identifying cross-caps and surfaces as developable and maximal. Besides, we demonstrate that if the spacelike ruled surface is developable, then the $ z $-parameter curve is an asymptotic curve if and only if the ruled surface is maximal.</p></abstract>

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Generalized osculating‐type ruled surfaces of singular curves

Cited by 3 publications

References 23 publications

Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space

Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space

Framed Curve Families Induced by Real and Complex Coupled Dispersionless-Type Equations

Special non-lightlike ruled surfaces in Minkowski 3-space

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