Space curves defined by curvature-torsion relations and associated helices

The relationships between certain families of special curves, including the general helices, slant helices, rectifying curves, Salkowski curves, spherical curves, and centrodes, are analyzed. First, characterizations of proper slant helices and Salkowski curves are developed, and it is shown that, for any given proper slant helix with principal normal n, one may associate a unique general helix whose binormal b coincides with n. It is also shown that centrodes of Salkowski curves are proper slant helices. Moreover, with each unit-speed non-helical Frenet curve in the Euclidean space E3, one may associate a unique circular helix, and characterizations of the slant helices, rectifying curves, Salkowski curves, and spherical curves are presented in terms of their associated circular helices. Finally, these families of special curves are studied in the context of general polynomial/rational parameterizations, and it is observed that several of them are intimately related to the families of polynomial/rational Pythagorean-hodograph curves.

Circular Helix-Like Curve: An Effective Tool of Biological Sequence Analysis and Comparison

This paper constructed a novel injection from a DNA sequence to a 3D graph, named circular helix-like curve (CHC). The presented graphical representation is available for visualizing characterizations of a single DNA sequence and identifying similarities and differences among several DNAs. A 12-dimensional vector extracted from CHC, as a numerical characterization of CHC, was applied to analyze phylogenetic relationships of 11 species, 74 ribosomal RNAs, 48 Hepatitis E viruses, and 18 eutherian mammals, respectively. Successful experiments illustrated that CHC is an effective tool of biological sequence analysis and comparison.

Bertrand Curves of AW(k)-Type in the Equiform Geometry of the Galilean Space

We consider curves of AW(k)-type (1≤k≤3) in the equiform geometry of the Galilean spaceG3. We give curvature conditions of curves of AW(k)-type. Furthermore, we investigate Bertrand curves in the equiform geometry ofG3. We have shown that Bertrand curve in the equiform geometry ofG3is a circular helix. Besides, considering AW(k)-type curves, we show that there are Bertrand curves of weak AW(2)-type and AW(3)-type. But, there are no such Bertrand curves of weak AW(3)-type and AW(2)-type.

A generalized space curve meshing equation for arbitrary intersecting gear

This article derives the space curve meshing equations for arbitrary intersecting gear mechanism, which could achieve continuous and smooth transmission between arbitrary angle intersecting axes, and then establishes the central curves equations of the driving and driven tines on the basis of space curve meshing equations. According to the equations, a calculation example is given and material prototype samples are made to experimentally validate the kinematic performance. The result shows that if the central curve of the driving tine is a circular helix, the central curve of the driven tine is close to a conical helix. This article will supply a basic theory for the application of the arbitrary intersecting gear mechanism in various areas.

LEGENDRIAN HELIX AND CABLE LINKS

Traynor ([11]) has described an example of a two-component Legendrian "circular helix link" Λ0 ⊔ Λ1 in the 1-jet space J1(S1) of the circle (with its canonical contact structure) that is topologically but not Legendrian isotopic to the link Λ1 ⊔ Λ0. We give a complete classification of the Legendrian realizations of this topological link type, as well as all other "cable links" in J1(S1).

Legendrian circular helix links

Examples are given of legendrian links in the manifold of cooriented contact elements of the plane, or equivalently, in the 1-jet space of the circle which are not equivalent via an isotopy of contact diffeomorphisms. These examples have generalizations to linked legendrian spheres in contact manifolds diffeomorphic to ℝn×Sn−1. These links are distinguished by applying the theory of generating functions to contact manifolds.

Determination of phase velocities of a circular helix using the method of internal constraints

A procedure is described to determine phase velocities of a circular helix. Computations are made for three helix configurations using the method of internal constraints. The results so obtained are compared with the results obtained by using the elementary theory proposed by J. H. Michell, and the extended elementary theory proposed by A. B. Basset.