Abstract
The differential geometry of plane curves has many applications in physics especially in mechanics. The curvature of a plane curve plays a role in the centripetal acceleration and the centripetal force of a particle traversing a curved path in a plane. In this paper, we introduce the concept of the -curves associated with a plane curve which are more general than the well-known curves such as involute, evolute, parallel, symmetry set, and midlocus. In fact, we introduce the -curves associated with a plane curve via its normal and tangent for both the cases, a Frenet curve and a Legendre curve. Moreover, the curvature of an -curve has been obtained in several approaches.
1. Introduction
The differential geometry of a plane curve is an attractive area of research for geometers and physicists, owing to its applications in several areas such as mechanics, computer graphics, computer vision, and medical imaging. In mechanics, for example, the differential geometry of plane curves is used to study the motion of a particle in a plane. Moreover, the curvature of a curved path is used for computing the centripetal acceleration and the centripetal force of a particle moving along that path (cf. [1]).
In this paper, we define the -curve associated with a given plane curve, for both the cases, a Frenet curve and a Legendre curve. Note that Legendre curves are more general than Frenet curves. Recently, the geometry of Legendre plane curves has been quite extensively studied, and in particular, their evolutes and involutes have been investigated (cf. [2–5]). An important achievement of this paper is that what we find a neat expression for the curvature of an -curve.
This paper consists of five main sections. The first section is introductory, giving a general idea about the paper. The second section contains basic concepts of the differential geometry of Frenet plane curves and Legendre plane curves which will be used in the rest of this paper. In the third section, we introduce the concept of the -curves associated with Frenet and Legendre curves via their normals and we study their curvatures in several cases. In the fourth section, we introduce and study the -curves associated with a Frenet curve and a Legendre curve in a plane via their tangents. Moreover, we give formulae for the curvature of the -curve associated with a plane curve in both the cases, a Frenet curve and a Legendre curve. In the fifth section, we give nontrivial examples of the -curve associated with a regular curve via its normal and we draw these curves using the Maple.
2. Preliminaries
In this section, we are going to review basic concepts of the differential geometry of plane curves. For more detail about plane curves and their properties, we refer the reader to [6, 7]. A smooth plane curve is a map given by such that and are smooth functions on , where is an open interval of . If is a regular parametrized curve (i.e., for all ), then we define the unit tangent vector by and the unit normal vector by , where is the counterclockwise rotation by and . Now, the Frenet formula is given by where prime is the derivative with respect to the parameter and is the curvature of which is given by . The pair is called the moving frame of the regular curve . If is parametrized by its arc-length , then the Frenet formula is given by where is the unit tangent vector, is the unit normal vector, and . Also, the curvature is defined by , where is the angle function between the horizontal lines and the tangent of and is the arc length of .
Definition 1. Let be a regular parametrized curve and , where is a constant. Then, and are parallel curves.
Definition 2. Let be a regular parametrized curve with nonvanishing curvature. Then, the evolute of is given by
It is a well-known result that is regular at if and only if .
In the rest of this section, we review some basic concepts of Legendre plane curves, and for more information, we refer the reader to [2–5, 8].
Definition 3. The map is called a Legendre curve if for all , where is the unit circle and is a smooth unit vector field. The map is a Legendre immersion if it is an immersion.
We call a frontal if there exists a smooth function such that is a Legendre curve, and we call a wavefront if there exists a smooth function such that is a Legendre immersion.
For a Frenet curve , if has a singular point, then the moving frame is not well-defined. For a Legendre curve, , an alternative frame is well-defined at any point. This frame is given by , where . Also, we have the following formula:
where . We call the pair a moving frame of a Legendre curve . In addition, there exists a smooth function such that . The curvature of the Legendre curve is .
Definition 4 (see [4]). Suppose that is a frontal with the curvature . If there exists a unique smooth function such that for all , then the evolute of is given by . Moreover, is a frontal with the curvature . This means that is a Legendre curve, where is the counterclockwise rotation by on .
Definition 5 (see [4]). Suppose that is a frontal with the curvature . The involute of frontal at is given by , and is a Legendre curve with curvature , where is the clockwise rotation by on .
3. -Curve via the Normal Vector Associated with a Plane Curve
3.1. Regular Curve
In this section, we study the -curve via the normal vector associated with a regular plane curve. Also, the curvature of this curve will be obtained in two different ways.
Definition 6. Let be a regular parametrized curve. Then, the -curve via the normal vector associated with is defined by , where is a smooth function.
In the following lemma, we give the necessary and sufficient condition for the curve to be a regular curve.
Lemma 7. Let be a unit speed curve. Then, the curve is regular if and only if or .
Proof. The proof of this lemma is obvious.
Remark 8. From Lemma 7, it can be easily obtained that all singular points of the -curve via the normal vector associated with lie on the evolute of .
The following theorem provides a useful formula for the curvature of the -curve via the normal vector associated with in the case of its regularity.
Theorem 9. Let be a unit speed curve and be its associated -curve via the normal vector. (1)If , then , where is the angle between the tangent vector of and the unit tangent vector of (2)If , then , where is the angle between the tangent vector of and the unit normal vector of
Proof. Let be a unit speed curve and be its associated -curve via the normal vector. Then, we have Case 1. If , then from Lemma 7, is a regular curve, and we have which gives So, we have Equation (8) can be rewritten as From equation (5), we have Now, using equation (8) in equation (10), we get So, Substituting (8) and (12) in (5), we have The unit tangent vector of , , can be written as and the unit normal vector of , , can be written as Now, Hence, Case 2. If , then from Lemma 7, is a regular curve, and we have which gives So, we have Equation (20) can be rewritten as Now, using equation (20) in equation (10), we get Now, Substituting (20) and (23) in (5), we get So, Now, Hence,
As an application of Theorem 9, the curvatures of parallel curves and evolute become special cases of this theorem. Precisely, we have the following corollary.
Corollary 10. Let be a unit speed curve with nonvanishing curvature and be its associated -curve via the normal vector. (1)If is a constant and , then are parallel curves and (2)If and , then is the evolute of and
3.2. Legendre Curve
In this section, we consider the case when is a Legendre curve.
Definition 11. Let be a Legendre curve. Then, the -curve via the vector field associated with is given by , where is a smooth function.
Lemma 12. Let be a Legendre curve. Then, the -curve via the vector field associated with is regular if and only if or .
Proof. Let be associated with -curve via the vector field , then we have Since and , we have Now, if and only if or .
Theorem 13. Let be a Legendre curve and be its associated -curve via the vector field . (1)If , then , where is the angle between the tangent vector of and the unit vector field (2)If , then , where is the angle between the tangent vector of and the unit vector field
Proof. Let be a Legendre curve and be its associated -curve via the vector field . Then, we have Case 1. If , then from Lemma 12, is a regular curve, and we have which gives So, we have Equation (34) can be rewritten as From equation (31), we have Now, using equation (34) in (36), we get So, From (34) and (38), equation (31) becomes Now, Thus, Hence, Case 2. If , then from Lemma 12, is a regular curve, and we have which gives So, we have Equation (45) can be rewritten as Now, using equation (45) in (36), we have So, Substituting (45) and (48) in (31), we get Now, Thus, Hence,
Now, we have the following corollary of Theorem 13.
Corollary 14. Let be a frontal and be its associated -curve via the vector field . (1)If is a unique smooth function such that and , then is the evolute of and it is a regular curve with (2)If is a constant and , then
4. -Curve via the Tangent Vector Associated with a Plane Curve
4.1. Regular Curve
Definition 15. Let be a regular parametrized curve. Then, the -curve via the tangent vector associated with is given by , where is a smooth function.
Now, we state the following lemma.
Lemma 16. Let be a unit speed curve. Then, the curve associated with is regular if and only if or .
Remark 17. From Lemma 16, if has a singularity at , then has an inflexion point at or .
Theorem 18. Let be a unit speed curve and be its associated -curve via the tangent vector. (1)If , then , where is the angle between the tangent vector of and the unit tangent vector of (2)If , then , where is the angle between the tangent vector of and the unit normal vector of
Proof. Let be a unit speed curve and be its associated -curve via the tangent vector. Then, we have Case 1. If , then from Lemma 16, is a regular curve, and we have which gives From equation (53), we have Substituting equation (55) in equation (56), we obtain that From equation (55) and equation (57), equation (53) becomes Therefore, Hence, Case 2. If , then from Lemma 16, is a regular curve, and we have which gives Substituting (62) in (56), we obtain that From equation (62) and equation (63), equation (53) becomes Therefore, Hence,
The following corollary can be easily obtained from Theorem 18.
Corollary 19. Let be a unit speed curve and be its associated -curve via the tangent vector. If (that is, for some constant ) and , then is the involute of and .
From Theorem 18, it can be easily obtained a simple and neat formula for the curvature of the regular part of the midlocus associated with a regular part of the symmetry set of a plane curve. The symmetry set of a plane curve is the closure of the locus of centers of the bitangent circles, and the associated midlocus is the set of all midpoints of the chords joining the tangency points. For more detail in the symmetry set of a plane curve and the associated midlocus, we refer the reader to [9–12].
In [9], the first author of this paper obtained the curvature of the midlocus associated with the regular part of the symmetry set of a plane curve. This curvature is given by the following formula: where is the angle between the normal of a given curve and the tangent of its symmetry set and is the angle between the tangent of the symmetry set and the tangent of midlocus and prime denotes the derivative with respect to the arc length of the symmetry set. This formula contains more factors than the following formula in the next corollary which gives a simple formula of the curvature of the midlocus. In the following corollary, is the symmetry set of a given curve, is the radius function of the bitangent circles, and the associated -curve via the tangent of is the associated midlocus.
Corollary 20. Let be a unit speed curve and be its associated -curve via the tangent vector such that . (1)If , then (2)If , then
4.2. Legendre Curve
Definition 21. Let be a Legendre curve. The -curve via the vector filed associated with is given by , where is a smooth function.
Lemma 22. Let be a Legendre curve. The -curve via the vector field associated with is regular if and only if or .
Theorem 23. Let be a Legendre curve and be its associated -curve via the vector field . (1)If , then , where is the angle between the tangent vector of and the unit vector field (2)If , then , where is the angle between the tangent vector of and the unit vector field
Proof. Let be a Legendre curve and be its associated -curve via the vector field . Then, we have Case 1. If , then from Lemma 22, is a regular curve, and we have which gives From equation (68), we have Substituting (70) in (71), it can be shown that From (70) and (72), equation (68) becomes Now, Hence, Case 2. If , then from Lemma 22, is a regular curve, and we have which gives Using equation (77) in (71), it can be shown that From (77) and (78), equation (68) becomes Now, Hence,
Corollary 24. Let be a Legendre curve with the curvature and be its associated -curve via the vector field . If and , then is the involute of the frontal and it is a regular curve with .
5. Examples
In this section, we give an example in the case of a regular curve and we repeat the process to the resulting curve but with different function . We use the Maple for calculation and drawing pictures. First, let us consider the curve , . It is clear that is a closed curve (see Figure 1(a)), and we choose . The -curve via the normal vector associated with is given by , where
(a) The curve
(b) The curve
(c) The curve
The graph of is shown in Figure 1(b), and we may call this graph Clown’s face.
In what follows, we consider the -curve via the normal vector associated with with ; this curve is given by . The calculation becomes very tedious, and the forms of and are too long and very complicated. We just use the Maple for drawing the graph of this curve which is shown in Figure 1(c). Again, the resulting graph is interesting as well as the graph of and we may call this graph Tiger’s face.
Remark 25. From this section, one can observe the importance of -curves associated with a plane curve in the theory of computer graphics. We hope this method for creating curves from a given curve will find its applications to computer graphics and related topics soon.
6. Final Remark
Throughout this paper, we introduce the concept of the -curves associated with a plane curve in the cases of Frenet and Legendre curves. The curvatures of these new curves have been obtained in several ways. This work has direct applications to the motion of particles in the plane. Also, it is useful for calculating the centripetal acceleration and the centripetal force of a particle traversing a curved path such as -curve in the plane.
Data Availability
No external data has been used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.