# Some Geometric Characterizations off-Curves Associated with a Plane Curve via Vector Fields

Authors: Azeb Alghanemi; Abeer AlGhawazi
Journal: Advances in Mathematical Physics
 (2022)

---

## 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 thef-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.


---

## Body

## 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 thef-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 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 thef-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.


## 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 γ is a map γ:I⟶ℝ2 given by γt=γ1t,γ2t such that γ1t and γ2t are smooth functions on I, where I is an open interval of ℝ. If γ is a regular parametrized curve (i.e., γ′t=0 for all t∈I), then we define the unit tangent vector by Tγt=γ′t/γ′t and the unit normal vector by Nγt=JTγt, where J is the counterclockwise rotation by π/2 and γ′t=γ′t·γ′t. Now, the Frenet formula is given by
(1)Tγ′tNγ′t=0γ′tκγt−γ′tκγt0TγtNγt,where prime is the derivative with respect to the parameter t and κγt is the curvature of γ which is given by κγt=Tγ′t·Nγt/γ′t. The pair Tγt,Nγt 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
(2)Tγ′sNγ′s=0κγs−κγs0TγsNγs,where Tγs=γ′s is the unit tangent vector, Nγs=JTγs is the unit normal vector, and κγs=±Tγ′s. Also, the curvature is defined by κγs=dχs/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 1.
Letγ:I⟶ℝ2 be a regular parametrized curve and γ~t=γt+λNγt, where λ is a constant. Then, γ~ and γ are parallel curves.Definition 2.
Letγ:I⟶ℝ2 be a regular parametrized curve with nonvanishing curvature. Then, the evolute of γ is given by
(3)Eγt=γt+1κγtNγt.
It is a well-known result thatEγ is regular at t0∈I if and only if κγ′t0≠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 [2–5, 8].Definition 3.
The mapγ,ω:I⟶ℝ2×S1 is called a Legendre curve if γ′t·ωt=0 for all t∈I, where S1 is the unit circle and ω:I⟶S1 is a smooth unit vector field. The map γ,ω is a Legendre immersion if it is an immersion.
We callγ:I⟶ℝ2 a frontal if there exists a smooth function ω:I⟶S1 such that γ,ω is a Legendre curve, and we call γ:I⟶ℝ2 a wavefront if there exists a smooth function ω:I⟶S1 such that γ,ω is a Legendre immersion.
For a Frenet curveγ, if γ has a singular point, then the moving frame is not well-defined. For a Legendre curve, γ,ω:I⟶ℝ2×S1, an alternative frame is well-defined at any point. This frame is given by ω,μ, where μt=Jωt. Also, we have the following formula:
(4)ω′tμ′t=0ℓt−ℓt0ωtμt,where ℓt=ω′t·μt. We call the pair ωt,μt a moving frame of a Legendre curve γ. In addition, there exists a smooth function βt such that βt=γ′t·μt. The curvature of the Legendre curve is ℓt,βt.Definition 4 (see [4]).
Suppose thatγ is a frontal with the curvature ℓ,β. If there exists a unique smooth function ρ:I⟶ℝ such that βt=ρtℓt for all t∈I, then the evolute Eγ:I⟶ℝ2 of γ is given by Eγt=γt−ρtωt. Moreover, Eγ is a frontal with the curvature ℓt,ρ′t. This means that Eγ,Jω:I⟶ℝ2×S1 is a Legendre curve, where J is the counterclockwise rotation by π/2 on ℝ2.Definition 5 (see [4]).
Suppose thatγ is a frontal with the curvature ℓ,β. The involute of frontal γ at t0∈I is given by Iγt0=γt−∫t0tβudu, and Iγ,J−1ω:I⟶ℝ2×S1 is a Legendre curve with curvature ℓt,∫t0tβuduℓ, where J−1 is the clockwise rotation by π/2 on ℝ2.


## 3.f-Curve via the Normal Vector Associated with a Plane Curve

### 3.1. Regular Curve

In this section, we study thef-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.
Letγ:I⟶ℝ2 be a regular parametrized curve. Then, the f-curve via the normal vector associated with γ is defined by αt=γt+ftNγt, where f:I⟶ℝ is a smooth function.In the following lemma, we give the necessary and sufficient condition for the curveα to be a regular curve.Lemma 7.
Letγ:I⟶ℝ2 be a unit speed curve. Then, the curve α is regular if and only if fκγ≠1 or f′≠0.Proof.
The proof of this lemma is obvious.Remark 8.
From Lemma7, it can be easily obtained that all singular points of the f-curve via the normal vector associated with γ lie on the evolute of γ.The following theorem provides a useful formula for the curvature of thef-curve via the normal vector associated with γ in the case of its regularity.Theorem 9.
Letγ:I⟶ℝ2 be a unit speed curve and α be its associated f-curve via the normal vector.
(1)
Iffκγ≠1, then κα=κγ±θ′cosθ/1−fκγ, where θ is the angle between the tangent vector of α and the unit tangent vector of γ(2)
Iff′≠0, then κα=κγ∓ϕ′cosϕ/f′, where ϕ is the angle between the tangent vector of α and the unit normal vector of γProof.
Letγ:I⟶ℝ2 be a unit speed curve and α be its associated f-curve via the normal vector. Then, we have
(5)α′=1−fκγTγ+f′Nγ.
Case 1. Iffκγ=1, then from Lemma 7, α is a regular curve, and we have
(6)α′·Tγ=1−fκγ,which gives
(7)α′Tγcosθ=1−fκγ.
So, we have(8)α′cosθ=1−fκγ.
Equation (8) can be rewritten as
(9)α′=1−fκγsecθ.
From equation (5), we have
(10)α′2=1−fκγ2+f′2.
Now, using equation (8) in equation (10), we get
(11)α′2=α′2cos2θ+f′2.
So,(12)f′=±α′sinθ.
Substituting (8) and (12) in (5), we have
(13)α′=α′cosθTγ±α′sinθNγ.
The unit tangent vector ofα, Tα, can be written as
(14)Tα=cosθTγ±sinθNγ,and the unit normal vector of α, Nα, can be written as
(15)Nα=∓sinθTγ+cosθNγ.
Now,(16)Tα′·Nα=κγ±θ′.
Hence,(17)κα=Tα′·Nαα′=κγ±θ′cosθ1−fκγ.
Case 2. Iff′=0, then from Lemma 7, α is a regular curve, and we have
(18)α′·Nγ=f′,which gives
(19)αNγcosϕ=f′.
So, we have(20)α′cosϕ=f′.
Equation (20) can be rewritten as
(21)α′=f′secϕ.
Now, using equation (20) in equation (10), we get
(22)α′2=1−fκγ2+α′2cos2ϕ.
Now,(23)1−fκγ=±α′sinϕ.
Substituting (20) and (23) in (5), we get
(24)α′=±α′sinϕTγ+α′cosϕNγ.
So,(25)Tα=±sinϕTγ+cosϕNγ,(26)Nα=−cosϕTγ±sinϕNγ.
Now,(27)Tα′·Nα=κγ∓ϕ′.
Hence,(28)κα=κγ∓ϕ′cosϕf′.As an application of Theorem9, the curvatures of parallel curves and evolute become special cases of this theorem. Precisely, we have the following corollary.Corollary 10.
Letγ:I⟶ℝ2 be a unit speed curve with nonvanishing curvature and α be its associated f-curve via the normal vector.
(1)
Iff is a constant and f≠1/κγ, then γ,α are parallel curves and κα=κγ/1−fκγ(2)
Iff′≠0 and f=1/κγ, then α is the evolute of γ and κα=∓κγ3/κγ′


### 3.2. Legendre Curve

In this section, we consider the case whenγ is a Legendre curve.Definition 11.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve. Then, the f-curve via the vector field ω associated with γ is given by α~t=γt+ftωt, where f:I⟶ℝ is a smooth function.Lemma 12.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve. Then, the f-curve via the vector field ω associated with γ is regular if and only if f≠−β/ℓ or f′≠0.Proof.
Letα~=γ+fω be associated with f-curve via the vector field ω, then we have
(29)α~′=γ′+f′ω+fω′.
Sinceγ′=βμ and ω′=ℓμ, we have
(30)α~′=f′ω+β+fℓμ.
Now,α~′≠0⟶ if and only if β+fℓ≠0 or f′≠0.Theorem 13.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve and α~ be its associated f-curve via the vector field ω.
(1)
Iff≠−β/ℓ, then κα~=ℓ∓ψ′cosψ/β+ℓf, where ψ is the angle between the tangent vector of α~ and the unit vector field μ(2)
Iff′≠0, then κα~=ℓ±ξ′cosξ/f′, where ξ is the angle between the tangent vector of α~ and the unit vector field ωProof.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve and α~ be its associated f-curve via the vector field ω. Then, we have
(31)α~′=f′ω+β+fℓμ.
Case 1. Iff=−β/ℓ, then from Lemma 12, α~ is a regular curve, and we have
(32)α~′·μ=β+fℓ,which gives
(33)α~′μcosψ=β+fℓ.
So, we have(34)α~′cosψ=β+fℓ.
Equation (34) can be rewritten as
(35)α~′=β+fℓsecψ.
From equation (31), we have
(36)α~′2=β+fℓ2+f′2.
Now, using equation (34) in (36), we get
(37)α~′2=α~′2cos2ψ+f′2.
So,(38)f′=±α~′sinψ.
From (34) and (38), equation (31) becomes
(39)α~′=±α~′sinψω+α~′cosψμ.
Now,(40)Tα~=±sinψω+cosψμ,Nα~=−cosψω±sinψμ.
Thus,(41)Tα~′·Nα~=ℓ∓ψ′.
Hence,(42)κα~=ℓ∓ψ′cosψβ+ℓf.
Case 2. Iff′=0, then from Lemma 12, α~ is a regular curve, and we have
(43)α~′·ω=f′,which gives
(44)α~′ωcosξ=f′.
So, we have(45)α~′cosξ=f′.
Equation (45) can be rewritten as
(46)α~′=f′secξ.
Now, using equation (45) in (36), we have
(47)α~′2=β+ℓf2+α~′2cos2ξ.
So,(48)β+ℓf=±α~′sinξ.
Substituting (45) and (48) in (31), we get
(49)α~′=α~′cosξω±α~′sinξμ.
Now,(50)Tα~=cosξω±sinξμ,Nα~=∓sinξω+cosξμ.
Thus,(51)Tα~′·Nα~=ℓ±ξ′.
Hence,(52)κα~=ℓ±ξ′cosξf′.Now, we have the following corollary of Theorem13.Corollary 14.
Letγ be a frontal and α~ be its associated f-curve via the vector field ω.
(1)
Iff is a unique smooth function such that f=−β/ℓ and f′=0, then α~ is the evolute of γ and it is a regular curve with κα~=±ℓ3/ℓ′β−ℓβ′(2)
Iff is a constant and f≠−β/ℓ, then κα~=±ℓ/β+fℓ


## 3.1. Regular Curve

In this section, we study thef-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.
Letγ:I⟶ℝ2 be a regular parametrized curve. Then, the f-curve via the normal vector associated with γ is defined by αt=γt+ftNγt, where f:I⟶ℝ is a smooth function.In the following lemma, we give the necessary and sufficient condition for the curveα to be a regular curve.Lemma 7.
Letγ:I⟶ℝ2 be a unit speed curve. Then, the curve α is regular if and only if fκγ≠1 or f′≠0.Proof.
The proof of this lemma is obvious.Remark 8.
From Lemma7, it can be easily obtained that all singular points of the f-curve via the normal vector associated with γ lie on the evolute of γ.The following theorem provides a useful formula for the curvature of thef-curve via the normal vector associated with γ in the case of its regularity.Theorem 9.
Letγ:I⟶ℝ2 be a unit speed curve and α be its associated f-curve via the normal vector.
(1)
Iffκγ≠1, then κα=κγ±θ′cosθ/1−fκγ, where θ is the angle between the tangent vector of α and the unit tangent vector of γ(2)
Iff′≠0, then κα=κγ∓ϕ′cosϕ/f′, where ϕ is the angle between the tangent vector of α and the unit normal vector of γProof.
Letγ:I⟶ℝ2 be a unit speed curve and α be its associated f-curve via the normal vector. Then, we have
(5)α′=1−fκγTγ+f′Nγ.
Case 1. Iffκγ=1, then from Lemma 7, α is a regular curve, and we have
(6)α′·Tγ=1−fκγ,which gives
(7)α′Tγcosθ=1−fκγ.
So, we have(8)α′cosθ=1−fκγ.
Equation (8) can be rewritten as
(9)α′=1−fκγsecθ.
From equation (5), we have
(10)α′2=1−fκγ2+f′2.
Now, using equation (8) in equation (10), we get
(11)α′2=α′2cos2θ+f′2.
So,(12)f′=±α′sinθ.
Substituting (8) and (12) in (5), we have
(13)α′=α′cosθTγ±α′sinθNγ.
The unit tangent vector ofα, Tα, can be written as
(14)Tα=cosθTγ±sinθNγ,and the unit normal vector of α, Nα, can be written as
(15)Nα=∓sinθTγ+cosθNγ.
Now,(16)Tα′·Nα=κγ±θ′.
Hence,(17)κα=Tα′·Nαα′=κγ±θ′cosθ1−fκγ.
Case 2. Iff′=0, then from Lemma 7, α is a regular curve, and we have
(18)α′·Nγ=f′,which gives
(19)αNγcosϕ=f′.
So, we have(20)α′cosϕ=f′.
Equation (20) can be rewritten as
(21)α′=f′secϕ.
Now, using equation (20) in equation (10), we get
(22)α′2=1−fκγ2+α′2cos2ϕ.
Now,(23)1−fκγ=±α′sinϕ.
Substituting (20) and (23) in (5), we get
(24)α′=±α′sinϕTγ+α′cosϕNγ.
So,(25)Tα=±sinϕTγ+cosϕNγ,(26)Nα=−cosϕTγ±sinϕNγ.
Now,(27)Tα′·Nα=κγ∓ϕ′.
Hence,(28)κα=κγ∓ϕ′cosϕf′.As an application of Theorem9, the curvatures of parallel curves and evolute become special cases of this theorem. Precisely, we have the following corollary.Corollary 10.
Letγ:I⟶ℝ2 be a unit speed curve with nonvanishing curvature and α be its associated f-curve via the normal vector.
(1)
Iff is a constant and f≠1/κγ, then γ,α are parallel curves and κα=κγ/1−fκγ(2)
Iff′≠0 and f=1/κγ, then α is the evolute of γ and κα=∓κγ3/κγ′


## 3.2. Legendre Curve

In this section, we consider the case whenγ is a Legendre curve.Definition 11.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve. Then, the f-curve via the vector field ω associated with γ is given by α~t=γt+ftωt, where f:I⟶ℝ is a smooth function.Lemma 12.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve. Then, the f-curve via the vector field ω associated with γ is regular if and only if f≠−β/ℓ or f′≠0.Proof.
Letα~=γ+fω be associated with f-curve via the vector field ω, then we have
(29)α~′=γ′+f′ω+fω′.
Sinceγ′=βμ and ω′=ℓμ, we have
(30)α~′=f′ω+β+fℓμ.
Now,α~′≠0⟶ if and only if β+fℓ≠0 or f′≠0.Theorem 13.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve and α~ be its associated f-curve via the vector field ω.
(1)
Iff≠−β/ℓ, then κα~=ℓ∓ψ′cosψ/β+ℓf, where ψ is the angle between the tangent vector of α~ and the unit vector field μ(2)
Iff′≠0, then κα~=ℓ±ξ′cosξ/f′, where ξ is the angle between the tangent vector of α~ and the unit vector field ωProof.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve and α~ be its associated f-curve via the vector field ω. Then, we have
(31)α~′=f′ω+β+fℓμ.
Case 1. Iff=−β/ℓ, then from Lemma 12, α~ is a regular curve, and we have
(32)α~′·μ=β+fℓ,which gives
(33)α~′μcosψ=β+fℓ.
So, we have(34)α~′cosψ=β+fℓ.
Equation (34) can be rewritten as
(35)α~′=β+fℓsecψ.
From equation (31), we have
(36)α~′2=β+fℓ2+f′2.
Now, using equation (34) in (36), we get
(37)α~′2=α~′2cos2ψ+f′2.
So,(38)f′=±α~′sinψ.
From (34) and (38), equation (31) becomes
(39)α~′=±α~′sinψω+α~′cosψμ.
Now,(40)Tα~=±sinψω+cosψμ,Nα~=−cosψω±sinψμ.
Thus,(41)Tα~′·Nα~=ℓ∓ψ′.
Hence,(42)κα~=ℓ∓ψ′cosψβ+ℓf.
Case 2. Iff′=0, then from Lemma 12, α~ is a regular curve, and we have
(43)α~′·ω=f′,which gives
(44)α~′ωcosξ=f′.
So, we have(45)α~′cosξ=f′.
Equation (45) can be rewritten as
(46)α~′=f′secξ.
Now, using equation (45) in (36), we have
(47)α~′2=β+ℓf2+α~′2cos2ξ.
So,(48)β+ℓf=±α~′sinξ.
Substituting (45) and (48) in (31), we get
(49)α~′=α~′cosξω±α~′sinξμ.
Now,(50)Tα~=cosξω±sinξμ,Nα~=∓sinξω+cosξμ.
Thus,(51)Tα~′·Nα~=ℓ±ξ′.
Hence,(52)κα~=ℓ±ξ′cosξf′.Now, we have the following corollary of Theorem13.Corollary 14.
Letγ be a frontal and α~ be its associated f-curve via the vector field ω.
(1)
Iff is a unique smooth function such that f=−β/ℓ and f′=0, then α~ is the evolute of γ and it is a regular curve with κα~=±ℓ3/ℓ′β−ℓβ′(2)
Iff is a constant and f≠−β/ℓ, then κα~=±ℓ/β+fℓ


## 4.f-Curve via the Tangent Vector Associated with a Plane Curve

### 4.1. Regular Curve

Definition 15.
Letγ:I⟶ℝ2 be a regular parametrized curve. Then, the f-curve via the tangent vector associated with γ is given by Ωt=γt+ftTγt, where f:I⟶ℝ is a smooth function.Now, we state the following lemma.Lemma 16.
Letγ:I⟶ℝ2 be a unit speed curve. Then, the curve Ω associated with γ is regular if and only if f′≠−1 or fκγ≠0.Remark 17.
From Lemma16, if Ω has a singularity at t0, then γ has an inflexion point at t0 or Ωt0=γt0.Theorem 18.
Letγ:I⟶ℝ2 be a unit speed curve and Ω be its associated f-curve via the tangent vector.
(1)
Iff′≠−1, then κΩ=κγ±ζ′cosζ/1+f′, where ζ is the angle between the tangent vector of Ω and the unit tangent vector of γ(2)
Iffκγ≠0, then κΩ=κγ∓Ψ′cosΨ/fκγ, where Ψ is the angle between the tangent vector of Ω and the unit normal vector of γProof.
Letγ:I⟶ℝ2 be a unit speed curve and Ω be its associated f-curve via the tangent vector. Then, we have
(53)Ω′=1+f′Tγ+fκγNγ.
Case 1. Iff′≠−1, then from Lemma 16, Ω is a regular curve, and we have
(54)Ω′·Tγ=1+f′,which gives
(55)Ω′cosζ=1+f′.
From equation (53), we have
(56)Ω′2=1+f′2+fκγ2.
Substituting equation (55) in equation (56), we obtain that
(57)fκγ=±Ω′sinζ.
From equation (55) and equation (57), equation (53) becomes
(58)Ω′=Ω′cosζTγ±Ω′sinζNγ.
Therefore,(59)TΩ=cosζTγ±sinζNγ,NΩ=∓sinζTγ+cosζNγ.
Hence,(60)κΩ=TΩ′·NΩΩ′=κγ±ζ′cosζ1+f′.
Case 2. Iffκγ=0, then from Lemma 16, Ω is a regular curve, and we have
(61)Ω′·Nγ=fκγ,which gives
(62)Ω′cosΨ=fκγ.
Substituting (62) in (56), we obtain that
(63)1+f′=±ΩsinΨ.
From equation (62) and equation (63), equation (53) becomes
(64)Ω′=±Ω′sinΨTγ+Ω′cosΨNγ.
Therefore,(65)TΩ=±sinΨTγ+cosΨNγ,NΩ=−cosΨTγ±sinΨNγ.
Hence,(66)κΩ=TΩ′·NΩΩ′=κγ∓Ψ′cosΨfκγ.The following corollary can be easily obtained from Theorem18.Corollary 19.
Letγ:I⟶ℝ2 be a unit speed curve and Ω be its associated f-curve via the tangent vector. If f′=−1 (that is, fs=C−s for some constant C) and f≠0, then Ω is the involute of γ and κΩ=singκγ/f.From Theorem18, 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:
(67)κΩ=cosΘκγ+Θ′sinφsinφ+rφ′,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.
Letγ:I⟶ℝ2 be a unit speed curve and Ω be its associated f-curve via the tangent vector such that f=−rr′.
(1)
Iff′≠−1, then κΩ=κγ±ζ′cosζ/1−r′2−rr′′(2)
Iffκγ≠0, then κΩ=−κγ∓Ψ′cosΨ/rr′κγ


### 4.2. Legendre Curve

Definition 21.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve. The f-curve via the vector filed μ associated with γ is given by Ω~t=γt+ftμt, where f:I⟶ℝ is a smooth function.Lemma 22.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve. The f-curve via the vector field μ associated with γ is regular if and only if β+f′≠0 or ℓf=0.Theorem 23.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve and Ω~ be its associated f-curve via the vector field μ.
(1)
Ifβ+f′≠0, then κΩ~=ℓ∓η′cosη/β+f′, where η is the angle between the tangent vector of Ω~ and the unit vector field μ(2)
Ifℓf≠0, then κΩ~=ℓ±ε′cosε/ℓf, where ε is the angle between the tangent vector of Ω~ and the unit vector field ωProof.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve and Ω~ be its associated f-curve via the vector field μ. Then, we have
(68)Ω~′=β+f′μ−ℓfω.
Case 1. Ifβ+f′=0, then from Lemma 22, Ω~ is a regular curve, and we have
(69)Ω~′·μ=β+f′,which gives
(70)Ω~′cosη=β+f′.
From equation (68), we have
(71)Ω~′2=β+f2+ℓf2.
Substituting (70) in (71), it can be shown that
(72)ℓf=±Ω~′sinη.
From (70) and (72), equation (68) becomes
(73)Ω~′=Ω~′cosημ∓Ω~′sinηω.
Now,(74)TΩ~=∓sinηω+cosημ,NΩ~=−cosηω∓sinημ.
Hence,(75)κΩ~=ℓ±η′cosηβ+ℓf′.
Case 2. Ifℓf≠0, then from Lemma 22, Ω~ is a regular curve, and we have
(76)Ω~′·ω=−ℓf,which gives
(77)Ω~′cosε=−ℓf.
Using equation (77) in (71), it can be shown that
(78)β+f′=±Ω~′sinε.
From (77) and (78), equation (68) becomes
(79)Ω~′=Ω~′cosεω±Ω~′sinεμ.
Now,(80)TΩ~=cosεω±sinεμ,NΩ~=∓sinεω+cosεμ.
Hence,(81)κΩ~=ℓ±ε′cosεℓf.Corollary 24.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve with the curvature ℓ,β and Ω~ be its associated f-curve via the vector field μ. If f=−∫t0tβudu and ℓf≠0, then Ω~ is the involute of the frontal and it is a regular curve with κΩ~=±1/∫t0tβudu.


## 4.1. Regular Curve

Definition 15.
Letγ:I⟶ℝ2 be a regular parametrized curve. Then, the f-curve via the tangent vector associated with γ is given by Ωt=γt+ftTγt, where f:I⟶ℝ is a smooth function.Now, we state the following lemma.Lemma 16.
Letγ:I⟶ℝ2 be a unit speed curve. Then, the curve Ω associated with γ is regular if and only if f′≠−1 or fκγ≠0.Remark 17.
From Lemma16, if Ω has a singularity at t0, then γ has an inflexion point at t0 or Ωt0=γt0.Theorem 18.
Letγ:I⟶ℝ2 be a unit speed curve and Ω be its associated f-curve via the tangent vector.
(1)
Iff′≠−1, then κΩ=κγ±ζ′cosζ/1+f′, where ζ is the angle between the tangent vector of Ω and the unit tangent vector of γ(2)
Iffκγ≠0, then κΩ=κγ∓Ψ′cosΨ/fκγ, where Ψ is the angle between the tangent vector of Ω and the unit normal vector of γProof.
Letγ:I⟶ℝ2 be a unit speed curve and Ω be its associated f-curve via the tangent vector. Then, we have
(53)Ω′=1+f′Tγ+fκγNγ.
Case 1. Iff′≠−1, then from Lemma 16, Ω is a regular curve, and we have
(54)Ω′·Tγ=1+f′,which gives
(55)Ω′cosζ=1+f′.
From equation (53), we have
(56)Ω′2=1+f′2+fκγ2.
Substituting equation (55) in equation (56), we obtain that
(57)fκγ=±Ω′sinζ.
From equation (55) and equation (57), equation (53) becomes
(58)Ω′=Ω′cosζTγ±Ω′sinζNγ.
Therefore,(59)TΩ=cosζTγ±sinζNγ,NΩ=∓sinζTγ+cosζNγ.
Hence,(60)κΩ=TΩ′·NΩΩ′=κγ±ζ′cosζ1+f′.
Case 2. Iffκγ=0, then from Lemma 16, Ω is a regular curve, and we have
(61)Ω′·Nγ=fκγ,which gives
(62)Ω′cosΨ=fκγ.
Substituting (62) in (56), we obtain that
(63)1+f′=±ΩsinΨ.
From equation (62) and equation (63), equation (53) becomes
(64)Ω′=±Ω′sinΨTγ+Ω′cosΨNγ.
Therefore,(65)TΩ=±sinΨTγ+cosΨNγ,NΩ=−cosΨTγ±sinΨNγ.
Hence,(66)κΩ=TΩ′·NΩΩ′=κγ∓Ψ′cosΨfκγ.The following corollary can be easily obtained from Theorem18.Corollary 19.
Letγ:I⟶ℝ2 be a unit speed curve and Ω be its associated f-curve via the tangent vector. If f′=−1 (that is, fs=C−s for some constant C) and f≠0, then Ω is the involute of γ and κΩ=singκγ/f.From Theorem18, 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:
(67)κΩ=cosΘκγ+Θ′sinφsinφ+rφ′,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.
Letγ:I⟶ℝ2 be a unit speed curve and Ω be its associated f-curve via the tangent vector such that f=−rr′.
(1)
Iff′≠−1, then κΩ=κγ±ζ′cosζ/1−r′2−rr′′(2)
Iffκγ≠0, then κΩ=−κγ∓Ψ′cosΨ/rr′κγ


## 4.2. Legendre Curve

Definition 21.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve. The f-curve via the vector filed μ associated with γ is given by Ω~t=γt+ftμt, where f:I⟶ℝ is a smooth function.Lemma 22.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve. The f-curve via the vector field μ associated with γ is regular if and only if β+f′≠0 or ℓf=0.Theorem 23.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve and Ω~ be its associated f-curve via the vector field μ.
(1)
Ifβ+f′≠0, then κΩ~=ℓ∓η′cosη/β+f′, where η is the angle between the tangent vector of Ω~ and the unit vector field μ(2)
Ifℓf≠0, then κΩ~=ℓ±ε′cosε/ℓf, where ε is the angle between the tangent vector of Ω~ and the unit vector field ωProof.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve and Ω~ be its associated f-curve via the vector field μ. Then, we have
(68)Ω~′=β+f′μ−ℓfω.
Case 1. Ifβ+f′=0, then from Lemma 22, Ω~ is a regular curve, and we have
(69)Ω~′·μ=β+f′,which gives
(70)Ω~′cosη=β+f′.
From equation (68), we have
(71)Ω~′2=β+f2+ℓf2.
Substituting (70) in (71), it can be shown that
(72)ℓf=±Ω~′sinη.
From (70) and (72), equation (68) becomes
(73)Ω~′=Ω~′cosημ∓Ω~′sinηω.
Now,(74)TΩ~=∓sinηω+cosημ,NΩ~=−cosηω∓sinημ.
Hence,(75)κΩ~=ℓ±η′cosηβ+ℓf′.
Case 2. Ifℓf≠0, then from Lemma 22, Ω~ is a regular curve, and we have
(76)Ω~′·ω=−ℓf,which gives
(77)Ω~′cosε=−ℓf.
Using equation (77) in (71), it can be shown that
(78)β+f′=±Ω~′sinε.
From (77) and (78), equation (68) becomes
(79)Ω~′=Ω~′cosεω±Ω~′sinεμ.
Now,(80)TΩ~=cosεω±sinεμ,NΩ~=∓sinεω+cosεμ.
Hence,(81)κΩ~=ℓ±ε′cosεℓf.Corollary 24.
Letγ,ω:I⟶ℝ2×S1 be a Legendre curve with the curvature ℓ,β and Ω~ be its associated f-curve via the vector field μ. If f=−∫t0tβudu and ℓf≠0, then Ω~ is the involute of the frontal and it is a regular curve with κΩ~=±1/∫t0tβudu.


## 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 functionf. We use the Maple for calculation and drawing pictures. First, let us consider the curve γt=cost,sin3t, t∈0,2π. It is clear that γ is a closed curve (see Figure 1(a)), and we choose f1t=3sint−5esint. The f1-curve via the normal vector associated with γ is given by γ~t=γt+f1tNγt=γ~1t,γ~2t, where
(82)γ~1t=cost+15cos3tesint−9cos3tsint144cos6t−216cos4t+80cos2t+1,γ~2t=sint+5sintesint+3cos2t−3144cos6t−216cos4t+80cos2t+1.Figure 1
The curveγ and its resulting f-curves.
(a)
The curveγ(b)
The curveγ~(c)
The curveγ~~The graph ofγ~ is shown in Figure 1(b), and we may call this graph Clown’s face.In what follows, we consider thef-curve via the normal vector associated with γ~ with ft=f2t=−5−2esin9t; this curve is given by γ~~t=γ~t+f2tNγ~. The calculation becomes very tedious, and the forms of γ~~1 and γ~~2 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 γ~ and we may call this graph Tiger’s face.Remark 25.
From this section, one can observe the importance off-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 thef-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 f-curve in the plane.



---

*Source: 9881237-2022-04-27.xml*