Abstract
In this paper, we consider the Darboux frame of a curve lying on an arbitrary regular surface and we use its unit osculator Darboux vector , unit rectifying Darboux vector , and unit normal Darboux vector to define some direction curves such as -direction curve, -direction curve, and -direction curve, respectively. We prove some relationships between 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 , , and are, respectively, zero. Finally, we enrich our study by giving some examples.
1. Introduction
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. The 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].
Recently, the problem of deriving a new curve from a given curve and obtaining new characterizations for them has taken its place among popular topics. This category of curves called associated curve has been investigated in various research. 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]. They 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 . However, Deshmukh et al. defined in [12] the natural mate curve and the conjugate mate curve which are similar to the principal-direction curve and binormal-direction curve, respectively, from algebraic viewpoint, but are more accurate and comprehensive from the geometric viewpoint since the integral curve is defined only for vector fields on a region which contains a curve, not along a curve. Then, they derived some new characterizations of helices, slant helices, spherical curves, and rectifying curves. Furthermore, 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. These direction curves were used to give a new approach to construct slant helices and C-slant helices. In [14], Macit and Duldul defined W-direction curves and W-rectifying curves of a Frenet curve in by utilizing the Darboux vector of the curve. They also introduced V-direction curve of a given curve on a surface by using the Darboux frame. This curve was used to characterize the relatively normal-slant helix in [3].
Motivated by this, in the present study, we consider the Darboux frame of a curve lying on an arbitrary regular surface, and we define in Euclidean 3-dimensional space the -direction curve, the -direction curve and the -direction curve of as the integral curve of the unit osculator Darboux vector , the unit rectifying Darboux vector , and the unit normal Darboux vector , respectively. Then, we give some relationships between the curve and each direction curve. Especially, we obtain necessary and sufficient conditions for that these direction curves be a general helix, a spherical curve, and a curve with constant torsion. Beside, we discuss the cases where the Darboux invariants , , and are, respectively, zero. Finally, two examples are illustrated.
2. Preliminaries
In this section, we recall some basic concepts and properties on classical differential geometry of curves lying on a regular surface, in the Euclidean 3-dimensional space.(i)We denote by the Euclidean 3-dimensional space, with the usual metric where and are two vectors of .(ii)Let be a regular curve in , with nonnull curvature, we also assume that is parametrized by arc-length , i.e., for all . The Frenet frame along the curve is an orthonormal frame where is the unit tangent, is the unit principal normal, is the unit binormal, and and are, respectively, the curvature and the torsion of the curve , given by The curve is called a Frenet curve if . It is known that a Frenet curve in is a spherical curve if and only if holds, where , . Moreover, if the Frenet curve is a spherical curve lying on a sphere of radius , then we have [15](iii)Let be a regular surface, and be a unit speed curve on the surface . The Darboux frame along the curve is an orthonormal frame , where is the unit tangent, is the unit normal on the surface M, and . Then, the Darboux equations are given by the following relations:where , , and are, respectively, the geodesic curvature, the normal curvature, and the geodesic torsion of the curve and are given bywith denote the angle between the surface normal and the normal .
The curve is called a general helix (resp., a relatively normal slant helix and an isophote curve) if the vector field T (resp., V and U) makes a constant angle with a fixed direction, i.e., there exists a fixed unit vector and a constant angle such that (resp., and ).
Let us give some theorems characterizing these curves.
Theorem 1 (see [2]). A curve is a general helix if and only if the following functionis constant.
Theorem 2 (see [3]). A unit speed curve on a surface M with is a relatively normal slant helix if and only if the following functionis constant.
Theorem 3 (see [4]). A unit speed curve on a surface M with is an isophote curve if and only if the following functionis constant.
3. Direction Curves Associated with Darboux Vector Fields
In this section, we introduce the direction curves associated with Darboux vector fields of a curve lying on a regular surface, and we determine some of their properties.
Let be a unit speed regular curve lying on a regular surface . We denote by its Darboux frame, , , and the normal curvature, the geodesic curvature, and the geodesic torsion of the curve and by the unit osculator Darboux vector with and the Darboux invariant associated with the Darboux frame.
Definition 1. The integral curve (resp., and of the osculator (resp., rectifying and normal) Darboux vectors field (resp., and ) is called -direction (resp., -direction and direction) curve of . In other words,
3.1. Frenet Frame of Direction Curves Associated with Darboux Vector Fields
In this paragraph, we determine the Frenet frame of , , and , respectively, assuming for each case that the curve is well defined.
Theorem 4. Let be the -direction curve of . If , the Frenet vector fields , curvature , and torsion of are given bywhere .
Proof. From the definition of the curve , we have .
Using (6) and after calculation, we getThen,So, by taking the norm of (15), we obtain , where . Thus,The cross production of and leads us the binormal vector as follows:On the other hand, we have
Theorem 5. Let be the -direction curve of . If , the Frenet vector fields , curvature , and torsion of are given bywhere .
Proof. We have .
Differentiating , we getTherefore, we have , where . Thus,On the other hand, we have
Theorem 6. Let be the -direction curve of . We denote by and the curvature and the torsion of . If , the Frenet vector fields , curvature , and torsion of are given bywhere .
Proof. In this case, we have .
Differentiating by using (6), we get .
By using (7), we prove thatThen,So, by taking the norm of (30), we have , where . Consequently,On the other hand, we haveUsing Theorems 1–3, we conclude the following results.
Corollary 1. (1) is a general helix if and only if is an isophote curve(2) is a general helix if and only if is a relatively normal-slant helix(3) is a general helix if and only if is a general helix
Proof. The proof can be done by using (13), (21), and (28), respectively.
3.2. Spherical Direction Curves of Darboux Vector Fields
In this paragraph, we propose to determine a necessary and sufficient condition for the direction curve (resp., and ) to be a spherical curve.
Theorem 7. Let be a unit speed regular curve lying on a regular surface , such that and its -direction curve. The curve is a spherical curve if and only ifwhere is the radius of the sphere.
Proof. From (5), since the curve is lying on a sphere of radius a, we haveIt follows thatand by using the expression of and , we obtainWe have ; otherwise, , and as is assumed spherical, necessarily [1], which is absurd because we assumed . We getTherefore, we getand it means thatConversely, suppose that , we haveThen,So, . Hence the result from (4).
By reasoning similarity, we obtain the following theorem.
Theorem 8. Let be a unit speed regular curve lying on a regular surface , such that and its -direction curve. The curve is a spherical curve if and only ifwhere is the radius of the sphere.
Theorem 9. Let be a unit speed regular curve lying on a regular surface , such that and its -direction curve. The curve is a spherical curve if and only ifwhere is the radius of the sphere, and and are the curvature and the torsion of , respectively.
Proof. From (5), since is lying on a sphere of radius a, we haveIt follows thatby using (28), we obtainSince , we writeTherefore, we getConversely, suppose that , we haveThen, it is easy to see that . Hence the result from (4).
3.3. Direction Curves of Darboux Vector Fields with Constant Torsion
In this paragraph, we propose to study the case where the direction curve (resp., and ) has a constant torsion.
From formula (13), we can conclude that if there exists a function such that the normal curvature and the geodesic torsion of the curve satisfywhere c is a constant, then the -direction curve of the curve has a constant torsion.
Conversely, we give the following theorem.
Theorem 10. Let be a unit speed regular curve lying on a regular surface , such that and its -direction curve. If the curve has a constant torsion, then the normal curvature and the geodesic torsion of satisfy the following equalities:where and
Proof. We takeBy differentiating (56), we obtainOn the other hand, we have(i)If , by multiplying (58) by , and using (57), we get Since , we can reformulate the given expression of as or equivalently By integrating (61), we obtain so immediately, we find(ii)In a similar way, we make sure of the result for . Then, by multiplying (58) by and using (57), we get Since , we can reformulate the given expression of as or equivalently By integrating (66), we obtain so immediately, we find By taking , we get the result as desired.Similarly, from formula (21), we can conclude that if there exists a function such that the geodesic curvature and the geodesic torsion of the curve satisfywhere is a constant, then the -direction curve of the curve has a constant torsion.
Conversely, we can prove the following theorem, using the same reasoning as Theorem 10.
Theorem 11. Let be a unit speed regular curve lying on a regular surface , such that and its -direction curve. If the curve has a constant torsion, then the geodesic curvature and the geodesic torsion of satisfy the following equalities:where andUsing formula (28), we can state the following theorem without proof.
Theorem 12. Let be a unit speed regular curve lying on a regular surface , with torsion and its -direction curve. has a constant torsion if and only if the curvature of is constant.
4. Necessary and Sufficient Condition for the Darboux Invariant (Resp., and to Be Zero
In Section 3, we defined the direction curve , when it exists, and we have given its Frenet frame as well as some properties. All this work was done under the condition , that is, it was assumed that the curvature of is nonzero. In this section, we propose to study the special case where is zero.
Theorem 13. Let be a unit speed regular curve lying on a regular surface , and its -direction curve. Then, if and only if the position vector of always lies in the osculating plane of .
Proof. Suppose that , then from (15), we getConsequently, .
Conversely, suppose thatwhere and are differentiable functions. Since is the -direction curve of , by differentiating (73) with respect to s, we obtainwhich gives us the following system:In this case, we assume that . If (we notice that the result is the same if ), from the third equation of (75), we haveBy replacing (76) in the first equation of system (75), we obtain the following differential equation:We solve the differential equation, and we findwhere K is an integration constant. So, from (76), we writeand by differentiating (79), we obtainTherefore, by replacing (78) and (80) in the second equation of system (75), we findHence , which completes the proof.
By doing the same, we get the following two theorems.
Theorem 14. Let be a unit speed regular curve lying on a regular surface , and its -direction curve. Then, if and only if the position vector of always lies in the rectifying plane of .
Theorem 15. Let be a unit speed regular curve lying on a regular surface and its -direction curve. Then, if and only if the position vector of always lies in the normal plane of .
5. Examples
Example 1. Let be a curve lying on the surface given by the following parametrization (see Figure 1):The Darboux frame of isWe can notice that is an isophote curve, and according to Corollary 1, is a general helix.
We have , , and . It followsConsequently, the -direction curve, -direction curve, and -direction curve are given, respectively, by , , and as follows (see Figures 1–3):where , are real constants.
Example 2. Let be a curve lying on the surface given by the following parametrization (see Figure 4):The Darboux frame of isIn this example, the curve is a general helix, a relatively normal slant helix, and an isophote curve. Therefore, according to Corollary 1, the curves , , and are general helices.
We have It followsConsequently, the -direction curve, -direction curve, and -direction curve are given, respectively, by , , and as follows (see Figures 5–7):where , , is a real constant.
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare no conflicts of interest.