Space curves defined by curvature-torsion relations and associated helices

Abstract

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 E 3 , 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.

Characterizations of Loxodromes on Rotational Surfaces in Euclidean 3-Space

“…A curve α is a general helix if and only if the ratio of κ and τ is a constant. When both κ and τ are constants, α is called a circular helix (see [3]).…”

“…Similar as loxodromes, helices are important in navigation, too. For some interesting and magical applications of helices, we refer to [2,3] and references therein.…”

Position Vectors of the Natural Mate and Conjugate of a Space Curve

“…In this section, we review some basic concepts of the differential geometry of curves in Euclidean 3-space, and for more detail, we refer the reader to [2,3,[11][12][13][14][15][16]. First, we start with the definition of a smooth space curve.…”

“…The rectifying curves are used to analyze joint kinematics (cf. [2,3]). The Salkowski curves are useful in constructing closed curves with constant curvature and continuous torsion such as knotted curves (cf.…”

Bishop Frames of Salkowski Curves in E3

“…The Darboux and pole vectors belonging to the Frenet frame and modified frames of Salkowski curves 3 E are studied in [24]. Other some studies on Salkowski curves in 3 E can be looked at from [25][26][27][28]. In this study, alternative, type-1, type-2 and N-Bishop frames of Salkowski curves are calculated and curvatures, Darboux and pole vectors belonging to the frames are investigated.…”