The local classifications of the ruled surfaces of normals and binormals of a regular space curve

We investigate the relation between two types of space curves, the Mannheim curves and constant-pitch curves and primarily explicate a method of deriving Mannheim curves and constant-pitch curves from each other by means of a suitable deformation of a space curve. We define a “radius” function and a “pitch” function for any arbitrary regular space curve and use these to characterize the two classes of curves. A few non-trivial examples of both Mannheim and constant pitch curves are discussed. The geometric nature of Mannheim curves is established by using the notion of osculating helices. The Frenet–Serret motion of a rigid body in theoretical kinematics is studied for the special case of a Mannheim curve and the axodes in this case are deduced. In particular, we show that the fixed axode is developable if and only if the motion trajectory is a Mannheim curve.

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Ruled invariants and associated ruled surfaces of a space curve

Applied Mathematics and Computation ◽

10.1016/j.amc.2018.12.011 ◽

2019 ◽

Vol 348 ◽

pp. 479-486 ◽

Cited By ~ 1

Author(s):

Huili Liu ◽

Yixuan Liu ◽

Seoung Dal Jung

Keyword(s):

Space Curve ◽

Ruled Surfaces

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On geometry of the midlocus associated to a smooth curve in plane and space

Filomat ◽

10.2298/fil1808977a ◽

2018 ◽

Vol 32 (8) ◽

pp. 2977-2990

Author(s):

Azeb Alghanemi ◽

Peter Giblin

Keyword(s):

Plane Curve ◽

Smooth Curve ◽

Space Curve ◽

Regular Space ◽

Geometric Conditions ◽

Special Cases ◽

Smooth Plane ◽

Surface Singularities

The singularities of the midpoint map associated to a smooth plane curve, which is a map from the plane to the plane, are classified. The midlocus associated to a regular space curve is introduced. The geometric conditions for the midlocus of a space curve to have a crosscap or an S?1 singularities are investigated. A more general map, the ?-point map, associated to a space curve is introduced and many known surface singularities are realized as a special cases of this construction.

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A Different View on Dynamics of Space Curves Geometry

Journal of Nonlinear Mathematical Physics ◽

10.1007/s44198-021-00023-8 ◽

2021 ◽

Author(s):

Yılmaz Tunçer

Keyword(s):

Space Curve ◽

Regular Space ◽

Position Vector ◽

Space Curves

AbstractIn this study, we define the X-torque curves, $$X-$$ X - equilibrium curves, X-moment conservative curves, $$X-$$ X - gyroscopic curves as new curves derived from a regular space curve by using the Frenet vectors of a space curve and its position vector, where $$X\in \left\{ T\left( s\right) , N\left( s\right) , B\left( s\right) \right\} $$ X ∈ T s , N s , B s and we examine these curves and we give their properties.

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