Abstract
In this paper, we consider the Darboux frame of a curve α lying on an arbitrary regular surface and we use its unit osculator Darboux vector D ¯ o , unit rectifying Darboux vector D ¯ r , and unit normal Darboux vector D ¯ n to define some direction curves such as D ¯ o -direction curve, D ¯ r -direction curve, and D ¯ n -direction curve, respectively. We prove some relationships between α and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants δ o , δ r , and δ n are, respectively, zero. Finally, we enrich our study by giving some examples.
Highlights
In the theory of curves in differential geometry, characterizing the curves and giving general information about their structure in terms of curvature is a very interesting and important problem that is developed by many differential geometers in different ambient spaces. e most popular and important curves that have been studied in many types of research are spherical curve [1], general helix [2], relatively normal slant helix [3], isophote curve [4], Salkowski curve [5], and anti-Salkowski curve [6]
The problem of deriving a new curve from a given curve and obtaining new characterizations for them has taken its place among popular topics. is category of curves called associated curve has been investigated in various research
In the Euclidean 3-dimensional space, a new version of the associated curve called direction curve was introduced by Choi and Kim in [11]. ey defined the principal-direction curve and binormal-direction curve as the integral curve of principal normal N and binormal B of a Frenet curve, respectively, and they use this concept to characterize general helices, slant helices, and PD-rectifying curve in E3
Summary
In the theory of curves in differential geometry, characterizing the curves and giving general information about their structure in terms of curvature is a very interesting and important problem that is developed by many differential geometers in different ambient spaces. e most popular and important curves that have been studied in many types of research are spherical curve [1], general helix [2], relatively normal slant helix [3], isophote curve [4], Salkowski curve [5], and anti-Salkowski curve [6]. Bertrand curve [7], involute-evolute curve [8], Mannheim curve [9], and spherical indicatrix [10] are among the leading examples In this sense, in the Euclidean 3-dimensional space, a new version of the associated curve called direction curve was introduced by Choi and Kim in [11]. Ey defined the principal-direction curve and binormal-direction curve as the integral curve of principal normal N and binormal B of a Frenet curve, respectively, and they use this concept to characterize general helices, slant helices, and PD-rectifying curve in E3. In [13], the authors expressed new direction curves such as evolute direction curves, Bertrand direction curves, and Mannheim directon curves by means of a vector field generated by Frenet vectors of normal indicatrix of a given curve.
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