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Research article

Circular evolutes and involutes of spacelike framed curves and their duality relations in Minkowski 3-space

  • Received: 30 December 2023 Revised: 16 January 2024 Accepted: 19 January 2024 Published: 30 January 2024
  • MSC : 53A04, 53A05, 57R45

  • In the present paper, we defined the circular evolutes and involutes for a given spacelike framed curve with respect to Bishop directions in Minkowski 3-space. Then, we studied the essential duality relations among parallel curves, normal surfaces, and circular evolutes and involutes. Furthermore, we also studied the duality relations of their singularities. Based on these studies, we found that it is crucially important to consider the duality relations among different geometric objects for the research of submanifolds with singularities.

    Citation: Wei Zhang, Pengcheng Li, Donghe Pei. Circular evolutes and involutes of spacelike framed curves and their duality relations in Minkowski 3-space[J]. AIMS Mathematics, 2024, 9(3): 5688-5707. doi: 10.3934/math.2024276

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  • In the present paper, we defined the circular evolutes and involutes for a given spacelike framed curve with respect to Bishop directions in Minkowski 3-space. Then, we studied the essential duality relations among parallel curves, normal surfaces, and circular evolutes and involutes. Furthermore, we also studied the duality relations of their singularities. Based on these studies, we found that it is crucially important to consider the duality relations among different geometric objects for the research of submanifolds with singularities.



    Differential geometry of curves and surfaces is considered to be the beginning of modern geometry. Starting with the beautiful and essential work of Gauss [10], a perfect theory of regular curves and surfaces with moving frame method was established. Now with the development of singularity theory [1,2,3,31,34], non-regular curves and surfaces have been studied in Euclidean 3-space [22,26]. On the other hand, due to the necessity of physics, the above methods are used in study of the semi-Riemannnian manifolds [17,21,27,29].

    It is well known that evolutes and involutes are important research objects in differential geometry and mathematical physics going back to in 1673. In Huygens' book Horologium oscillatorium, the elementary properties of the evolutes and involutes of regular plane curves was studied [11]. From the viewpoint of the differential geometry of curves and surfaces with singularities, it has gradually been established for the evolutes and involutes of singular curves [1,3,6,14,16], such as fronts and frontals. In recent decades, Fukunaga, Honda and Takahashi introduced the concept of framed curves and framed surfaces in Euclidean 3-space [9,12]. Thereafter, Fukunaga and Takahashi defined the evolutes and involutes of fronts which are plane curves and allowed to contain singular points in Euclidean 2-space [6,7,8]. Tunçer, Ünal, and Karacan studied the properties the spherical indicatrices of evolutes and involutes of a space curve [33]. Further, Şekerci and Izumiya considered the evolutoids of frontals in the Minkowski plane [4,5]. Through the work of [18], we can give the definitions of evolutes and focal surfaces for (1,k)-type curves with respect to Bishop frames in Euclidean 3-space and discuss their singularities and the classification theorem.

    Recently, through the work of Honda and Takahashi [13], we have known the definitions of circular evolutes and involutes of framed curves with respect to Bishop frames in Euclidean 3-space and the duality relations among parallel curves, normal surfaces, and circular evolutes and involutes. We can also learn the local singularity behavior of the circular evolutes and involutes of framed curves through the work of Honda and Takahashi [13]. Furthermore, we studied the nullcone fronts of spacelike framed curves in Minkowski 3-space. We defined the moving frame of a spacelike framed curve, and gave the appropriate Frenet type formula [19,20]. So, there is natural question of what the phenomena of a circular evolute will be in Minkowski 3-space. In the present paper, we will give a complete answer to this question for spacelike framed curves in Minkowski 3-space.

    In the present paper, we research circular evolutes and involutes with a viewpoint of singularity theory in Minkowski 3-space. First, we define the Bishop frame in Minkowski 3-space, and give the relation of the curvature functions between spacelike frames and spacelike Bishop frames. In Section 3, we give the necessary and sufficient condition that a parallel curve is a spacelike or timelike framed curve under causal character in Minkowski 3-space and give the corresponding curvature functions. In Section 4, we give the necessary and sufficient condition that the normal surface of a given spacelike framed curve is a spacelike or timelike framed surface under causal character in Minkowski 3-space and the corresponding invariant functions, as well as give a necessary and sufficient condition that a singularity of a normal surface is a cross cap. Our main contributions are focused on two aspects in Section 5. On the one hand, we study the duality relations among parallel curves, normal surfaces, and circular evolutes and involutes for a given spacelike framed curve in Minkowski 3-space. On the other hand, we study the relations among the sigularities of spacelike framed curves, normal surfaces, and circular evolutes and involutes. Finally, we give an example to illustrate the duality behavior which has been studied in this paper.

    All the maps and manifolds considered here are C.

    We briefly review some essential concepts of Minkowski 3-space, which are discussed in detail in [27,29]. Let R31 be the Minkowski 3-space equipped with the canonical pseudo-scalar product x,y=x1y1+x2y2+x3y3, where x=(x1,x2,x3) and y=(y1,y2,y3). We define the norm of x by x=|x,x| and the pseudo-vector product of x and y by

    xy=det(e1e2e3x1x2x3y1y2y3),

    where {e1,e2,e3} is the canonical basis of R31.

    Definition 2.1. A vector xR31 is said to be

    1) spacelike, if x,x>0 or x=0,

    2) timelike, if x,x<0,

    3) lightlike, if x,x=0, x0.

    It is similar to the concept of the unit sphere in Euclidean 3-space. We can also discuss the pseudo-sphere in Minkowski 3-space. The de Sitter 2-space is defined by

    S21={xR31 |x,x=1},

    the hyperbolic 2-space is defined by

    H20={xR31 |x,x=1}.

    We briefly review the theory of framed curves and framed surfaces in Minkowski 3-space. A (smooth) curve is a differentiable map γ:IRR31 where I is an open interval. We say that a curve γ(t) is a spacelike, timelike, or lightlike curve if ˙γ(t) is spacelike, timelike, or lightlike, where ˙γ(t)=ddtγ(t).

    Definition 2.2. ([19]) Let γ:IR31 be a spacelike curve. Then, the C map (γ,ν1,ν2):IR31×Δ is called a spacelike framed curve if

    ˙γ(t),ν1(t)=0,˙γ(t),ν2(t)=0,tI,

    where

    Δ={(ν1,ν2)S21×H20ν1(t),ν2(t)=0},

    or

    Δ={(ν1,ν2)H20×S21ν1(t),ν2(t)=0}.

    Moreover, γ:IR31 is said to be a spacelike framed base curve if there is a C map (γ,ν1,ν2):IR31×Δ such that (γ,ν1,ν2) is a spacelike framed curve.

    Let (γ(t),ν1(t),ν2(t)) be a spacelike framed curve. We denote δ(t)=sign(ν1(t))=ν1(t),ν1(t). We define μ(t)=ν1(t)ν2(t), which means μ(t) is a unit spacelike vector field along γ(t). Then, we can have a smooth function α(t) satisfying ˙γ(t)=α(t)μ(t). Furthermore, we have the following Frenet type formulae.

    (˙ν1(t)˙ν2(t)˙μ(t))=(0(t)m(t)(t)0n(t)δ(t)m(t)δ(t)n(t)0)(ν1(t)ν2(t)μ(t)), (2.1)

    where (t)=˙ν1(t),ν2(t),m(t)=˙ν1(t),μ(t), and n(t)=˙ν2(t),μ(t). We call the functions (α(t),(t),m(t),n(t)) the curvature of a spacelike framed curve in Minkowski 3-sapce. Then, we consider the frame {ν,ω,μ} which is obtained by

    (ν(t)ω(t))=(coshθ(t)sinhθ(t)sinhθ(t)coshθ(t))(ν1(t)ν2(t)),μ(t)=ν(t)ω(t).

    Then, we have the Frenet type formulae of the frame {ν,ω,μ} as follows:

    (˙ν(t)˙ω(t)˙μ(t))=(0¯(t)¯m(t)¯(t)0¯n(t)δ(t)¯m(t)δ(t)¯n(t)0)(ν(t)ω(t)μ(t)), (2.2)

    where ¯(t)=(t)+˙θ(t), ¯m(t)=m(t)coshθ(t)+n(t)sinhθ(t), and ¯n(t)=m(t)sinhθ(t)+n(t)coshθ(t).

    Remark 2.3. Let γ:IR31 be a timelike curve. Then, the C map (γ,ν1,ν2):IR31×Δ5 is called a timelike framed curve if

    ˙γ(t),ν1(t)=0,˙γ(t),ν2(t)=0,tI,

    where

    Δ5={(ν1,ν2)S21×S21ν1(t),ν2(t)=0}.

    Similar to the discussion of a spacelike framed curve above, we have the Frenet type formulae of a timelike framed curve in Minkowski 3-space as follows:

    (˙ν1(t)˙ν2(t)˙μ(t))=(0(t)m(t)(t)0n(t)m(t)n(t)0)(ν1(t)ν2(t)μ(t)),

    where (t)=˙ν1(t),ν2(t),m(t)=˙ν1(t),μ(t), n(t)=˙ν2(t),μ(t), and ˙γ(t)=α(t)μ(t). We call the functions (α(t),(t),m(t),n(t)) the curvature of a timelike framed curve in Minkowski 3-space.

    Definition 2.4. For a spacelike framed curve (γ,ν,ω):IR31×Δ, if there exists a smooth function β:IR31 such that ˙ν(t)=β(t)μ(t), where μ(t)=ν(t)ω(t), then we call ν(t) a Bishop direction. If ν(t) and ω(t) are Bishop directions, then we call the moving frame {ν,ω,μ} a Bishop frame.

    By the Frenet type formulae of a spacelike framed curve, we can have that there is a function θ(t):IR such that ¯(t)=0, that is, θ(t)=(t), which means that we can always take a frame {ν,ω,μ} to be a Bishop frame by a suitable θ(t) for a given moving frame {ν1,ν2,μ} of a spacelike framed curve in Minkowski 3-space.

    In the following, we discuss the appropriate moving frame of a surface in Minkowski 3-space. For more detailed discussion, please refer to [9,32].

    Definition 2.5. Let x:UR31 be a spacelike surface. Then, the C map (x,n,s):UR31×Δ1 is said to be spacelike framed surface if xu(u,v),n(u,v)=0, xv(u,v),n(u,v)=0 for all (u,v)U, where xu(u,v)=(x/u)(u,v), xv(u,v)=(x/v)(u,v) and

    Δ1={(n,s)H20×S21n(u,v),s(u,v)=0}.

    Moreover, x:UR31 is said to be a spacelike framed base surface if there is a C map (x,n,s):UR31×Δ1 such that (x,n,s) is a spacelike framed surface.

    Definition 2.6. Let x:UR31 be a timelike surface. Then, the C map (x,n,s):UR31×˜Δ is said to be a timelike framed surface if xu(u,v),n(u,v)=0, xv(u,v),n(u,v)=0 for all (u,v)U, where xu(u,v)=(x/u)(u,v), xv(u,v)=(x/v)(u,v) and

    ˜Δ={(n,s)S21×S21n(u,v),s(u,v)=0},

    or

    ˜Δ={(n,s)S21×H20n(u,v),s(u,v)=0}.

    Moreover, x:UR31 is said to be a timelike framed base surface if there is a C map (x,n,s):UR31×˜Δ such that (x,n,s) is a timelike framed surface.

    We denote {a1,b1,a2,b2,e1,f1,g1,e2,f2,g2} as the invariant functions of a spacelike or timelike framed surface (x,n,s), where t(u,v)=n(u,v)s(u,v) and

    a1(u,v)=xu(u,v),s(u,v),b1(u,v)=xu(u,v),t(u,v),a2(u,v)=xv(u,v),s(u,v),b2(u,v)=xv(u,v),t(u,v),e1(u,v)=nu(u,v),s(u,v),f1(u,v)=nu(u,v),t(u,v),g1(u,v)=su(u,v),t(u,v),e2(u,v)=nv(u,v),s(u,v),f2(u,v)=nv(u,v),t(u,v),g2(u,v)=sv(u,v),t(u,v).

    Let (γ,ν,ω) be a spacelike framed curve in Minkowski 3-space. We will discuss the parallel curve of γ(t) with respect to the direction of ω(t) in this section.

    Definition 3.1. We define a curve called a parallel curve of γ(t) in Minkowski 3-space as

    Pγ[ω](t)=γ(t)+λw(t),λR{0}.

    By Definition 3.1, we have

    ˙Pγ[ω](t)=λ¯(t)v(t)+(α(t)+λ¯n(t))μ(t).

    So, t0R is a singularity of the curve Pγ[ω](t) if and only if ¯(t0)=0 and α(t0)+λ¯n(t0)=0.

    Proposition 3.2. Let w(t) be a timelike vector. Then, (Pγ[ω],n,ω):IR31×Δ is a spacelike framed curve where n:IS21 if and only if there is a φ(t):IR such that

    λ¯(t)cosφ(t)+(α(t)+λ¯n(t))sinφ(t)=0,tI

    for a fixed λR{0}.

    Proof. Suppose (Pγ[ω],n,ω) is a framed curve, ω(t) is timelike, and n(t),ω(t)=0. So, the vector n(t) is contained in the spacelike plane SpanR{ν(t),μ(t)}. Then, we have

    n(t)=cosφ(t)ν(t)+sinφ(t)μ(t).

    Furthermore,

    ˙Pγ[ω](t),n(t)=λ¯(t)cosφ(t)+(α(t)+λ¯n(t))sinφ(t)=0,tI.

    Conversely, if we have the above equation, then we can define n:IS21 by n(t)=cosφ(t)ν(t)+sinφ(t)μ(t). It is clear that (Pγ[ω],n,ω) satisfies the definition of a spacelike framed curve. This concludes the proof.

    Proposition 3.3. Let w be a spacelike vector. Then, we have the following:

    1) (Pγ[ω],n,ω):IR31×Δ is a spacelike framed curve where n:IH20 if and only if there is a φ(t):IR such that

    (λ¯(t)+α(t)+λ¯n(t))e2φ(t)=(λ¯(t)+α(t)+λ¯n(t)),tI (3.1)

    for a fixed λR{0}.

    2) (Pγ[ω],n,ω):IR31×Δ5 is a timelike framed curve where n:IS21 if and only if there is a φ(t):IR such that

    (λ¯(t)+α(t)+λ¯n(t))e2φ(t)=λ¯(t)+α(t)+λ¯n(t),tI (3.2)

    for a fixed λR{0}.

    Proof. Suppose (Pγ[ω],n,ω) is a spacelike or timelike framed curve in Minkowski 3-space, ω(t) is spacelike, and n(t),ω(t)=0. So, the vector n is contained in the timelike plane SpanR{ν(t),μ(t)}. As is known, there are four connected components of a timelike plane with respect to hyperbolic isometries. Then, the vector n has four cases.

    1) If n(t)SpanR{ν(t),μ(t)} is a spacelike vector, then we have n(t)=sinhφ(t)ν(t)+coshφ(t)μ(t) or n(t)=sinhφ(t)ν(t)coshφ(t)μ(t). By Definition 2.2, we have

    ˙Pγ[ω](t),n(t)=λ¯(t)coshφ(t)+(α(t)+λ¯n(t))sinhφ(t)=0,tI.

    So,

    (λ¯(t)+α(t)+λ¯n(t))e2φ(t)=(λ¯(t)+α(t)+λ¯n(t)),tI.

    Conversely, if n(t) satisfies Eq (3.1), we can define n:IH20 by the method in the proof of Proposition 3.2, which satisfies that (Pγ[ω],n,ω):IR31×Δ is a spacelike framed curve. Then, we conclude the proof of conclusion 1).

    2) If n(t)SpanR{ν(t),μ(t)} is a timelike vector, then we have n(t)=coshφ(t)ν(t)+sinhφ(t)μ(t) or n(t)=coshφ(t)ν(t)sinhφ(t)μ(t). By calculation, we obtain

    ˙Pγ[ω](t),n(t)=λ¯(t)coshφ(t)+(α(t)+λ¯n(t))sinhφ(t)=0,tI.

    So,

    (λ¯(t)+α(t)+λ¯n(t))e2φ(t)=λ¯(t)+α(t)+λ¯n(t),tI.

    Conversely, if n(t) satisfies Eq (3.2), we can define n:IS21 by the method in the proof of Proposition 3.2, which satisfies that (Pγ[ω],n,ω):IR31×Δ5 is a timelike framed curve. Then, we conclude the proof of conclusion 2).

    If ¯(t)=0, the frame {ν,ω,μ} is a Bishop frame, and we can then obviously see that (Pγ[w],ν,ω) is a spacelike or timelike framed curve by taking φ(t)=0 in Propositions 3.2 and 3.3.

    For a spacelike or timelike framed base curve γ(t), the curvature functions are the fundamental invariants of γ(t). Therefore, we will give the relations between the curvature functions of γ(t) and the parallel curves in the followings:

    Proposition 3.4. Let w be a timelike vector. If (Pγ[w],n,w) is a spacelike framed curve, then the curvature functions (P,mP,nP,αP) of (Pγ[w],n,w) saitisfy

    αP(t)=˙Pγ[ω](t),μP(t)=λ¯(t)sinφ(t)+(α(t)+λ¯n(t))cosφ(t),P(t)=˙ω(t),n(t)=¯(t)cosφ(t)+¯n(t)sinφ(t),mP(t)=˙n(t),μP(t)=¯m(t)+˙φ(t),nP(t)=˙ω(t),μP(t)=¯(t)sinφ(t)+¯n(t)cosφ(t).

    Proof. If ω is a timetike vector and (Pγ[w],n,w) is a spacelike framed curve, then by Proposition 3.2 we have a φ(t):IR satisfying

    λ¯(t)cosφ(t)+(α(t)+λ¯m(t))sinφ(t)=0,tI.

    We can define n:IS21 by n(t)=cosφ(t)v(t)+sinφ(t)μ(t). Then, μP(t)=n(t)ω(t)=sinφ(t)ν(t)+cosφ(t)μ(t). By Definition 2.2, we have αP(t)=˙Pγ[ω](t),μP(t), P(t)=˙ω(t),n(t), mP(t)=˙n(t),μP(t), and nP(t)=˙ω(t),μP(t). Then, by calculation, we can conclude the proof.

    Remark 3.5. Let w be a spacelike vector. If (Pγ[w],n,w) is a spacelike framed curve, by Proposition 3.3, we have a φ(t):IR satisfying

    (λ¯(t)+α(t)+λ¯n(t))e2φ(t)=(λ¯(t)+α(t)+λ¯n(t)),tI.

    We can define n:IH20 by

    n(t)=coshφ(t)ν(t)+sinhφ(t)μ(t).

    Then, we have the following curvature functions (P,mP,nP,αP) of (Pγ[w],n,w):

    μP(t)=n(t)ω(t)=sinhφ(t)ν(t)+coshφ(t)μ(t),αP(t)=˙Pγ[ω](t),μP(t)=λ¯(t)sinhφ(t)+(α(t)+λ¯n(t))coshφ(t),P(t)=˙ω(t),n(t)=¯(t)coshφ(t)+¯n(t)sinhφ(t),mP(t)=˙n(t),μP(t)=¯m(t)+˙φ(t),nP(t)=˙ω(t),μP(t)=¯(t)sinhφ(t)+¯n(t)coshφ(t).

    We can also define n:IH20 by

    n(t)=coshφ(t)ν(t)sinhφ(t)μ(t).

    Then, we have the following curvature functions (P,mP,nP,αP) of (Pγ[w],n,w):

    μP(t)=n(t)ω(t)=sinhφ(t)ν(t)coshφ(t)μ(t),αP(t)=˙Pγ[ω](t),μP(t)=λ¯(t)sinhφ(t)(α(t)+λ¯n(t))coshφ(t),P(t)=˙ω(t),n(t)=¯(t)coshφ(t)¯n(t)sinhφ(t),mP(t)=˙n(t),μP(t)=¯m(t)+˙φ(t),nP(t)=˙ω(t),μP(t)=¯(t)sinhφ(t)¯n(t)coshφ(t).

    Let w be a timelike vector. If (Pγ[w],n,w) is a timelike framed curve, by Proposition 3.3 we have a φ(t):IR satisfying

    (λ¯(t)+α(t)+λ¯n(t))e2φ(t)=λ¯(t)+α(t)+λ¯n(t),tI.

    We can define n:IS21 by

    n(t)=sinhφ(t)ν(t)+coshφ(t)μ(t).

    Then, we have the following curvature functions (P,mP,nP,αP) of (Pγ[w],n,w):

    μP(t)=n(t)ω(t)=coshφ(t)ν(t)+sinhφ(t)μ(t),αP(t)=˙Pγ[ω](t),μP(t)=λ¯(t)coshφ(t)+(α(t)+λ¯n(t))sinhφ(t),P(t)=˙ω(t),n(t)=¯(t)sinhφ(t)+¯n(t)coshφ(t),mP(t)=˙n(t),μP(t)=¯m(t)˙φ(t),nP(t)=˙ω(t),μP(t)=¯(t)coshφ(t)+¯n(t)sinhφ(t).

    We can also define n:IS21 by

    n(t)=sinhφ(t)ν(t)coshφ(t)μ(t).

    Then, we have the following curvature functions (P,mP,nP,αP) of (Pγ[w],n,w):

    μP(t)=n(t)ω(t)=coshφ(t)ν(t)sinhφ(t)μ(t),αP(t)=˙Pγ[ω](t),μP(t)=λ¯(t)coshφ(t){α(t)+λ¯n(t)}sinhφ(t),P(t)=˙ω(t),n(t)=¯(t)sinhφ(t)¯n(t)coshφ(t),mP(t)=˙n(t),μP(t)=¯m(t)˙φ(t),nP(t)=˙ω(t),μP(t)=¯(t)coshφ(t)¯n(t)sinhφ(t).

    In this section, we will discuss some surfaces constructed by a given spacelike framed curve γ(t). First, we will introduce a special ruled surface referred to as a normal surface. Then, we give some essential arguments of such normal surfaces which we use in the next section.

    Definition 4.1. Let (γ,ν1,ν2):IR31×Δ be a spacelike framed curve with frame {ν,ω,μ}. Then, we define a surface NSγ[w]:I×RR31 called a normal surface by

    NSγ[w](t,λ)=γ(t)+λw(t),(t,λ)I×R.

    Then, det(˙γ(t),ω(t),˙ω(t))=α(t)¯(t). Therefore, NSγ[ω](t,λ) is developable if and only if α(t)¯(t)=0. We see that if the frame {ν,ω,μ} is a Bishop frame, then the normal surface with repect to the direction of ω is always a developable surface on the regular part of NSγ[ω](t,λ). We call this situation Bishop normal developable on the regular part of NSγ[ω](t,λ).

    By Definition 4.1, we have

    NSγ[ω](t,λ)tNSγ[ω](t,λ)λ=λ¯(t)μ(t)δ(t)(α(t)+λ¯n(t))ν(t). (4.1)

    Therefore, by Eq (4.1) we have that (t0,λ0)I×R is a singularity of NSγ[ω](t,λ) if and only if λ0¯(t0)=0 and α(t0)+λ0¯n(t0)=0.

    With we have done in Propositions 3.2 and 3.3, we can have a similar discussion for the problem of whether a normal surface is a framed surface. Because the proof is similar to before, here we will directly give our propositions and omit the proof.

    Proposition 4.2. Let w be a timelike vector. Then, (NSγ[w],n,w):I×RR31×˜Δ is a timelike framed surface where n:I×RS21 if and only if there is a φ(t,λ):I×RR such that

    λ¯(t)cosφ(t,λ)+(α(t)+λ¯m(t))sinφ(t,λ)=0,(t,λ)I×R.

    Proposition 4.3. Let w be a spacelike vector. Then, we have the following:

    1) (NSγ[w],n,w):I×RR31×˜Δ is a timelike framed surface n:I×RS21 if and only if there is a φ(t,λ):I×RR such that

    (λ¯(t)+α(t)+λ¯n(t))e2φ(t,λ)=(λ¯(t)+α(t)+λ¯n(t)),(t,λ)I×R.

    2) (NSγ[w],n,w):I×RR31×Δ1 is a spacelike framed surface n:I×RH20 if and only if there is a φ(t,λ):I×RR such that

    (λ¯(t)+α(t)+λ¯n(t))e2φ(t,λ)=λ¯(t)+α(t)+λ¯n(t),(t,λ)I×R.

    If ¯(t)=0, the frame {ν,ω,μ} is a Bishop frame, and we then obviously see that NSγ[ω](t,λ) is always a spacelike or timelike framed base surface in Minkowski 3-space by taking φ(t,λ)=0 in Propositions 4.2 and 4.3.

    In the following theorem, we will show the local behavior of the singularities of the normal surface for a given spacelike framed curve.

    Theorem 4.4. Suppose that (t0,λ0)I×R is a singualarity of NSγ[ω](t,λ). Then, the singularity (t0,λ0) of NSγ[ω](t,λ) is a cross cap if and only if

    α(t0)˙¯(t0)+˙α(t0)¯(t0)0.

    Proof. By Whitney's theorem [34], the sufficient and necessary condition that a singularity (t0,λ0) of NSγ[ω](t,λ) is a cross cap is det(NSγ[ω]λ,NSγ[ω]λt,NSγ[ω]tt)0. According to the calculation results, we have

    NSγ[ω]λ=ω(t),%NSγ[ω]λt=¯(t)ν(t)+¯n(t)μ(t),NSγ[ω]t=α(t)μ(t)+λ¯ν(t)+λ¯n(t)μ(t),NSγ[ω]tt=(˙α(t)+λ¯(t)¯m(t)+λ˙¯n(t))μ(t)+(δ(t)α(t)¯m(t)+λ˙¯(t))λδ(t)¯n(t)¯m(t))ν(t)+(δ(t)α(t)¯n(t)+λδ(t)¯n2(t)+λ¯2(t))ω(t)).

    On the other hand, if (t0,λ0) is a singularity of NSγ[ω](t), then we have λ0¯(t0)=0 and α(t0)+λ0¯n(t0)=0. So, we can get

    NSγ[ω]tt|(t0,λ0)=(˙α(t0)+λ0˙¯n(t0))μ(t0)+λ0˙¯(t0)ν(t0).

    Therefore,

    det(NSγ[ω]λ,NSγ[ω]λt,NSγ[ω]tt)|(t0,λ0)=¯(t0)(˙α(t0)+λ0¯n(t0))¯n(t0)λ0˙¯(t0)=α(t0)˙¯(t0)+˙α(t0)¯(t0).

    This completes the proof.

    If the frame {ν,ω,μ} is a Bishop frame, which means ¯(t)0, then α(t0)˙¯(t0)+˙α(t0)¯(t0)0. Therefore, by Theorem 4.4, we have the following corollary:

    Corollary 4.5. Let (γ(t),ν,ω) be a spacelike framed curve with a Bishop frame of {ν,ω,μ}. Then, the singularity of NSγ[ω](t,λ) can not be a cross cap.

    Remark 4.6. We already know that the classification of singularities is well established not only for frontals or fronts in Euclidean 3-space, but also non-lightlike frontals or fronts in Minkowski 3-space [15,30,31]. Roughly speaking, the classification of singularities here consists of two parts. The first part is about non-degenerate singularities. For the case of fronts about non-degenerate singularities, we can have the necessary and sufficient conditions for the recognization of the singularities of a cuspidal edge, swallowtail, and cuspidal butterfly [9,15]. For the case of frontals about non-degenerate singularities, we can have the necessary and sufficient conditions for the recognization of the singularities of a cuspidal cross cap [15]. In the case of fronts about degenerate singularities with the corank one condition, we can have the necessary and sufficient conditions for the recognization of the singularities of cuspidal lips and cuspidal beaks [15]. In the case of frontals about degenerate singularities with the corank one condition, we can have the necessary and sufficient conditions for the recognization of the Chen Matumoto Mond ± singularities [28,30]. In further related work, we will give detailed classification results, but it is not the main theme of this article. Thus, we will not go into the details in this article.

    In this section, we will discuss the circular evolutes and involutes of a spacelike framed curve in Minkowski 3-space with respect to a Bishop frame. First, we give the definition of circular evolutes.

    Definition 5.1. Let (γ,ν1,ν2):IR31×Δ be a spacelike framed curve with a Bishop frame {ν,ω,μ}, that is, ¯(t)=0 for all tI. We assume that ¯n(t)0 for all tI. Then, we define a curve Eγ[ω]:IR31 in Minkowski 3-space called a circular evolute by

    Eγ[ω](t)=γ(t)α(t)¯n(t)ω(t).

    Then, in the following two propositions, we will study the relations between circular evolutes and normal surfaces for a given spacelike framed curve.

    Proposition 5.2. Let (γ,ν1,ν2):IR31×Δ be a spacelike framed curve with a frame {ν,ω,μ} which is Bishop. Then, the circular evolute of γ(t) is the striction curve of the normal surface NSγ[ω](t,λ).

    Proof. Suppose that σ(t):IR31 is the striction of NSγ[ω]. Then, we have

    σ(t)=γ(t)˙γ(t),˙ω(t)˙ω(t),˙ω(t)ω(t).

    Because {ν,ω,μ} is a Bishop frame, then we have

    σ(t)=γ(t)α(t)μ(t),¯(t)ν(t)+¯n(t)μ(t)¯(t)ν(t)+¯n(t)μ(t),¯(t)ν(t)+¯n(t)μ(t)ω(t)=γ(t)α(t)¯n(t)ω(t).

    This concludes the proof.

    Proposition 5.3. Let (γ,ν1,ν2):IR31×Δ be a spacelike framed curve with a Bishop frame {ν,ω,μ}, that is, ¯(t)=0 for all tI. Then, the singular point set of NSγ[ω](t,λ) is the circular evolute of γ(t).

    Proof. (t0,λ0)I×R is a singularity of NSγ[ω] if and only if

    λ0¯(t0)=0,α(t0)+λ0¯n(t0)=0. (5.1)

    Because {ν,ω,μ} is a Bishop frame, Eq (5.1) is equivalent to α(t0)+λ0¯n(t0)=0. If (t0,λ0) is a singularity of NSγ[ω], we have

    NSγ[ω](t0,λ0)=γ(t0)α(t0)¯n(t0)=Eγ[ω](t0).

    This concludes the proof.

    By Definition 5.1, we have

    ˙Eγ[ω](t)=α(t)μ(t)α(t)¯n(t)¯n(t)μ(t))ddt(α(t)¯n(t))ω(t)=ddt(α(t)¯n(t))ω(t).

    Obviously, we can find ˙Eγ[ω](t),ν(t)=0, ˙Eγ[ω](t),μ(t)=0, and ν(t)μ(t)=δ(t)ω(t). So, (Eγ[ω](t),ν,μ):IR31×Δ is a spacelike or timelike framed curve in Minkowski 3-space with moving frame {ν,μ,δω}. We can get the corresponding invariant functions as follows:

    αE(t)=˙γ(t),δ(t)ω(t)=ddt(α(t)¯n(t)),E(t)=˙ν(t),μ(t)=¯m(t),mE(t)=˙ν(t),δ(t)ω(t)=0,nE(t)=˙μ(t),δ(t)ω(t)=δ(t)¯n(t).

    For the convenience of expression, we denote the ω-evolute of γ(t) as Eγ[ω](t), and denote the ω-parallel curve of γ(t) as Pγ[ω](t). Then, we have the following duality relation between a spacelike framed curve γ(t) and the parallel curve with respect to ω-evolute:

    Proposition 5.4. Let (γ,ν,ω) be a spacelike framed curve. Then, we have

    EPγ[ω](t)=Eγ[ω](t).

    Proof. Because here {ν,ω,μ} is a Bishop frame, we have ¯(t)=0. Then, if ω is a timelike vector, by Proposition 3.2, we can take φ(t)=0. By Proposition 3.4, we have the following:

    EPγ[ω](t)=Pγ[ω](t)αP(t)nP(t)ω(t)=γ(t)+λω(t)α(t)+λ¯n(t)¯n(t)ω(t)=Eγ[ω](t).

    If ω is a spacelike vector, we can also substitute the corresponding αP(t) and nP(t) into the ω-evolutes of Pγ[ω](t). Then, we have EPγ[ω](t)=Eγ[ω](t).

    In the following, we consider the singular points of circular evolutes.

    Definition 5.5. Let γ:IR31 be a smooth curve in Lorentz-Minkowski 3-space. tI is said to be an (n,n+1)-cusp singularity of γ(t) if rank(γ(n)(t),γ(n+1)(t))=2 and ˙γ(t)=¨γ(t)=γ(3)(t)==γ(n1)(t)=0.

    Theorem 5.6. Let (γ,ν1,ν2):IR31×Δ be a spacelike framed curve, and let the frame {ν,ω,μ} be a Bishop frame. We also assume that ¯n(t)0 for all tI. Let t0 be a singularity of γ(t), which means α(t0)=0. Then, we have the following conclusions:

    1) t0 is a (2,3)-cusp singularty of γ(t) if and only if t0 is a regular point of Eγ[ω](t).

    2) t0 is an (n+1,n+2)-cusp singularity of γ(t) if and only if t0 is an (n,n+1)-cusp of Eγ[ω](t) for any n2,nN.

    Proof. 1) By Eq (2.2), we have

    ¨γ(t0)=˙α(t0)μ(t0)+α(t0)˙μ(t0),γ (5.2)

    If t_0 is a (2, 3) -cusp of \mathit{\boldsymbol{\gamma}}(t) , then we have \dot{ \mathit{\boldsymbol{\gamma}}}(t_0) = \mathit{\boldsymbol{0}} , rank (\ddot{ \mathit{\boldsymbol{\gamma}}}(t_0), \dddot{ \mathit{\boldsymbol{\gamma}}}(t_0)) = 2 , and \overline{n}(t_0) \neq 0 . Therefore, the singularity of \mathit{\boldsymbol{\gamma}}(t) is a (2, 3) -cusp if and only if \dot{\alpha}(t_0) \neq 0 .

    On the other hand,

    \dot{E}_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}](t_0) = \left[-\frac{d}{dt}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)\mathit{\boldsymbol{\omega}}(t)\right]_{t = t_0} = -\frac{1}{\overline{n}^{2}(t_0)}\left(\dot{\alpha}(t_0)\overline{n}(t_0)-\alpha(t_0)\dot{\overline{n}}(t_0)\right).

    Then, we have that E_{\mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}](t) is regular if and only if \dot{\alpha}(t_0) \neq 0 . Therefore, we have completed the first part of the proof.

    2) According to calculations, we have

    \begin{align*} \label{prof-sing-evo} \mathit{\boldsymbol{\gamma}}'(t)& = \alpha(t)\mathit{\boldsymbol{\mu}}(t),\\ \ \mathit{\boldsymbol{\gamma}}''(t)& = \alpha(t)\mathit{\boldsymbol{\mu}}'(t)+\alpha'(t)\mathit{\boldsymbol{\mu}}(t),\\ \ \mathit{\boldsymbol{\gamma}}'''(t)& = \alpha(t)\mathit{\boldsymbol{\mu}}''(t)+2\alpha'(t)\mathit{\boldsymbol{\mu}}'(t)+\alpha''(t)\mathit{\boldsymbol{\mu}}(t),\\ \ &\cdots \cdots\\ \mathit{\boldsymbol{\gamma}}^{(n+1)}(t) & = C_{n}^0\alpha(t) \mathit{\boldsymbol{\mu}}^{(n)}(t)+C_n^1\alpha'(t)\mathit{\boldsymbol{\mu}}^{(n-1)}(t)\\ &+\cdots+C_n^{n-1}\alpha^{(n-1)}(t)\mathit{\boldsymbol{\mu}}'(t)+C_n^n\alpha^{(n)}(t)\mathit{\boldsymbol{\mu}}(t),\\ \mathit{\boldsymbol{\gamma}}^{(n+2)}(t) & = C_{n+1}^0\alpha(t)\mathit{\boldsymbol{\mu}}^{(n+1)}(t)+C_{n+1}^1\alpha'(t)\mathit{\boldsymbol{\mu}}^{(n)}(t)\\ &+\cdots+C_{n+1}^{n}\alpha^{(n)}(t)\mathit{\boldsymbol{\mu}}'(t)+C_{n+1}^{n+1}\alpha^{(n+1)}(t)\mathit{\boldsymbol{\mu}}(t)\\ & = C_{n+1}^0\alpha(t)\mathit{\boldsymbol{\mu}}^{(n+1)}(t)+C_{n+1}^1\alpha'(t)\mathit{\boldsymbol{\mu}}^{(n)}(t)\\ &+\cdots+C_{n+1}^{n}\alpha^{(n)}(t)\delta(t)\left(-\overline{m}(t)\mathit{\boldsymbol{\nu}}(t)+\overline{n}(t)\mathit{\boldsymbol{\omega}}(t)\right)+C_{n+1}^{n+1}\alpha^{(n+1)}(t)\mathit{\boldsymbol{\mu}}(t). \end{align*}

    If t_0 is an (n+1, n+2) -cusp singularity of \mathit{\boldsymbol{\gamma}}(t) , then we have

    \begin{equation*} \left\{ \begin{array}{ll} &{\rm rank}\left( \mathit{\boldsymbol{\gamma}}^{(n+1)}(t), \mathit{\boldsymbol{\gamma}}^{(n+2)}(t)\right) = 2,\\ & \mathit{\boldsymbol{\gamma}}'(t_0) = \mathit{\boldsymbol{\gamma}}''(t_0) = \cdots = \mathit{\boldsymbol{\gamma}}^{(n)}(t_0) = \textbf{0}. \end{array} \right. \end{equation*}

    By the above equations, we can get that t_0 is an (n+1, n+2) -cusp of \mathit{\boldsymbol{\gamma}}(t) if and only if

    \begin{equation*} \left\{ \begin{array}{ll} &\alpha^{(n)}(t_0) \neq 0,\\ &\alpha(t_0) = \alpha'(t_0) = \cdots = \alpha^{(n-1)}(t_0) = 0. \end{array} \right. \end{equation*}

    On the other hand,

    \begin{align*} E_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}]'(t) = &-\frac{d}{dt}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)\mathit{\boldsymbol{\omega}}(t),\\ E_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}]''(t) = &\left(-\left(\frac{\alpha(t)}{\overline{n}(t)}\right)'\mathit{\boldsymbol{\omega}}(t)\right)' = -\left[\left(\frac{\alpha(t)}{\overline{n}(t)}\right)'\mathit{\boldsymbol{\omega}}'(t)+\left(\frac{\alpha(t)}{\overline{n}(t)}\right)''\mathit{\boldsymbol{\omega}}(t)\right],\\ &\cdots \cdots\\ E_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}]^{(n)}(t) = &-\Big[C_{n-1}^{0}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)'\mathit{\boldsymbol{\omega}}^{(n-1)}(t)+C_{n-1}^{1}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)''\mathit{\boldsymbol{\omega}}^{(n-2)}(t) \\ &+\cdots+C_{n-1}^{n-2}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)^{(n-1)}\mathit{\boldsymbol{\omega}}'(t)+C_{n-1}^{0}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)^{(n)}\mathit{\boldsymbol{\omega}}(t)\Big],\\ E_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}]^{(n+1)}(t) = &-\Big[C_{n}^{0}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)'\mathit{\boldsymbol{\omega}}^{(n)}(t)+C_{n}^{1}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)''\mathit{\boldsymbol{\omega}}^{n-1}(t) \\ &+\cdots+C_{n}^{n-1}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)^{(n)}(\overline{\ell}(t)\mathit{\boldsymbol{\nu}}(t)+\overline{n}(t)\mathit{\boldsymbol{\mu}}(t))+C_{n}^{0}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)^{(n+1)}\mathit{\boldsymbol{\omega}}(t) \Big]. \end{align*}

    If the singular point t_0 of E_{\mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}] is an (n, n+1) -cusp, by the above equations it is equivalent to

    \begin{equation} \begin{aligned} -\frac{d}{dt}\left(\frac{\alpha(t)}{\overline{n}(t)}\right)|_{t = t_0} = \left(\frac{\alpha(t_0)}{\overline{n}(t_0)}\right)'' & = \cdots = \left(\frac{\alpha(t_0)}{\overline{n}(_0)}\right)^{(n-1)} = 0,\\ \left(\frac{\alpha(t_0)}{\overline{n}(_0)}\right)^{(n)} &\neq 0. \end{aligned} \end{equation} (5.3)

    Furthermore,

    \begin{align*} &\left(\frac{\alpha(t)}{\overline{n}(t)}\right)' = \alpha(t)\left(\frac{1}{\overline{n}(t)}\right)'+\alpha'(t)\frac{1}{\overline{n}(t)},\\ &\left(\frac{\alpha(t)}{\overline{n}(t)}\right)'' = \alpha(t)\left(\frac{1}{\overline{n}(t)}\right)''+2\alpha'(t)\left(\frac{1}{\overline{n}(t)}\right)'+\alpha''(t)\frac{1}{\overline{n}(t)},\\ &\; \; \; \; \; \; \; \; \; \; \cdots \cdots \\ &\left(\frac{\alpha(t)}{\overline{n}(t)}\right)^{(n-1)} = C_{n-1}^{0}\alpha\left(\frac{1}{\overline{n}}\right)^{(n-1)}+C_{n-1}^{1}\alpha'\left(\frac{1}{\overline{n}}\right)^{(n-2)}+\cdots+C_{n-1}^{n-1}\alpha^{(n-1)}\left(\frac{1}{\overline{n}}\right),\\ &\left(\frac{\alpha(t)}{\overline{n}(t)}\right)^{(n)} = C_{n}^{0}\alpha\left(\frac{1}{\overline{n}}\right)^{(n)}+C_{n}^{1}\alpha' \left(\frac{1}{\overline{n}}\right)^{(n-1)}+\cdots+C_{n}^{n}\alpha^{(n)}\left(\frac{1}{\overline{n}}\right). \end{align*} (5.4)

    Because \alpha(t_0) = 0 , \overline{n}(t_0) \neq 0 , by Eqs (5.3) and (5.4), we can get that the singular point t_0 of E_{\mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}] is an (n, n+1) -cusp if and only if

    \begin{equation*} \left\{ \begin{array}{ll} &\alpha^{(n)}(t_0) \neq 0,\\ &\alpha(t_0) = \alpha'(t_0) = \cdots = \alpha^{(n-1)}(t_0) = 0. \end{array} \right. \end{equation*}

    Then, we have completed the second part of the proof.

    In the following, we will study the relations between circular evolutes and involutes for a given spacelike framed curve.

    Definition 5.7. Let (\mathit{\boldsymbol{\gamma}}, \mathit{\boldsymbol{\nu}}_1, \mathit{\boldsymbol{\nu}}_2): I \to \mathbb{R}_1^3 \times \Delta be a spacelike framed curve with m^2(t)-n^2(t) > 0 for all t \in I . Then, we define a curve I_{ \mathit{\boldsymbol{\gamma}}}[t_0](t): I\to \mathbb{R}_1^3 in Minkowski 3-space called an involute of \mathit{\boldsymbol{\gamma}}(t) with respect to a fixed t_0 \in I by

    I_{ \mathit{\boldsymbol{\gamma}}}[t_0](t) = \mathit{\boldsymbol{\gamma}}(t)-\left(\int_{t_0}^{t} \alpha(t) dt\right)\mathit{\boldsymbol{\mu}}(t)

    for a fixed t_0 \in I .

    We define \mathit{\boldsymbol{\xi}}(t) , \mathit{\boldsymbol{\eta}}(t) by

    \mathit{\boldsymbol{\xi}}(t) = \frac{n(t)\mathit{\boldsymbol{\nu}}_1(t)-m(t)\mathit{\boldsymbol{\nu}}_2(t)}{\sqrt{ m^2(t)-n^2(t)}},\; \mathit{\boldsymbol{\eta}}(t) = \mathit{\boldsymbol{\xi}}(t)\wedge\mathit{\boldsymbol{\mu}}(t) = \delta(t)\frac{-m(t)\mathit{\boldsymbol{\nu}}_1(t)+n(t)\mathit{\boldsymbol{\nu}}_2(t)}{\sqrt{m^2(t)-n^2(t)}}.

    Then, we have

    \begin{align*} \dot{\mathit{\boldsymbol{\xi}}}(t) = &\bigg( \frac{\dot{n}(t)(m^2(t)-n^2(t))-m\dot{m}(t)n(t)+n^2(t)\dot{n}(t)}{(m^2(t)-n^2(t))^{\frac{3}{2}}}- \frac{m(t)\ell(t)}{\sqrt{m^2(t)-n^2(t)}} \bigg)\mathit{\boldsymbol{\nu}}_1(t)\\ &+\bigg(\frac{\dot{m}(t)(m^2(t)-n^2(t))-m^2(t)\dot{m}(t)+m(t)n(t)\dot{n}(t)}{(m^2(t)-n^2(t))^{\frac{3}{2}}}+\frac{n(t)\ell(t)}{\sqrt{m^2(t)-n^2(t)}}\bigg)\mathit{\boldsymbol{\nu}}_2(t), \\ \dot{I}_{ \mathit{\boldsymbol{\gamma}}}[t_0](t)& = \left(\int_{t_0}^{t}\alpha(t) dt\right)\bigg(\delta(t)m(t)\mathit{\boldsymbol{\nu}}_1(t)-\delta(t)n(t)\mathit{\boldsymbol{\nu}}_2(t)\bigg), \end{align*}

    and \langle \dot{I}_{ \mathit{\boldsymbol{\gamma}}}[t_0](t), \mathit{\boldsymbol{\xi}}(t)\rangle = \langle \dot{I}_{ \mathit{\boldsymbol{\gamma}}}[t_0](t), \mathit{\boldsymbol{\mu}}(t) \rangle = 0 . Therefore, (I_{ \mathit{\boldsymbol{\gamma}}}[t_0](t), \mathit{\boldsymbol{\xi}}(t), \mathit{\boldsymbol{\mu}}(t)): I\to \mathbb{R}_1^3 \times \Delta is a spacelike or timelike framed curve with the curvature (\alpha_I, \ell_I, m_I, n_I) as follows:

    \begin{align*} \alpha_I(t) & = \langle \dot{I}_{ \mathit{\boldsymbol{\gamma}}}[t_0](t),\mathit{\boldsymbol{\eta}}(t) \rangle = -\delta(t)\left(\int_{t_0}^{t}\alpha(t)dt\right)\sqrt{m^2(t)-n^2(t)},\\ \ell_I(t) & = \langle \dot{\mathit{\boldsymbol{\xi}}}(t),\mathit{\boldsymbol{\mu}}(t) \rangle = 0,\\ m_I(t) & = \langle \dot{\mathit{\boldsymbol{\xi}}}(t),\mathit{\boldsymbol{\eta}}(t) \rangle = \frac{-m(t)\dot{n}(t)-\dot{m}(t)n(t)+(m^2(t)-n^2(t)\ell(t))}{m^2(t)-n^2(t)},\\ n_I(t) & = \langle \dot{\mathit{\boldsymbol{\mu}}}(t),\mathit{\boldsymbol{\eta}}(t) \rangle = \delta(t)\sqrt{m^2(t)-n^2(t)}. \end{align*}

    By Definition 2.4, we can see that \{\mathit{\boldsymbol{\xi}}, \mathit{\boldsymbol{\mu}}, \mathit{\boldsymbol{\eta}}\} is a Bishop frame along I_{ \mathit{\boldsymbol{\gamma}}}[t_0](t) .

    Proposition 5.8. Let (\mathit{\boldsymbol{\gamma}}, \mathit{\boldsymbol{\nu}}_1, \mathit{\boldsymbol{\nu}}_2): I \to \mathbb{R}_1^3 \times \Delta be a spacelike framed curve with m^2(t)-n^2(t) > 0 for all t \in I . Then, E_{I_{ \mathit{\boldsymbol{\gamma}}}[t_0]}[\mathit{\boldsymbol{\mu}}](t) = \mathit{\boldsymbol{\gamma}}(t) for any fixed t_0 \in I .

    Proof. By Definitions 5.1 and 5.7, we have

    \begin{align*} E_{I_{ \mathit{\boldsymbol{\gamma}}}[t_0]}[\mathit{\boldsymbol{\mu}}](t) & = I_{ \mathit{\boldsymbol{\gamma}}}[t_0](t)-\frac{\alpha_I(t)}{n_I(t)}\mathit{\boldsymbol{\mu}}(t)\\ & = I_{ \mathit{\boldsymbol{\gamma}}}[t_0](t)-\left(\int_{t_0}^{t} \alpha(t) dt\right)\mathit{\boldsymbol{\mu}}(t)-\frac{-\delta(t)\left(\int_{t_0}^{t}\alpha(t)dt\right)\sqrt{m^2(t)-n^2(t)}}{\delta(t)\sqrt{m^2(t)-n^2(t)}}\mathit{\boldsymbol{\mu}}(t)\\ & = \mathit{\boldsymbol{\gamma}}(t)-\left(\int_{t_0}^{t} \alpha(t) dt\right)\mathit{\boldsymbol{\mu}}(t)+\left(\int_{t_0}^{t} \alpha(t) dt\right)\mathit{\boldsymbol{\mu}}(t)\\ & = \mathit{\boldsymbol{\gamma}}(t). \end{align*}

    This concludes the proof.

    Proposition 5.9. Let (\mathit{\boldsymbol{\gamma}}, \mathit{\boldsymbol{\nu}}_1, \mathit{\boldsymbol{\nu}}_2): I \to \mathbb{R}_1^3 \times \Delta be a spacelike framed curve, and let \{\mathit{\boldsymbol{\nu}}, \mathit{\boldsymbol{\omega}}, \mathit{\boldsymbol{\mu}}\} be a Bishop frame of \mathit{\boldsymbol{\gamma}}(t) with \overline{n}(t) \neq 0 for all t \in I . Then, we have that I_{E_{ \mathit{\boldsymbol{\gamma}}}[\delta\mathit{\boldsymbol{\omega}}]}[t_0](t) is a parallel curve of \mathit{\boldsymbol{\gamma}}(t) . In particular, if t_0 is a singular point of \mathit{\boldsymbol{\gamma}}(t) , we have I_{E_{ \mathit{\boldsymbol{\gamma}}}[\delta\mathit{\boldsymbol{\omega}}]}[t_0](t) = \mathit{\boldsymbol{\gamma}}(t) .

    Proof. By Definitions 5.1 and 5.7, we have

    \begin{align*} I_{E_{ \mathit{\boldsymbol{\gamma}}}[\delta\mathit{\boldsymbol{\omega}}]}[t_0](t)& = E_{ \mathit{\boldsymbol{\gamma}}}[\delta(t) \mathit{\boldsymbol{\omega}}](t)- \left(\int_{t_0}^{t} \alpha_E (t) dt\right) \delta(t)\mathit{\boldsymbol{\omega}}(t)\\ & = \mathit{\boldsymbol{\gamma}}(t) - \frac{\alpha(t)}{\overline{n}(t)}\delta(t)\mathit{\boldsymbol{\omega}}(t)-\left(\int_{t_0}^{t}-\frac{d}{dt}\left(\frac{\alpha(t)}{\overline{n}(t)}\right) dt\right) \delta(t)\mathit{\boldsymbol{\omega}}(t)\\ & = \mathit{\boldsymbol{\gamma}}(t) - \frac{\alpha(t_0)}{\overline{n}(t_0)}\delta(t)\mathit{\boldsymbol{\omega}}(t). \end{align*}

    This concludes the proof.

    We now consider the singular points of involutes in the following:

    Theorem 5.10. Let (\mathit{\boldsymbol{\gamma}}, \mathit{\boldsymbol{\nu}}_1, \mathit{\boldsymbol{\nu}}_2): I \to \mathbb{R}_1^3 \times \Delta be a spacelike framed curve, and let the frame \{\mathit{\boldsymbol{\nu}}, \mathit{\boldsymbol{\omega}}, \mathit{\boldsymbol{\mu}}\} be a Bishop frame. We also assume that m^2(t)-n^2(t) > 0 for all t \in I , and let t_1 be a singularity of I_{ \mathit{\boldsymbol{\gamma}}}[t_0](t) , which means \alpha_{I}(t_1) = 0 . Then, we have the following conclusions:

    1) The point t_1 of \mathit{\boldsymbol{\gamma}}(t) is regular if and only if t_1 of I_{\mathit{\boldsymbol{\gamma}}}[t_0] is a (2, 3) -cusp.

    2) The singular point t_1 of \mathit{\boldsymbol{\gamma}}(t) is an (n, n+1) -cusp if and only if t_1 of I_{\mathit{\boldsymbol{\gamma}}}[t_0](t) is an (n+1, n+2) -cusp for any n\geqslant 2, n \in \mathbb{N} .

    Proof. 1) The point t_1 of \mathit{\boldsymbol{\gamma}}(t) is regular if and only if \alpha(t_1) \neq 0 . The singularity t_1 of I_{ \mathit{\boldsymbol{\gamma}}}[t_0](t) is a (2, 3) -cusp if and only if rank (\ddot{I}_{ \mathit{\boldsymbol{\gamma}}}[t_0](t_1), \dddot{I}_{ \mathit{\boldsymbol{\gamma}}}[t_0](t_1)) = 2 and \dot{I}_{ \mathit{\boldsymbol{\gamma}}}[t_0](t_1) = \textbf{0} , which means \alpha_{I}(t_1) = 0 and \dot{\alpha}_{I}(t_1) \neq 0 . Since

    \begin{align*} \dot{I}_{ \mathit{\boldsymbol{\gamma}}}[t_0](t)& = \delta(t)\alpha_{I}(t)\mathit{\boldsymbol{\eta}}(t),\\ \alpha_I (t) & = -\left(\delta(t)\int_{t_0}^{t}\alpha(t)dt\right)\sqrt{m^2(t)-n^2(t)}\; ,\\ \dot{\alpha}_{I}(t) & = -\delta(t)\alpha(t)\sqrt{m^2(t)-n^2(t)}-\left(\delta(t)\int_{t_0}^{t}\alpha(t)dt\right)\frac{d}{dt}\left(\sqrt{m^2(t)-n^2(t)}\right) \end{align*}

    and

    m^2(t) - n^2 (t) > 0,

    we can get the conclusion of 1).

    2) By the calculations of Theorem 5.6, we have

    \begin{align*} I^{(n)}_{ \mathit{\boldsymbol{\gamma}}}(t) & = \left(\delta\alpha_{I} \mathit{\boldsymbol{\eta}}\right)^{(n-1)} = C_{n-1}^{0}\delta\alpha_{I}\mathit{\boldsymbol{\eta}}^{(n-1)}+ \cdots + \left(\delta\alpha_{I}\right)^{(n-1)}\mathit{\boldsymbol{\eta}},\\ I^{(n+1)}_{ \mathit{\boldsymbol{\gamma}}}(t) & = C_{n}^{0}\delta\alpha_{I}\mathit{\boldsymbol{\eta}}^{(n)}+ \cdots + C_n ^{n-1}\left(\delta\alpha_{I}\right)^{(n-1)}\mathit{\boldsymbol{\eta}}'+C_n ^{n}\left(\delta\alpha_{I}\right)^{(n)}\mathit{\boldsymbol{\eta}},\\ I^{(n+2)}_{ \mathit{\boldsymbol{\gamma}}}(t) & = C_{n+1}^{0}\delta\alpha_{I}\mathit{\boldsymbol{\eta}}^{(n+1)}+ \cdots + C_{n+1}^{n}\left(\delta\alpha_{I}\right)^{(n)} \mathit{\boldsymbol{\eta}}'+ C_{n+1}^{n+1}\left(\delta\alpha_{I}\right)^{(n+1)}\mathit{\boldsymbol{\eta}}. \end{align*}

    Thus, t_1 is an (n+1, n+2) -cusp of I_{ \mathit{\boldsymbol{\gamma}}}[t_0](t) if and only if

    \begin{equation} \begin{cases} &\alpha_{I}(t_1) = \dot{\alpha}_I (t_1) = \cdots = \alpha_I ^{(n-1)}(t_1) = 0,\\ &\alpha_I ^{(n)}(t_1) \neq 0. \end{cases} \end{equation} (5.5)

    Furthermore,

    \begin{align*} -\delta(t)\alpha_{I}^{(n)}(t) = &C_{n}^{0}\left(\int_{t_0}^{t} \alpha(t) dt\right)\sqrt{m^2(t)-n^2(t)}^{(n)}+ C_n^1\alpha(t) \sqrt{m^2(t)-n^2(t)}^{(n-1)}+ \cdots \\ &+ C_n^{n-1}\alpha^{(n-2)}(t) \sqrt{m^2(t)-n^2(t)}^{'} + C_n^{n}\alpha(t)^{(n-1)} \sqrt{m^2(t)-n^2(t)} \end{align*}

    and

    m^2(t)- n^2 (t) > 0.

    We have that Eq (5.5) is equivalent to

    \begin{equation} \begin{cases} &\alpha(t_1) = \dot{\alpha} (t_1) = \cdots = \alpha^{(n-2)}(t_1) = 0,\\ &\alpha^{(n-1)}(t_1) \neq 0. \end{cases} \end{equation} (5.6)

    Namely, the singular point t_1 of \mathit{\boldsymbol{\gamma}}(t) is an (n, n+1) -cusp if and only if t_1 is an (n+1, n+2) -cusp of I_{\mathit{\boldsymbol{\gamma}}}[t_0](t) for any n\geqslant 2, n \in \mathbb{N} . This concludes the proof.

    In the following example, we give a spacelike framed curve in Minkowski 3-space. In this example, we will discuss its circular evolutes, involutes, normal surfaces, and their singularities. Then, we show the relationships among them by their geometric figure.

    Example 6.1. Let \mathit{\boldsymbol{\gamma}}(t) = (\sinh^3 t, \cosh^3 t, 1) . We can see that (0, 1, 1) is a (2, 3) -cusp of the curve \mathit{\boldsymbol{\gamma}}(t) , and \mathit{\boldsymbol{\gamma}}(t) is a spacelike framed curve with singularities.

    By \dot{ \mathit{\boldsymbol{\gamma}}}(t) = (3\sinh^2 t \cosh t, 3\cosh^2 t \sinh t, 0) , naturally we can take the Bishop frame \{\mathit{\boldsymbol{\nu}}, \mathit{\boldsymbol{\omega}}, \mathit{\boldsymbol{\mu}}\} of \mathit{\boldsymbol{\gamma}}(t) as \mathit{\boldsymbol{\mu}}(t) = (\sinh t, \cosh t, 0) , \mathit{\boldsymbol{\nu}}(t) = (\sqrt{2}\cosh t, \sqrt{2}\sinh t, -1) , \mathit{\boldsymbol{\omega}}(t) = (\cosh t, \sinh t, -\sqrt{2}) . Then, we have the Frenet formulae

    \begin{equation*} \begin{aligned} &\left( \begin{array}{ccc} \dot{\mathit{\boldsymbol{\nu}}}(t)\\ \dot{\mathit{\boldsymbol{\omega}}}(t)\\ \dot{\mathit{\boldsymbol{\mu}}}(t) \end{array} \right) = \begin{pmatrix} 0 & 0 &\sqrt{2}\\ 0 & 0 & 1 \\ \sqrt{2} & -1& 0 \end{pmatrix} \begin{pmatrix} \mathit{\boldsymbol{\nu}}(t)\\ \mathit{\boldsymbol{\omega}}(t)\\ \mathit{\boldsymbol{\mu}}(t) \end{pmatrix},\\ &\dot{ \mathit{\boldsymbol{\gamma}}}(t) = (3\sinh t \cosh t,3\cosh2 t \sinh t ,0)\mathit{\boldsymbol{\mu}}(t). \end{aligned} \end{equation*}

    By the definitions of normal surfaces and circular evolutes and involutes, we have

    \begin{align*} &E_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}](t) = (\sinh^3 t-3\sinh t\cosh^2 t,\cosh^3 t-3\sinh^2 t\cosh t,1+3\sqrt{2}\sinh t\cosh t),\\ &I_{ \mathit{\boldsymbol{\gamma}}}[0](t) = (\frac{3}{2}\sinh^3t,\cosh^3 t-\frac{3}{2}\sinh^2 t\cosh t,1),\\ &NS_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}] (t) = (\sinh^3 t+\lambda\cosh t,\cosh^3 t+\lambda\sinh t,1-\sqrt{2}\lambda). \end{align*}

    We show the geometric locus of \mathit{\boldsymbol{\gamma}}(t) , E_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}](t) , I_{ \mathit{\boldsymbol{\gamma}}}[0](t) , NS_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}] (t) in Figure 1. \mathit{\boldsymbol{\gamma}}(t) is the blue curve. The purple curve in Figure 1 is I_{ \mathit{\boldsymbol{\gamma}}}[0](t) . E_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}](t) is the red curve. The green surface NS_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}] (t) is the normal surface of \mathit{\boldsymbol{\gamma}}(t) , and this is a singular surface with a singularity type of cuspidal edge. We see that E_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}](t) lies in the singular set of NS_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}] (t) . We can also see that the black point is a regular point in E_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}](t) , but it is a (2, 3) -cusp of \mathit{\boldsymbol{\gamma}}(t) and a (3, 4) -cusp of I_{ \mathit{\boldsymbol{\gamma}}}[0](t) . Moreover, we find that the circular evolute of \mathit{\boldsymbol{\gamma}}(t) can be a regular curve, even if \mathit{\boldsymbol{\gamma}}(t) is a spacelike framed curve with singularities.

    Figure 1.  \mathit{\boldsymbol{\gamma}}(t) , E_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}](t) , I_{ \mathit{\boldsymbol{\gamma}}}[0](t) , NS_{ \mathit{\boldsymbol{\gamma}}}[\mathit{\boldsymbol{\omega}}] (t) .

    Through our research we have found that there are fancy duality relations not only among parallel curves, normal surfaces, and circular evolutes and involutes, but also for their singularities. Our example also shows more clearly that duality relations are a kind of relation that are very canonical and natural in our geometric imagination. Based on these studies, we can further consider the a family of curves and surfaces and research their related properties, such as the corresponding behaviors of one-parameter families of framed curves, or a family of curves that satisfies certain equations. On the other hand, although the equations are more complex with growth of dimensions, there has already been some related research [23,24,25]. Thus, it makes sense to further consider circular evolutes and involutes in higher dimensional space. In any case, we find that it is crucially important to consider the duality relations among different geometric objects for the research of submanifolds with singularities.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    We gratefully acknowledge the constructive comments from the editor and the anonymous referees. This work was supported by the National Natural Science Foundation of China (Grant No. 11671070).

    The authors declare that there are no conflicts of interest that may influence the publication of this work.



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