Some Geometric Characterizations of f -Curves Associated with a Plane Curve via Vector Fields

The di ﬀ erential 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 f -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 f -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 f -curve has been obtained in several approaches.


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 f -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][3][4][5]). An important achievement of this paper is that what we find a neat expression for the curvature of an f -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 f -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 f -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 f -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 f -curve associated with a regular curve via its normal and we draw these curves using the Maple.

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 γ is a map γ : I ⟶ ℝ 2 given by γðtÞ = ðγ 1 ðtÞ, γ 2 ðtÞÞ such that γ 1 ðtÞ and γ 2 ðtÞ are smooth functions on I, where I is an open interval of ℝ. If γ is a regular parametrized curve (i.e., γ ′ ðtÞ=0 for all t ∈ I), then we define the unit tangent vector by T γ ðtÞ = γ′ðtÞ/kγ′ðtÞk and the unit normal vector by N γ ðtÞ = JðT γ ðtÞÞ, where J is the counterclockwise rotation by π/2 and kγ′ðtÞk = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi γ ′ ðtÞ · γ ′ ðtÞ q . Now, the Frenet formula is given by where prime is the derivative with respect to the parameter t and κ γ ðtÞ is the curvature of γ which is given by κ γ ðtÞ = ð T γ ′ðtÞ · N γ ðtÞÞ/kγ′ðtÞk. The pair fT γ ðtÞ, N γ ðtÞg is called the moving frame of the regular curve γ. If γ is parametrized by its arc-length s, then the Frenet formula is given by where T γ ðsÞ = γ′ðsÞ is the unit tangent vector, N γ ðsÞ = JðT γ ðsÞÞ is the unit normal vector, and κ γ ðsÞ = ±kT γ ′ðsÞk. Also, the curvature is defined by κ γ ðsÞ = dχðsÞ/ds, where χ is the angle function between the horizontal lines and the tangent of γ and s is the arc length of γ.
Definition 2. Let γ : I ⟶ ℝ 2 be a regular parametrized curve with nonvanishing curvature. Then, the evolute of γ is given by It is a well-known result that E γ is regular at t 0 ∈ I if and only if κ γ ′ ðt 0 Þ ≠ 0.
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][3][4][5]8].
Definition 3. The map ðγ, ωÞ: I ⟶ ℝ 2 × S 1 is called a Legendre curve if γ ′ ðtÞ · ωðtÞ = 0 for all t ∈ I, where S 1 is the unit circle and ω : I ⟶ S 1 is a smooth unit vector field. The map ðγ, ωÞ is a Legendre immersion if it is an immersion.
We call γ : I ⟶ ℝ 2 a frontal if there exists a smooth function ω : I ⟶ S 1 such that ðγ, ωÞ is a Legendre curve, and we call γ : I ⟶ ℝ 2 a wavefront if there exists a smooth function ω : I ⟶ S 1 such that ðγ, ωÞ is a Legendre immersion.
For a Frenet curve γ, if γ has a singular point, then the moving frame is not well-defined. For a Legendre curve, ðγ , ωÞ: I ⟶ ℝ 2 × S 1 , an alternative frame is well-defined at any point. This frame is given by fω, μg, where μðtÞ = Jðωð tÞÞ. Also, we have the following formula: where ℓðtÞ = ω ′ ðtÞ · μðtÞ. We call the pair fωðtÞ, μðtÞg a moving frame of a Legendre curve γ. In addition, there exists a smooth function βðtÞ such that βðtÞ = γ′ðtÞ · μðtÞ. The curvature of the Legendre curve is ðℓðtÞ, βðtÞÞ.

f -Curve via the Normal Vector
Associated with a Plane Curve 3.1. Regular Curve. In this section, we study the f -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 γ : I ⟶ ℝ 2 be a regular parametrized curve. Then, the f -curve via the normal vector associated with γ is defined by αðtÞ = γðtÞ + f ðtÞN γ ðtÞ, where f : I ⟶ ℝ is a smooth function.
In the following lemma, we give the necessary and sufficient condition for the curve α to be a regular curve. Proof. The proof of this lemma is obvious.  Advances in Mathematical Physics The following theorem provides a useful formula for the curvature of the f -curve via the normal vector associated with γ in the case of its regularity. Theorem 9. Let γ : I ⟶ ℝ 2 be a unit speed curve and α be its associated f -curve via the normal vector. ( where θ is the angle between the tangent vector of α and the unit tangent vector of γ where ϕ is the angle between the tangent vector of α and the unit normal vector of γ Proof. Let γ : I ⟶ ℝ 2 be a unit speed curve and α be its associated f -curve via the normal vector. Then, we have Case 1. If f κ γ =1, then from Lemma 7, α 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 α, T α , can be written as and the unit normal vector of α, N α , can be written as Now, Hence, Case 2. If f ′ =0, then from Lemma 7, α 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,

Advances in Mathematical Physics
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 γ : I ⟶ ℝ 2 be a unit speed curve with nonvanishing curvature and α be its associated f -curve via the normal vector.

Legendre Curve.
In this section, we consider the case when γ is a Legendre curve.
Corollary 14. Let γ be a frontal andα be its associated f -curve via the vector field ω.
(1) If f is a unique smooth function such that f = −β/ℓ and f ′ =0, thenα is the evolute of γ and it is a regular curve with κα = ±ℓ 3 /ðℓ′β − ℓβ′Þ (1) If f ′ ≠ −1, then κ Ω = ððκ γ ± ζ ′ Þ cos ζÞ/ð1 + f ′ Þ, where ζ is the angle between the tangent vector of Ω and the unit tangent vector of γ where Ψ is the angle between the tangent vector of Ω and the unit normal vector of γ Proof. Let γ : I ⟶ ℝ 2 be a unit speed curve and Ω be its associated f -curve via the tangent vector. Then, we have Case 1. If f ′ ≠ −1, then from Lemma 16, Ω is a regular curve, and we have which gives From equation (53), we have Substituting equation (55) in equation (56), we obtain that
Corollary 19. Let γ : I ⟶ ℝ 2 be a unit speed curve and Ω be its associated f -curve via the tangent vector. If f ′ = −1 (that is, f ðsÞ = C − s for some constant C) and f ≠ 0, then Ω is the involute of γ and κ Ω = singðκ γ Þ/j f j.
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][10][11][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 φ is the angle between the normal of a given curve γ and the tangent of its symmetry set and Θ 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, γ is the symmetry set of a given curve, r is the radius function of the bitangent circles, and the associated f -curve via the tangent of γ is the associated midlocus.
Corollary 20. Let γ : I ⟶ ℝ 2 be a unit speed curve and Ω be its associated f -curve via the tangent vector such that f = −rr ′ .