Envelopes of legendre curves in the unit spherical bundle over the unit sphere

Envelopes of legendre curves in the unit spherical bundle over the unit sphere Abstract In this paper, we introduce a one-parameter family of Legendre curves in the unit spherical bundle over the unit sphere and the curvature. We give the existence and uniqueness theorems for one-parameter families of spherical Legendre curves by using the curvatures. Then we define an envelope for the one-parameter family of Legendre curves in the unit spherical bundle. We also consider the parallel curves and evolutes of one-parameter families of Legendre curves in the unit spherical bundle and their envelopes. Moreover, we give relationships among one-parameter families of Legendre curves in the unit spherical bundle over the unit sphere and one-parameter families of Legendre curves in the unit tangent bundle over the Euclidean plane. 1. Introduction An envelope of a family of curves in the Euclidean plane is a curve that is tangent to each member of the family at some point. If the curves are regular, then the tangent is well-defined (cf. [3–6, 9]). On the other hand, for singular plane curves, the classical definitions of envelopes are vague. In [11], the third author clarified the definition of the envelope for a family of singular curves in the unit tangent bundle over the Euclidean plane. This idea can be generalized to an envelope of a family of singular spherical curves. In [10], the third author gave a definition of Legendre curves in the unit spherical bundle, and established a moving frame of spherical Legendre curves. However, to the best of the authors' knowledge, no literature exists regarding the envelope of a family of singular curves in the unit sphere. In this paper, we use these approach and techniques to give the definition and investigate properties of envelopes for families of Legendre curves in the unit spherical bundle over the unit sphere. For basic results on the singularity theory, see [1, 2, 4, 7, 8]. In Section 2, we consider one-parameter families of Legendre curves in the unit spherical bundle and the curvatures. We give the existence and uniqueness theorems for one-parameter families of Legendre curves by using the curvatures. In Section 3, we define an envelope of a one-parameter family of Legendre curves in the unit spherical bundle. We obtained that the envelope is also a Legendre curve. The envelope of the dual is the dual of the envelope of the one-parameter family of Legendre curves. We also give the definitions of parallel curves and evolutes of one-parameter families of Legendre curves in the unit sphere bundle. We found that the parallel curves and evolutes are also one-parameter families of Legendre curves in the unit sphere bundle, the evolutes if exists. Then we consider the envelopes of the parallel curves and evolutes. The envelope of parallel curves of a one-parameter family of Legendre curves is equal to the parallel curve of the envelope of the one-parameter family of Legendre curves. Under a condition, the envelope of the evolute of a one-parameter family of Legendre curves is equal to the evolute of the envelope of the one-parameter family of Legendre curves. In Section 4, we give relationships among one-parameter families of Legendre curves in the unit spherical bundle over the unit sphere and one-parameter families of Legendre curves in the unit tangent bundle over the Euclidean plane. In Section 5, we give two examples and some pictures to illustrate our results. All maps and manifolds considered here are differential of class C∞. 2. Legendre curves in the unit spherical bundle over the unit sphere We first recall some definitions and theorems of Legendre curves in the unit spherical bundle over the unit sphere. For more detailed descriptions, see [10]. Let R3 be the 3-dimensional Euclidean space equipped with the inner product a·b=a1b1+a2b2+a3b3, where a=(a1,a2,a3) and b=(b1,b2,b3)∈R3. The vector product is given by a×b=∣e1e2e3a1a2a3b1b2b3∣, where e1,e2,e3 are the canonical basis on R3. Let S2={(x,y,z)∈R3∣x2+y2+z2=1} be the unit sphere. We denote the set {(a,b)∈S2×S2∣a·b=0} by Δ (cf. [10]). Then Δ is a 3-dimensional smooth manifold. We say that (γ,ν):I→Δ⊂S2×S2 is a Legendre curve (or, spherical Legendre curve) if γ˙(t)·ν(t)=0 for all t∈I, that is, (γ,ν) is an integrable curve with respect to the canonical contact 1-form on Δ. We call γ a frontal and ν a dual of γ. Moreover, if (γ,ν) is a Legendre immersion, we call γ a front. We define μ(t)=γ(t)×ν(t). By definition, μ(t)∈S2, γ(t)·μ(t)=0 and ν(t)·μ(t)=0 for all t∈I. It follows that {γ(t),ν(t),μ(t)} is a moving frame along the frontal γ(t). Let (γ,ν):I→Δ be a Legendre curve. We have the Frenet type formula. (γ˙(t)ν˙(t)μ˙(t))=(00m(t)00n(t)−m(t)−n(t)0)(γ(t)ν(t)μ(t)), where m(t)=γ˙(t)·μ(t) and n(t)=ν˙(t)·μ(t). We say that the pair of the functions (m,n) is the curvature of the Legendre curve (γ,ν):I→Δ. Definition 2.1. Let (γ,ν),(γ˜,ν˜):I→Δ be Legendre curves. We say that (γ,ν) and (γ˜,ν˜) are congruent as Legendre curves if there exists a special orthogonal matrix A∈SO(3) such that γ˜(t)=A(γ(t)), ν˜(t)=A(ν(t)) for all t∈I. Then we have the following existence and uniqueness theorems in terms of the curvature of the Legendre curve [10]. Theorem 2.2. (The Existence Theorem of spherical Legendre curves). Let (m,n):I→R×Rbe a smooth mapping. There exists a Legendre curve (γ,ν):I→Δ, whose associated curvature is (m,n). Theorem 2.3. (The Uniqueness Theorem of spherical Legendre curves). Let (γ,ν)and (γ˜,ν˜):I→Δbe Legendre curves whose curvatures (m,n)and (m˜,n˜), respectively. Then (γ,ν)and (γ˜,ν˜)are congruent as Legendre curves if and only if (m,n)and (m˜,n˜)coincide. We consider one-parameter families of Legendre curves in the unit spherical bundle Δ⊂S2×S2. Let I and Λ be intervals of R. Definition 2.4. Let (γ,ν):I×Λ→Δ be a smooth mapping. We say that (γ,ν) is a one-parameter family of spherical Legendre curves if γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. By definition, (γ(·,λ),ν(·,λ)):I→Δ is a Legendre curve for each fixed parameter λ∈Λ. We define μ(t,λ)=γ(t,λ)×ν(t,λ). Then {γ(t,λ),ν(t,λ),μ(t,λ)} is a moving frame along the frontal γ(t,λ) on S2. We have the Frenet type formula: (γt(t,λ)νt(t,λ)μt(t,λ))=(00m(t,λ)00n(t,λ)−m(t,λ)−n(t,λ)0)(γ(t,λ)ν(t,λ)μ(t,λ)), (γλ(t,λ)νλ(t,λ)μλ(t,λ))=(0L(t,λ)M(t,λ)−L(t,λ)0N(t,λ)−M(t,λ)−N(t,λ)0)(γ(t,λ)ν(t,λ)μ(t,λ)), where m(t,λ)=γt(t,λ)·μ(t,λ),n(t,λ)=νt(t,λ)·μ(t,λ),L(t,λ)=γλ(t,λ)·ν(t,λ),M(t,λ)=γλ(t,λ)·μ(t,λ),N(t,λ)=νλ(t,λ)·μ(t,λ). We denote the matrices A(t,λ)=(00m(t,λ)00n(t,λ)−m(t,λ)−n(t,λ)0),B(t,λ)=(0L(t,λ)M(t,λ)−L(t,λ)0N(t,λ)−M(t,λ)−N(t,λ)0). By γtλ(t,λ)=γλt(t,λ),νtλ(t,λ)=νλt(t,λ) and μtλ(t,λ)=μλt(t,λ), we have the integrability condition Aλ(t,λ)+A(t,λ)B(t,λ)=Bt(t,λ)+B(t,λ)A(t,λ), that is, (2.1) for all (t,λ)∈I×Λ. We call the tuple (m,n,L,M,N) with the integrability condition (2.1) the curvature of the one-parameter family of Legendre curves. Remark 2.5. Let (γ,ν):I×Λ→Δ be a one-parameter family of Legendre curves with the curvature (m,n,L,M,N). Then, it is easy to check that (γ,−ν), (−γ,ν) and (ν,γ) are also one-parameter family of Legendre curves. The curvatures are (−m,n,−L,−M,N), (m,−n,−L,M,−N) and (−n,−m,−L,−N,−M), respectively. Definition 2.6. Let (γ,ν) and (γ˜,ν˜):I×Λ→Δ be one-parameter families of Legendre curves. We say that (γ,ν) and (γ˜,ν˜) are congruent as one-parameter families of Legendre curves if there exists a special orthogonal matrix A∈SO(3) such that γ˜(t,λ)=A(γ(t,λ)) and ν˜(t,λ)=A(ν(t,λ)) for all (t,λ)∈I×Λ. Then we have the following existence and uniqueness theorems for one-parameter families of Legendre curves. Theorem 2.7. (The Existence Theorem for one-parameter families of spherical Legendre curves). Let (m,n,L,M,N):I×Λ→R5be a smooth mapping with the integrability condition. There exists a one-parameter family of Legendre curves (γ,ν):I×Λ→Δ, whose associated curvature is (m,n,L,M,N). Proof Choose any fixed value t=t0,λ=λ0 of the parameter. We consider an initial value problem, Ft(t,λ)=A(t,λ)F(t,λ),Fλ(t,λ)=B(t,λ)F(t,λ),F(t0,λ0)=I3, where F(t,λ)∈M(3), A(t,λ),B(t,λ) as the above, M(3) is the set of 3×3 matrices and I3 is the identity matrix. Then we consider Ftλ=AλF+AFλ=AλF+ABF=(Aλ+AB)F,Fλt=BtF+BFt=BtF+BAF=(Bt+BA)F. By the integrability condition Aλ+AB=Bt+BA, we have Ftλ=Fλt. Since I×Λ is simply connected, there exists a solution F(t,λ). Therefore, there exists a one-parameter family of Legendre curves (γ,ν):I×Λ→Δ whose associated curvature is (m,n,L,M,N).□ Lemma 2.8. Let (γ,ν)and (γ˜,ν˜):I×Λ→Δbe one-parameter families of Legendre curves having equal curvature, that is, (m(t,λ),n(t,λ),L(t,λ),M(t,λ),N(t,λ))=(m˜(t,λ),n˜(t,λ),L˜(t,λ),M˜(t,λ),N˜(t,λ))for all (t,λ)∈I×Λ. If there exist two parameters t=t0,λ=λ0for which (γ(t0,λ0),ν(t0,λ0))=(γ˜(t0,λ0),ν˜(t0,λ0)), then (γ,ν)and (γ˜,ν˜)coincide. Proof Define a smooth function f:I×Λ→R by f(t,λ)=γ(t,λ)·γ˜(t,λ)+ν(t,λ)·ν˜(t,λ)+μ(t,λ)·μ˜(t,λ). Since (m(t,λ),n(t,λ),L(t,λ),M(t,λ),N(t,λ))=(m˜(t,λ),n˜(t,λ),L˜(t,λ),M˜(t,λ),N˜(t,λ)), we have ft(t,λ)=(γt·γ˜+γ·γ˜t+νt·ν˜+ν·ν˜t+μt·μ˜+μ·μ˜t)(t,λ)=((mμ)·γ˜+γ·(m˜μ˜)+(nμ)·ν˜+ν·(n˜μ˜)+(−mγ−nν))·μ˜+μ·(−m˜γ˜−n˜ν˜)(t,λ)=((m−m˜)μ·γ˜+(m˜−m)γ·μ˜+(n−n˜)μ·ν˜+(n˜−n)ν·μ˜)(t,λ)=0,fλ(t,λ)=(γλ·γ˜+γ·γ˜λ+νλ·ν˜+ν·ν˜λ+μλ·μ˜+μ·μ˜λ)(t,λ)=((Lν+Mμ)·γ˜+γ·(L˜ν˜+M˜μ˜)+(−Lγ+Nμ)·ν˜+ν·(−L˜γ˜+N˜μ˜))+(−Mγ−Nν)·μ˜+μ·(−M˜γ˜−N˜ν˜)(t,λ)=((L−L˜)ν·γ˜+(L˜−L)γ·ν˜+(M−M˜)μ·γ˜+(M˜−M)γ·μ˜)+(N−N˜)μ·ν˜+(N˜−N)ν·μ˜(t,λ)=0 for all (t,λ)∈I×Λ. It follows that f is constant. By γ(t0,λ0)=γ˜(t0,λ0) and ν(t0,λ0)=ν˜(t0,λ0), we have f(t0,λ0)=3 and the function f is constant with value 3. By the Cauchy–Schwarz inequality, we have γ(t,λ)·γ˜(t,λ)≤∣γ(t,λ)∣∣γ˜(t,λ)∣=1,ν(t,λ)·ν˜(t,λ)≤∣ν(t,λ)∣∣ν˜(t,λ)∣=1,μ(t,λ)·μ˜(t,λ)≤∣μ(t,λ)∣∣μ˜(t,λ)∣=1. If one of these inequalities is strict, the value of f(t,λ) would be <3. It follows that these inequalities are equalities, and we have γ(t,λ)·γ˜(t,λ)=1,ν(t,λ)·ν˜(t,λ)=1,μ(t,λ)·μ˜(t,λ)=1 for all (t,λ)∈I×Λ. Then we have ∣γ(t,λ)−γ˜(t,λ)∣2=∣ν(t,λ)−ν˜(t,λ)∣2=∣μ(t,λ)−μ˜(t,λ)∣2=0. It follows that γ(t,λ)=γ˜(t,λ),ν(t,λ)=ν˜(t,λ),μ(t,λ)=μ˜(t,λ)forall(t,λ)∈I×Λ. □ Theorem 2.9. (The Uniqueness Theorem for one-parameter families of Legendre curves). Let (γ,ν)and (γ˜,ν˜):I×Λ→Δbe one-parameter families of Legendre curves with the curvatures (m,n,L,M,N)and (m˜,n˜,L˜,M˜,N˜), respectively. Then (γ,ν)and (γ˜,ν˜)are congruent as one-parameter family of Legendre curves if and only if (m,n,L,M,N)and (m˜,n˜,L˜,M˜,N˜)coincide. Proof Suppose that (γ,ν) and (γ˜,ν˜) are congruent as one-parameter families of Legendre curves. By a direct calculation, we have γ˜t(t,λ)=∂∂t(A(γ(t,λ)))=A(γt(t,λ))=m(t,λ)A(μ(t,λ))=m(t,λ)μ˜(t,λ),ν˜t(t,λ)=∂∂t(A(ν(t,λ)))=A(νt(t,λ))=n(t,λ)A(μ(t,λ))=n(t,λ)μ˜(t,λ),γ˜λ(t,λ)=∂∂λ(A(γ(t,λ)))=A(γλ(t,λ))=L(t,λ)A(ν(t,λ))+M(t,λ)A(μ(t,λ))=L(t,λ)ν˜(t,λ)+M(t,λ)μ˜(t,λ),ν˜λ(t,λ)=∂∂λ(A(ν(t,λ)))=A(νλ(t,λ))=−L(t,λ)A(γ(t,λ))+N(t,λ)A(μ(t,λ))=−L(t,λ)γ˜(t,λ)+N(t,λ)μ˜(t,λ). Therefore, the curvatures (m,n,L,M,N) and (m˜,n˜,L˜,M˜,N˜) coincide. Conversely, suppose that (m,n,L,M,N) and (m˜,n˜,L˜,M˜,N˜) coincide. Let (t0,λ0)∈I×Λ be fixed. By using a congruence as one-parameter family of Legendre curves, we may assume γ(t0,λ0)=γ˜(t0,λ0) and ν(t0,λ0)=ν˜(t0,λ0). By Lemma 2.8, we have γ(t,λ)=γ˜(t,λ) and ν(t,λ)=ν˜(t,λ) for all (t,λ)∈I×Λ.□ 3. Envelopes of one-parameter families of Legendre curves in the unit spherical bundle Let (γ,ν):I×Λ→Δ be a one-parameter family of Legendre curves with the curvature (m,n,L,M,N) and let e:U→I×Λ, e(u)=(t(u),λ(u)) be a smooth curve, where U is an interval of R. We denote Eγ=γ◦e:U→S2, Eγ(u)=γ◦e(u) and Eν=ν◦e:U→S2, Eν(u)=ν◦e(u). Definition 3.1. We call Eγ an envelope (and e a pre-envelope) for the one-parameter family of Legendre curves (γ,ν), when the following conditions are satisfied: The function λ is non-constant on any non-trivial subinterval of U. (The Variability Condition.) For all u, the curve Eγ is tangent at u to the curve γ(t,λ) at the parameter (t(u),λ(u)), meaning that the tangent vectors Eγ′(u)=(dE/du)(u) and μ(t(u),λ(u)) are linearly dependent. (The Tangency Condition.) Note that the tangency condition is equivalent to the condition Eγ′(u)·ν(t(u),λ(u))=Eγ′(u)·Eν(u)=0 for all u∈U. Therefore, we have the following Proposition. Proposition 3.2. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N). Suppose that e:U→I×Λ,e(u)=(t(u),λ(u))is a pre-envelope and Eγ=γ◦e:U→S2is an envelope of (γ,ν). Then Eγis a frontal. More precisely, (Eγ,Eν):U→Δis a Legendre curve with the curvature mEγ(u)=t′(u)m(e(u))+λ′(u)M(e(u)),nEγ(u)=t′(u)n(e(u))+λ′(u)N(e(u)). Proof By definition, Eγ(u)·Eν(u)=γ(e(u))·ν(e(u))=0 for all u∈U. Since Eγ is an envelope, Eγ′(u)·Eν(u)=0 for all u∈U. It follows that (Eγ,Eν):U→Δ is a Legendre curve. Then mEγ(u)=Eγ′(u)·μ(e(u))=(t′(u)γt(e(u))+λ′(u)γλ(e(u)))·μ(e(u))=t′(u)m(e(u))+λ′(u)M(e(u)),nEγ(u)=Eν′(u)·μ(e(u))=(t′(u)νt(e(u))+λ′(u)νλ(e(u)))·μ(e(u))=t′(u)n(e(u))+λ′(u)N(e(u)). □ By Proposition 3.2, the envelope of the dual is the dual of the envelope of the one-parameter family of Legendre curves. We have the envelope theorem as follows: Theorem 3.3. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves and let e:U→I×Λbe a smooth curve satisfying the variability condition. Then eis a pre-envelope of (γ,ν) (and Eγis an envelope) if and only if γλ(e(u))·ν(e(u))=0for all u∈U. Proof Suppose that e is a pre-envelope of (γ,ν). By the tangency condition, there exists a function c(u)∈R such that Eγ′(u)=c(u)μ(e(u)). By differentiating Eγ(u)=γ◦e(u), we have Eγ′(u)=t′(u)γt(e(u))+λ′(u)γλ(e(u)). It follows from γt(t,λ)=m(t,λ)μ(t,λ) that (t′(u)m(e(u))−c(u))μ(e(u))+λ′(u)γλ(e(u))=0. Then we have λ′(u)γλ(e(u))·ν(e(u))=0. By the variability condition, we have γλ(e(u))·ν(e(u))=0 for all u∈U. Conversely, suppose that γλ(e(u))·ν(e(u))=0 for all u∈U. Since Eγ′(u)·ν(e(u))=(t′(u)γt(e(u))+λ′(u)γλ(e(u)))·ν(e(u))=0, e is a pre-envelope of (γ,ν).□ By using the curvature of the one-parameter family of Legendre curves, we have the corollary of Theorem 3.3. Corollary 3.4. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N)and let e:U→I×Λbe a smooth curve satisfying the variability condition. Then e:U→I×Λis a pre-envelope of (γ,ν) (and Eγis an envelope) if and only if L(e(u))=0for all u∈U. Proposition 3.5. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves. Suppose that e:U→I×Λis a pre-envelope and Eγis an envelope of (γ,ν). Then e:U→I×Λis also a pre-envelope of (−γ,ν), (γ,−ν)and (ν,γ). Moreover, −Eγis an envelope of (−γ,ν), Eγis an envelope of (γ,−ν)and Eνis an envelope of (ν,γ). Proof Since e:U→I×Λ is a pre-envelope, we have γλ(e(u))·ν(e(u))=0 for all u∈U. It follows that −γλ(e(u))·ν(e(u))=0, γλ(e(u))·(−ν(e(u)))=0 and νλ(e(u))·γ(e(u))=0 for all u∈U. Thus e:U→I×Λ is also a pre-envelope of (−γ,ν), (γ,−ν) and (ν,γ). It follows that −Eγ=−γ◦e, Eγ=γ◦e and Eν=ν◦e are envelopes of (−γ,ν), (γ,−ν) and (ν,γ), respectively.□ Definition 3.6. We say that a map Φ:I˜×Λ˜→I×Λ is a one-parameter family of parameter change if Φ is a diffeomorphism and given by the form Φ(s,k)=(ϕ(s,k),φ(k)). Proposition 3.7. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N). Suppose that Φ:I˜×Λ˜→I×Λis a one-parameter family of parameter change. Then (γ˜,ν˜)=(γ◦Φ,ν◦Φ):I˜×Λ˜→Δis also a one-parameter family of Legendre curves with the curvature m˜(s,k)=m(Φ(s,k))ϕs(s,k),n˜(s,k)=n(Φ(s,k))ϕs(s,k),L˜(s,k)=L(Φ(s,k))φ′(k),M˜(s,k)=m(Φ(s,k))ϕk(s,k)+M(Φ(s,k))φ′(k),N˜(s,k)=n(Φ(s,k))ϕk(s,k)+N(Φ(s,k))φ′(k).If e:U→I×Λis a pre-envelope, Eγis an envelope, then Φ−1◦e:U→I˜×Λ˜is a pre-envelope and Eγis also an envelope of (γ˜,ν˜). Proof Since γ˜s(s,k)=γt(Φ(s,k))ϕs(s,k) and γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ, we have γ˜s(s,k)·ν˜(s,k)=0forall(s,k)∈I˜×Λ˜. Therefore, (γ˜,ν˜) is a one-parameter family of Legendre curves. Then we have m˜(s,k)=γ˜s(s,k)·μ˜(s,k)=γt(Φ(s,k))ϕs(s,k)·μ(Φ(s,k))=m(Φ(s,k))ϕs(s,k),n˜(s,k)=ν˜s(s,k)·μ˜(s,k)=νt(Φ(s,k))ϕs(s,k)·μ(Φ(s,k))=n(Φ(s,k))ϕs(s,k),L˜(s,k)=γ˜k(s,k)·ν˜(s,k)=(γt(Φ(s,k))ϕk(s,k)+γλ(Φ(s,k))φ′(k))·ν(Φ(s,k))=L(Φ(s,k))φ′(k),M˜(s,k)=γ˜k(s,k)·μ˜(s,k)=(γt(Φ(s,k))ϕk(s,k)+γλ(Φ(s,k))φ′(k))·μ(Φ(s,k))=m(Φ(s,k))ϕk(s,k)+M(Φ(s,k))φ′(k),N˜(s,k)=ν˜k(s,k)·μ˜(s,k)=(νt(Φ(s,k))ϕk(s,k)+νλ(Φ(s,k))φ′(k))·μ(Φ(s,k))=n(Φ(s,k))ϕk(s,k)+N(Φ(s,k))φ′(k). By the form of the diffeomorphism Φ(s,k)=(ϕ(s,k),φ(k)),Φ−1:I×Λ→I˜×Λ˜ is given by the form Φ−1(t,λ)=(ψ(t,λ),φ−1(λ)). It follows that Φ−1◦e(u)=(ψ(t(u),λ(u)),φ−1(λ(u))). Since (d/du)φ−1(λ(u))=φλ−1(λ(u))λ′(u), the variability condition holds. Moreover, we have γ˜k(s,k)·ν˜(s,k)=(γt(Φ(s,k))ϕk(s,k)+γλ(Φ(s,k))φ′(k))·ν(Φ(s,k))=φ′(k)γλ(Φ(s,k))·ν(Φ(s,k)). It follows that γ˜k(φ−1◦e(u))·ν˜(φ−1◦e(u))=φ′(φ−1(λ(u)))γλ(e(u))·ν(e(u))=0forallu∈U. By Theorem 3.3, Φ−1◦e is a pre-envelope of (γ˜,ν˜). Therefore, γ˜◦Φ−1◦e=γ◦Φ◦Φ−1◦e=γ◦e=Eγ is also an envelope of (γ˜,ν˜).□ Definition 3.8. Let (γ,ν):I×Λ→Δ be a one-parameter family of Legendre curves, we define the parallel curves of the one-parameter family of Legendre curves by γθ(t,λ)=cosθγ(t,λ)−sinθν(t,λ),νθ(t,λ)=sinθγ(t,λ)+cosθν(t,λ). (3.1) Proposition 3.9. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N), then (γθ,νθ):I×Λ→Δis also a one-parameter family of Legendre curves with the curvature mθ(t,λ)=cosθm(t,λ)−sinθn(t,λ),nθ(t,λ)=sinθm(t,λ)+cosθn(t,λ),Lθ(t,λ)=L(t,λ),Mθ(t,λ)=cosθM(t,λ)−sinθN(t,λ),Nθ(t,λ)=sinθM(t,λ)+cosθN(t,λ).If e:U→I×Λis a pre-envelope of (γ,ν), then e:U→I×Λis also a pre-envelope of (γθ,νθ). Moreover, we have (Eγθ(u),Eνθ(u))=(Eγθ(u),Eνθ(u))for all u∈U, where (Eγθ,Eνθ)is parallel curve of (Eγ,Eν)and (Eγθ,Eγθ)=(γθ◦e,νθ◦e). Proof Since (γ,ν):I×Λ→Δ is a one-parameter family of Legendre curves, γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. It follows that γtθ(t,λ)·νθ(t,λ)=0 for all (t,λ)∈I×Λ. Thus (γθ,νθ):I×Λ→Δ is also a one-parameter family of Legendre curves. By definition, μθ(t,λ)=γθ(t,λ)×νθ(t,λ)=(cosθγ(t,λ)−sinθν(t,λ))×(sinθγ(t,λ)+cosθν(t,λ))=μ(t,λ). Therefore, we have mθ(t,λ)=γtθ(t,λ)·μθ(t,λ)=cosθm(t,λ)−sinθn(t,λ),nθ(t,λ)=νtθ(t,λ)·μθ(t,λ)=sinθm(t,λ)+cosθn(t,λ),Lθ(t,λ)=γλθ(t,λ)·νθ(t,λ)=L(t,λ),Mθ(t,λ)=γλθ(t,λ)·μθ(t,λ)=cosθM(t,λ)−sinθN(t,λ),Nθ(t,λ)=γλθ(t,λ)·μθ(t,λ)=sinθM(t,λ)+cosθN(t,λ). Since e:U→I×Λ is a pre-envelope of (γ,ν), we have γλ(e(u))·ν(e(u))=0 for all u∈U. It follows that γλθ(e(u))·νθ(e(u))=0 for all u∈U. Thus, e:U→I×Λ is a pre-envelope of (γθ,νθ) by Theorem 3.3. Moreover, Eγθ(u)=cosθEγ(u)−sinθEν(u)=cosθγ◦e(u)−sinθν◦e(u),Eνθ(u)=sinθEγ(u)+cosθEν(u)=sinθγ◦e(u)+cosθν◦e(u),Eγθ(u)=γθ◦e(u)=(cosθγ−sinθν)◦e(u)=cosθγ◦e(u)−sinθν◦e(u),Eνθ(u)=νθ◦e(u)=(sinθγ+cosθν)◦e(u)=sinθγ◦e(u)+cosθν◦e(u). Thus, we have (Eγθ(u),Eνθ(u))=(Eγθ(u),Eνθ(u)) for all u∈U.□ In [10], the evolute of the spherical Legendre curve is defined. Now, we define the evolute of a one-parameter family of Legendre curves in the unit sphere bundle. Definition 3.10. Let (γ,ν):I×Λ→Δ be a one-parameter family of Legendre curves with the curvature (m,n,L,M,N). Suppose that (m(t,λ),n(t,λ))≠(0,0) for all (t,λ)∈I×Λ. We define the evolute of the one-parameter family of Legendre curves (γ,ν) by E(γ)(t,λ)=±n(t,λ)m2(t,λ)+n2(t,λ)γ(t,λ)∓m(t,λ)m2(t,λ)+n2(t,λ)ν(t,λ). (3.2) Proposition 3.11. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N). Suppose that (m(t,λ),n(t,λ))≠(0,0), for all (t,λ)∈I×Λ. Then the evolute (E(γ),μ):I×Λ→Δof (γ,ν)is also a one-parameter family of Legendre curves with the curvature (mE,nE,LE,ME,NE), where mE(t,λ)=mtn−mntm2+n2(t,λ),nE(t,λ)=±m2+n2(t,λ),LE(t,λ)=±nM−mNm2+n2(t,λ),ME(t,λ)=mλn−mnλ−L(m2+n2)m2+n2(t,λ),NE(t,λ)=±mM+nNm2+n2(t,λ).If e:U→I×Λis a pre-envelope of (γ,ν)and (nM−mN)◦(e(u))=0for all u∈U, then e:U→I×Λis also a pre-envelope of (E(γ),μ). Moreover, we have EE(γ)(u)=EEγ(u)for all u∈U, where EE(γ)is the envelope of E(γ), EEγis the evolute of Eγ. Proof Since (γ,ν):I×Λ→Δ is a one-parameter family of Legendre curves and {γ(t,λ),ν(t,λ),μ(t,λ)} is a moving frame along the frontal γ(t,λ), we have E(γ)(t,λ)·μ(t,λ)=0,Et(γ)(t,λ)·μ(t,λ)=0 for all (t,λ)∈I×Λ. Therefore, (E(γ),μ):I×Λ→Δ is a one-parameter family of Legendre curves. We denote (γE,νE)=(E(γ),μ). By definition μE(t,λ)=γE(t,λ)×νE(t,λ)=∓m(t,λ)m2(t,λ)+n2(t,λ)γ(t,λ)∓n(t,λ)m2(t,λ)+n2(t,λ)ν(t,λ). Thus, mE(t,λ)=γEt(t,λ)·μE(t,λ)=mtn−mntm2+n2(t,λ),nE(t,λ)=νEt(t,λ)·μE(t,λ)=±m2+n2(t,λ),LE(t,λ)=γEλ(t,λ)·νE(t,λ)=±nM−mNm2+n2(t,λ),ME(t,λ)=γEλ(t,λ)·μE(t,λ)=mλn−mnλ−L(m2+n2)m2+n2(t,λ),NE(t,λ)=νEλ(t,λ)·μE(t,λ)=±mM+nNm2+n2(t,λ). Since (nM−mN)◦(e(u))=0 for all u∈U, we have E(γ)λ(e(u))·μ(e(u))=±nM−mNm2+n2◦e(u)=0. By Theorem 3.3, e:U→I×Λ is a pre-envelope of (E(γ),μ). The envelope of E(γ) is given by EE(γ)(u)=E(γ)◦e(u)=(±mm2+n2γ∓nm2+n2ν)◦e(u)=±mm2+n2(e(u))γ(e(u))∓nm2+n2(e(u))ν(e(u)). On the other hand, by Proposition 3.2, the evolute of Eγ is given by EEγ(u)=±mEγ(u)mEγ2(u)+nEγ2(u)Eγ(u)∓nEγ(u)mEγ2(u)+nEγ2(u)Eν(u)=±t′m+λ′M(t′m+λ′M)2+(t′n+λ′N)2(e(u))γ(e(u))∓t′n+λ′N(t′m+λ′M)2+(t′n+λ′N)2(e(u))ν(e(u)). Since (nM−mN)(e(u))=0 for all u∈U, we have (t′m+λ′M)2(m2+n2)(e(u))=m2((t′m+λ′M)2+(t′n+λ′N)2)(e(u)),(t′n+λ′N)2(m2+n2)(e(u))=n2((t′m+λ′M)2+(t′n+λ′N)2)(e(u)). Then t′m+λ′M(t′m+λ′M)2+(t′n+λ′N)2(e(u))=mm2+n2(e(u)),t′n+λ′N(t′m+λ′M)2+(t′n+λ′N)2(e(u))=nm2+n2(e(u)). Thus, we have EE(γ)(u)=EEγ(u) for all u∈U.□ 4. Relationships among envelopes of Legendre curves in the spherical bundle over the unit sphere and the unit tangent bundle over the Euclidean plane We first recall the definition of the envelope of a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane. For more detailed descriptions, see [11]. Let (γ,ν):I×Λ→R2×S1 be a smooth mapping. We say that (γ,ν) is a one-parameter family of Legendre curves if γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. We denote J(a)=(−a2,a1) the anticlockwise rotation by π/2 of a vector a=(a1,a2). We define μ(t,λ)=J(ν(t,λ)). Since {ν(t,λ),μ(t,λ)} is a moving frame along γ(t,λ) on R2, we have the Frenet type formula: (νt(t,λ)μt(t,λ))=(0ℓ(t,λ)−ℓ(t,λ)0)(ν(t,λ)μ(t,λ)),(νλ(t,λ)μλ(t,λ))=(0m(t,λ)−m(t,λ)0)(ν(t,λ)μ(t,λ)),γt(t,λ)=β(t,λ)μ(t,λ), where ℓ(t,λ)=νt(t,λ)·μ(t,λ), m(t,λ)=νλ(t,λ)·μ(t,λ) and β(t,λ)=γt(t,λ)·μ(t,λ). By the integrability condition νtλ(t,λ)=νλt(t,λ), ℓ and m satisfy the condition ℓλ(t,λ)=mt(t,λ) for all (t,λ)∈I×Λ. We call the triple (ℓ,m,β) are the curvature of the one-parameter family of Legendre curves (γ,ν). Let (γ,ν):I×Λ→R2×S1 be a one-parameter family of Legendre curves with the curvature (ℓ,m,β) and let e:U→I×Λ, e(u)=(t(u),λ(u)) be a smooth curve, where U is an interval of R. We denote Eγ=γ◦e:U→R2, Eγ(u)=γ◦e(u) and Eν=ν◦e:U→R2, Eν(u)=ν◦e(u). We call Eγ an envelope (and e a pre-envelope) for the one-parameter family of Legendre curves (γ,ν):I×Λ→R2×S1, when the following conditions satisfy: The function λ is non-constant on any non-trivial subinterval of U. (The Variability Condition.) For all u, the curve Eγ is tangent at u to the curve γ(t,λ) at the parameter (t(u),λ(u)), meaning that the tangent vectors Eγ′(u)=(dE/du)(u) and μ(t(u),λ(u)) are linearly dependent. (The Tangency Condition.) We consider relationships among envelopes of Legendre curves in the spherical bundle over the unit sphere and the unit tangent bundle over the Euclidean plane. We denote a hemisphere S+={(x,y,z)∈S2∣z>0}. Now we consider the central projection ϕ:S+→R2 by ϕ(x,y,z)=(xz,yz). Proposition 4.1. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N)and γ(I×Λ)⊂S+. We denote γ(t,λ)=(x(t,λ),y(t,λ),z(t,λ)), ν(t,λ)=(a(t,λ),b(t,λ),c(t,λ)). Suppose that (a(t,λ),b(t,λ))≠(0,0)for all (t,λ)∈I×Λ. Then (γ˜,ν˜):I×Λ→R2×S1is a one-parameter family of Legendre curves with the curvature (ℓ˜,m˜,β˜), where γ˜(t,λ)=ϕ◦γ(t,λ)=(x(t,λ)z(t,λ),y(t,λ)z(t,λ)),ν˜(t,λ)=1a2(t,λ)+b2(t,λ)(a(t,λ),b(t,λ)),ℓ˜(t,λ)=nza2+b2(t,λ),m˜(t,λ)=Nza2+b2(t,λ),β˜(t,λ)=mz2+(xb−ya)ztz2a2+b2(t,λ). Proof Since (γ,ν):I×Λ→Δ is a one-parameter family of Legendre curves in the unit spherical bundle, we have γ(t,λ)·ν(t,λ)=0,γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. It follows that x(t,λ)at(t,λ)+y(t,λ)bt(t,λ)+z(t,λ)ct(t,λ)=0. By definition, we have μ(t,λ)=γ(t,λ)×ν(t,λ)=(y(t,λ)c(t,λ)−z(t,λ)b(t,λ),z(t,λ)a(t,λ)−x(t,λ)c(t,λ),x(t,λ)b(t,λ)−y(t,λ)a(t,λ)). By a direct calculation, we have m(t,λ)=γt(t,λ)·μ(t,λ)=−xtb+ytaz(t,λ),n(t,λ)=νt(t,λ)·μ(t,λ)=−atb+btaz(t,λ),N(t,λ)=νλ(t,λ)·μ(t,λ)=−aλb+bλaz(t,λ). By the assumption (a(t,λ),b(t,λ))≠(0,0), ν˜:I×Λ→S1 is a smooth mapping. Moreover, we have γ˜t(t,λ)=(xt(t,λ)z(t,λ)−x(t,λ)zt(t,λ),yt(t,λ)z(t,λ)−y(t,λ)zt(t,λ))/z2(t,λ)andγ˜t(t,λ)·ν˜(t,λ)=0. Therefore, (γ˜,ν˜):I×Λ→R2×S1 is a one-parameter family of Legendre curves. By definition, we have μ˜(t,λ)=J(ν˜(t,λ))=(−b(t,λ),a(t,λ))/a2(t,λ)+b2(t,λ)and the curvature ℓ˜(t,λ)=ν˜t(t,λ)·μ˜(t,λ)=−atb+btaa2+b2(t,λ)=nza2+b2(t,λ),m˜(t,λ)=ν˜λ(t,λ)·μ˜(t,λ)=−aλb+bλaa2+b2(t,λ)=Nza2+b2(t,λ),β˜(t,λ)=γ˜t(t,λ)·μ˜(t,λ)=(−xtb+yta)z+(xb−ya)ztz2a2+b2(t,λ)=mz2+(xb−ya)ztz2a2+b2(t,λ). □ Proposition 4.2. Under the same assumptions in Proposition4.1, suppose that e:U→I×Λis a pre-envelope of (γ,ν)and Eγ:U→S2is an envelope. Then e:U→I×Λis also a pre-envelope of (γ˜,ν˜):I×Λ→R2×S1. Moreover, we have Eγ˜(u)=E˜γ(u)for all u∈U, where Eγ˜=γ˜◦eand E˜γ=ϕ◦Eγ. Proof Since (γ,ν):I×Λ→Δ is a one-parameter family of Legendre curves in the unit spherical bundle and e:U→I×Λ is a pre-envelope of (γ,ν), we have γ(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ, γλ(e(u))·ν(e(u))=0 for all u∈U. It follows that (a(t,λ)(xλ(t,λ)z(t,λ)−x(t,λ)zλ(t,λ))+b(t,λ)(yλ(t,λ)z(t,λ)−y(t,λ)zλ(t,λ)))◦e(u)=0. Then we have γ˜λ(e(u))·ν˜(e(u))=0 for all u∈U. Therefore, e:U→I×Λ is a pre-envelope of (γ˜,ν˜) (cf. [11]). Moreover, we have Eγ˜(u)=γ˜◦e(u)=ϕ◦γ◦e(u)=ϕ(Eγ(u))=E˜γ(u) for all u∈U.□ Conversely, we have the following results. Proposition 4.3. Let (γ˜,ν˜):I×Λ→R2×S1be a one-parameter family of Legendre curves with the curvature (ℓ˜,m˜,β˜). We denote γ˜(t,λ)=(x(t,λ),y(t,λ)), ν˜(t,λ)=(a(t,λ),b(t,λ)). Then (γ,ν):I×Λ→Δ⊂S+×S2is a one-parameter family of Legendre curves in the unit spherical bundle with the curvature (m,n,L,M,N), where γ(t,λ)=ϕ−1◦γ˜(t,λ)=(x,y,1)1+x2+y2(t,λ),ν(t,λ)=(a,b,−xa−yb)1+(xa+yb)2(t,λ),m(t,λ)=β˜+(ytx−xty)(xa+yb)(1+x2+y2)1+(xa+yb)2(t,λ),n(t,λ)=ℓ˜1+x2+y21+(xa+yb)2(t,λ),L(t,λ)=xλa+yλb1+x2+y21+(xa+yb)2(t,λ),M(t,λ)=(yλx−xλy)(xa+yb)+yλa−xλb(1+x2+y2)1+(xa+yb)2(t,λ),N(t,λ)=m˜(1+x2+y2)+(xλa+yλb)(xb−ya)1+x2+y2(1+(xa+yb)2)(t,λ). Proof Since (γ˜,ν˜):I×Λ→R2×S1 is a one-parameter family of Legendre curves, then we have γ˜t(t,λ)·ν˜(t,λ)=(xta+ytb)(t,λ)=0forall(t,λ)∈I×Λ. By the definition, μ˜(t,λ)=J(ν˜(t,λ))=(−b(t,λ),a(t,λ)). It follows that ℓ˜(t,λ)=(−atb+abt)(t,λ),β˜(t,λ)=(−xtb+yta)(t,λ),m˜(t,λ)=(−aλb+abλ)(t,λ). By a direct calculation, we have γt(t,λ)=11+x2+y2((1+y2)xt−xyyt,(1+x2)yt−xxty,−xxt−yyt)(t,λ). Then γ(t,λ)·ν(t,λ)=0 and γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. Therefore, (γ,ν):I×Λ→Δ is a one-parameter family of Legendre curves in the unit spherical bundle. By definition, μ(t,λ) is given by μ(t,λ)=γ(t,λ)×ν(t,λ)=(−xya−(1+y2)b,(1+x2)a+xyb,xb−ya)(1+x2+y2)(1+(xa+yb)2)(t,λ). By a direct calculation, we have the curvature (m,n,L,M,N) of (γ,ν).□ Proposition 4.4. Under the same assumptions in Proposition4.3, suppose that e:U→I×Λis a pre-envelope of (γ˜,ν˜)and Eγ˜:U→R2is an envelope. Then e:U→I×Λis also a pre-envelope of (γ,ν):I×Λ→Δ⊂S+×S2. Moreover, we have ϕ−1◦Eγ˜(u)=Eγ(u)for all u∈U. Proof Since e:U→I×Λ is a pre-envelope of (γ˜,ν˜), we have γ˜λ(e(u))·ν˜(e(u))=0 for all u∈U (cf. [11]). It follows that (xλ(t,λ)·a(t,λ)+yλ(t,λ)·b(t,λ))◦e(u)=0. By a direct calculation, we have γλ(e(u))·ν(e(u))=0 for all u∈U. By Theorem 3.3, e:U→I×Λ is a pre-envelope of (γ,ν). Moreover, we have ϕ−1◦Eγ˜(u)=ϕ−1◦γ˜◦e(u)=γ◦e(u)=Eγ(u) for all u∈U.□ Also, we consider the canonical projection π:S+→D2⊂R2 by π(x,y,z)=(x,y), where D2={(x,y)∈R2∣x2+y2<1}. Proposition 4.5. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves in the unit spherical bundle with the curvature (m,n,L,M,N)and γ(I×Λ)⊂S+. We denote γ(t,λ)=(x(t,λ),y(t,λ),z(t,λ))and ν(t,λ)=(a(t,λ),b(t,λ),c(t,λ)). Then (γ˜,ν˜):I×Λ→D2×S1is a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane with the curvature (ℓ˜,m˜,β˜), where γ˜(t,λ)=π◦γ(t,λ)=(x(t,λ),y(t,λ)),ν˜(t,λ)=(za−xc,zb−yc)(za−xc)2+(zb−yc)2(t,λ),ℓ˜(t,λ)=nz+xyt−xty(za−xc)2+(zb−yc)2(t,λ),m˜(t,λ)=Nz+xyλ−xλy(za−xc)2+(zb−yc)2(t,λ),β˜(t,λ)=m−(xb−ya)zt(za−xc)2+(zb−yc)2(t,λ). Proof If z(t,λ)a(t,λ)−x(t,λ)c(t,λ)=0 and z(t,λ)b(t,λ)−y(t,λ)c(t,λ)=0, then a(t,λ)=x(t,λ)c(t,λ)/z(t,λ)andb(t,λ)=y(t,λ)c(t,λ)/z(t,λ). Since ν(t,λ)∈S2, we have c2(t,λ)=z2(t,λ) and hence c(t,λ)=±z(t,λ). It follows that a(t,λ)=±x(t,λ)andb(t,λ)=±y(t,λ). This contradicts the fact that γ(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. Hence, ν˜:I×Λ→S1 is a smooth mapping. By (xta+ytb+ztc)(t,λ)=0 and (xtx+yty+ztz)(t,λ)=0, we have γ˜t(t,λ)·ν˜(t,λ)=0forall(t,λ)∈I×Λ. Therefore, (γ˜,ν˜):I×Λ→D2×S1 is a one-parameter family of Legendre curves. By a similar calculation as in Proposition 4.1, we have the curvature (ℓ˜,m˜,β˜) of (γ˜,ν˜).□ Proposition 4.6. Under the same assumptions in Proposition4.5, suppose that e:U→I×Λis a pre-envelope of (γ,ν)and Eγ:U→S2is an envelope. Then e:U→I×Λis also a pre-envelope of (γ˜,ν˜):I×Λ→D2×S1. Moreover, we have Eγ˜(u)=E˜γ(u)for all u∈U, where Eγ˜=γ˜◦eand E˜γ=π◦Eγ. Proof Since e:U→I×Λ is a pre-envelope of (γ,ν), we have γλ(e(u))·ν(e(u))=0 for all u∈U. It follows that (xλ(t,λ)(a(t,λ)z(t,λ)−x(t,λ)c(t,λ))+yλ(t,λ)(b(t,λ)z(t,λ)−y(t,λ)c(t,λ)))◦e(u)=0. By a direct calculation, we have γ˜λ(e(u))·ν˜(e(u))=0 for all u∈U. Therefore, e:U→I×Λ is a pre-envelope of (γ˜,ν˜) (cf. [11]). Moreover, we have Eγ˜(u)=γ˜◦e(u)=π◦γ◦e(u)=π(Eγ(u))=E˜γ(u) for all u∈U.□ Conversely, we have the following results. Proposition 4.7. Let (γ˜,ν˜):I×Λ→D2×S1be a one-parameter family of Legendre curves with the curvature (ℓ˜,m˜,β˜). We denote γ˜(t,λ)=(x(t,λ),y(t,λ)),ν˜(t,λ)=(a(t,λ),b(t,λ)).Then we have (γ,ν):I×Λ→Δ⊂S+×S2is a one-parameter family of Legendre curves, where γ(t,λ)=π−1◦γ˜(t,λ)=(x(t,λ),y(t,λ)z(t,λ)),ν(t,λ)=11−(xa+yb)2(a−x(xa−yb),b−y(xa−yb),−z(xa+yb))(t,λ).Here we put z(t,λ)=1−x(t,λ)2−y(t,λ)2. Proof Since γ˜(t,λ)·γ˜(t,λ)<1 and ν˜(t,λ)·ν˜(t,λ)=1, we have x(t,λ)a(t,λ)+y(t,λ)b(t,λ)<1 for all (t,λ)∈I×Λ. Therefore, ν:I×Λ→S2 is a smooth mapping. By the same argument as in Proposition 4.5, we have ℓ˜(t,λ)=−at(t,λ)b(t,λ)+a(t,λ)bt(t,λ),β˜(t,λ)=−xt(t,λ)b(t,λ)+yt(t,λ)a(t,λ),m˜(t,λ)=−aλ(t,λ)b(t,λ)+a(t,λ)bλ(t,λ). Since γt(t,λ)=(xt(t,λ),yt(t,λ),−x(t,λ)xt(t,λ)−y(t,λ)yt(t,λ)z(t,λ)),x2(t,λ)+y2(t,λ)+z2(t,λ)=1, we have γ(t,λ)·ν(t,λ)=0 and γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. Therefore, (γ,ν):I×Λ→Δ⊂S+×S2 is a one-parameter family of Legendre curves.□ Remark 4.8. We can detect to the curvature (m,n,L,M,N) of (γ,ν) in Proposition 4.7. However, the description is a little bit long, we omit it. Proposition 4.9. Under the same assumptions in Proposition4.7, suppose that e:U→I×Λis a pre-envelope of (γ˜,ν˜)and Eγ˜:U→R2is an envelope. Then e:U→I×Λis also a pre-envelope of (γ,ν):I×Λ→Δ⊂S+×S2. Moreover, we have π−1◦Eγ˜(u)=Eγ(u) for all u∈U. Proof Since e:U→I×Λ is a pre-envelope of (γ˜,ν˜), we have γ˜λ(e(u))·ν˜(e(u))=0 for all u∈U (cf. [11]). It follows that (xλ(t,λ)·a(t,λ)+yλ(t,λ)·b(t,λ))◦e(u)=0. By a direct calculation, we have γλ(e(u))·ν(e(u))=0 for all u∈U. By Theorem 3.3, e:U→I×Λ is a pre-envelope of (γ,ν). Moreover, we have π−1◦Eγ˜(u)=π−1◦γ˜◦e(u)=γ◦e(u)=Eγ(u) for all u∈U.□ 5. Examples Example 5.1. Let (γ,ν):[0,2π)×[0,2π)→Δ, γ(t,θ)=(cosθ(34cost−14cos3t)−32sinθcost,34sint−14sin3t,32cosθcost+sinθ(34cost−14cos3t)),ν(t,θ)=(cosθ(−34sint−14sin3t)−32sinθsint,34cost+14cos3t,32cosθsint−sinθ(34sint+14sin3t)). Then (γ,ν) is a one-parameter family of Legendre curves. By definition μ(t,θ)=γ(t,θ)×ν(t,θ)=(−334cosθcos2t−38cosθcos2t+338cosθ−12sinθ,−3sintcost,−334sinθcos2t−38sinθcos2t+338sinθ+12cosθ). Then the curvature is given by m(t,θ)=332(96sintcos4t−54sintcos2t+9sintcostcos3t−33sin3tcos2t−20sint+12sin3t),n(t,θ)=38(6costcos2t+5cost−3cos3t),L(t,θ)=3sintcost,M(t,θ)=98(2cos2t−1)−18cos3t,N(t,θ)=34sintcos2t+116sin3t−1516sint. If we take e:[0,2π)→[0,2π)×[0,2π),e(u)=(0,u),(π/2,u),(π,u),(3π/2,u), then we got L(e(u))=0 for all u∈[0,2π). Thus, e are pre-envelopes of (γ,ν). Hence, the envelopes Eγ:[0,2π)→S2 are given by Eγ=(1/2cosu−3/2sinu,0,1/2sinu+3/2cosu),(0,1,0),(−1/2cosu+3/2sinu,0,−1/2sinu−3/2cosu),(0,−1,0), see Fig. 1. Example 5.2. Let n,m and k be natural numbers with m=k+n. We give a mapping (γ,ν):R×[0,2π)→Δ by γ(t,θ)=1t2m+t2n+1(cosθ−tnsinθ,sinθ+tncosθ,tm),ν(t,θ)=1k2t2m+m2t2k+n2(ktmcosθ+mtksinθ,ktmsinθ−mtkcosθ,n). Then (γ,ν):R×[0,2π)→Δ is a one-parameter family of Legendre curves. By definition, μ(t,θ)=γ(t,θ)×ν(t,θ)=1t2m+t2n+1k2t2m+m2t2k+n2(tn(mt2k+n)cosθ+(−kt2m+n)sinθ,tn(mt2k+n)sinθ+(kt2m−n)cosθ,−kt2n+k−mtk). Then the curvature is given by m(t,θ)=−tn−1k2t2m+m2t2k+n2t2m+t2n+1,n(t,θ)=mnktk−1t2m+t2n+1k2t2m+m2t2k+n2,L(t,θ)=−mtk+ktm+nt2m+t2n+1k2t2m+m2t2k+n2,M(t,θ)=−nk2t2m+m2t2k+n2,N(t,θ)=tmt2m+t2n+1. If we take e:[0,2π)→R×[0,2π), e(u)=(0,u), then we got L(e(u))=0 for all u∈[0,2π). Thus, e is a pre-envelope of (γ,ν). Hence, the envelope Eγ:[0,2π)→S2 is given by Eγ(u)=(cosu,sinu,0). For example, when n=2,m=3,k=1, then we have γ(t,θ)=1t6+t4+1(cosθ−t2sinθ,sinθ+t2cosθ,t3),ν(t,θ)=1t6+9t2+4(t3cosθ+3tsinθ,t3sinθ−3tcosθ,2). By definition, μ(t,θ)=γ(t,θ)×ν(t,θ)=1t6+t4+1t6+9t2+4(t2(3t2+2)cosθ+(−t6+2)sinθ,t2(3t2+2)sinθ+(t6−2)cosθ,−t5−3t). Then the curvature is given by m(t,θ)=−tt6+9t2+4t6+t4+1,n(t,θ)=6t6+t4+1t6+9t2+4,L(t,θ)=−t5−3tt6+9t2+4t6+t4+1,M(t,θ)=−2t6+9t2+4,N(t,θ)=t3t6+t4+1. The envelope Eγ:[0,2π)→S2 is given by Eγ(u)=(cosu,sinu,0), see Fig. 2. The parallel curve of the envelope Eγ(u) for λ=π/6 is Eγπ6(u)=(3/2cosu,3/2sinu,−1/2). The parallel curves of (γ,ν) is γπ6(t,θ)=cosπ6γ(t,θ)−sinπ6ν(t,θ)=(3(cosθ−t2sinθ)2t6+t4+1−t3cosθ+3tsinθ2t6+9t2+4,3(sinθ+t2cosθ)2t6+t4+1−t3sinθ−3tcosθ2t6+9t2+4,3t32t6+t4+1−1t6+9t2+4). Then the envelope of these parallel curves is given by Eγπ6(u)=(3/2cosu,3/2sinu,−1/2). Thus, we have Eγπ6(u)=Eγπ6(u) for all u∈U. Figure 3 put these curves in the one sphere. Figure 1. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelopes. Figure 1. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelopes. Figure 2. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelope. Figure 2. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelope. Figure 3. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelope, parallel curves and the envelope of these parallel curves. Figure 3. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelope, parallel curves and the envelope of these parallel curves. Funding This work was supported by the National Natural Science Foundation of China (D.P.) (Grant no. 11671070), China Scholarship Council (Y.L.) (Grant no. 201606620072) and JSPS KAKENHI (M.T.) (Grant no. JP 17K05238). Conflict of interests The authors declare that there is no conflicts of interests in this work. Acknowledgements The authors would like to thank the referee for helpful comments to improve the original manuscript. References 1 V. I. Arnol’d , Singularities of Caustics and Wave Fronts. Mathematics and Its Applications Vol. 62 , Kluwer Academic Publishers , Dordrecht , 1990 . Google Scholar CrossRef Search ADS 2 V. I. Arnol’d , S. M. Gusein-Zade and A. N. Varchenko , Singularities of Differentiable Maps, 1986 . 3 J. W. Bruce and P. J. Giblin , What is an envelope? Math. Gaz. 65 ( 1981 ), 186 – 192 . Google Scholar CrossRef Search ADS 4 J. W. Bruce and P. J. Giblin , Curves and Singularities. A geometrical introduction to singularity theory , 2nd edn , Cambridge University Press , Cambridge , 1992 . Google Scholar CrossRef Search ADS 5 C. G. Gibson , Elementary Geometry of Differentiable Curves. An undergraduate introduction , Cambridge University Press , Cambridge , 2001 . Google Scholar CrossRef Search ADS 6 A. Gray , E. Abbena and S. Salamon , Modern differential geometry of curves and surfaces with Mathematica. Studies in Advanced Mathematics , 3rd edn , Chapman and Hall/CRC , Boca Raton, FL , 2006 . 7 G. Ishikawa , Singularities of Curves and Surfaces in Various Geometric Problems, CAS Lecture Notes 10, Exact Sciences. 2015 . 8 S. Izumiya , M. C. Romero-Fuster , M. A. S. Ruas and F. Tari , Differential Geometry from a Singularity Theory Viewpoint , World Scientific Pub. Co Inc , Hackensack , 2015 . Google Scholar CrossRef Search ADS 9 J. W. Rutter , Geometry of Curves, Chapman & Hall/CRC Mathematics , Chapman & Hall/CRC , Boca Raton, FL , 2000 . 10 M. Takahashi , Legendre Curves in the Unit Spherical Bundle over the Unit Sphere and Evolutes, Real and Complex Singularities, 337–355, Contemp. Math., 675, Amer. Math. Soc., Providence, RI, 2016 . 11 M. Takahashi , Envelopes of Legendre curves in the unit tangent bundle over the Euclidean plane , Results Math. 71 ( 2017 ), 1473 – 1489 . DOI:10.1007/s00025-016-0619-7 . Google Scholar CrossRef Search ADS © The Author 2017. Published by Oxford University Press. All rights reserved. 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Envelopes of legendre curves in the unit spherical bundle over the unit sphere

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Abstract

Abstract In this paper, we introduce a one-parameter family of Legendre curves in the unit spherical bundle over the unit sphere and the curvature. We give the existence and uniqueness theorems for one-parameter families of spherical Legendre curves by using the curvatures. Then we define an envelope for the one-parameter family of Legendre curves in the unit spherical bundle. We also consider the parallel curves and evolutes of one-parameter families of Legendre curves in the unit spherical bundle and their envelopes. Moreover, we give relationships among one-parameter families of Legendre curves in the unit spherical bundle over the unit sphere and one-parameter families of Legendre curves in the unit tangent bundle over the Euclidean plane. 1. Introduction An envelope of a family of curves in the Euclidean plane is a curve that is tangent to each member of the family at some point. If the curves are regular, then the tangent is well-defined (cf. [3–6, 9]). On the other hand, for singular plane curves, the classical definitions of envelopes are vague. In [11], the third author clarified the definition of the envelope for a family of singular curves in the unit tangent bundle over the Euclidean plane. This idea can be generalized to an envelope of a family of singular spherical curves. In [10], the third author gave a definition of Legendre curves in the unit spherical bundle, and established a moving frame of spherical Legendre curves. However, to the best of the authors' knowledge, no literature exists regarding the envelope of a family of singular curves in the unit sphere. In this paper, we use these approach and techniques to give the definition and investigate properties of envelopes for families of Legendre curves in the unit spherical bundle over the unit sphere. For basic results on the singularity theory, see [1, 2, 4, 7, 8]. In Section 2, we consider one-parameter families of Legendre curves in the unit spherical bundle and the curvatures. We give the existence and uniqueness theorems for one-parameter families of Legendre curves by using the curvatures. In Section 3, we define an envelope of a one-parameter family of Legendre curves in the unit spherical bundle. We obtained that the envelope is also a Legendre curve. The envelope of the dual is the dual of the envelope of the one-parameter family of Legendre curves. We also give the definitions of parallel curves and evolutes of one-parameter families of Legendre curves in the unit sphere bundle. We found that the parallel curves and evolutes are also one-parameter families of Legendre curves in the unit sphere bundle, the evolutes if exists. Then we consider the envelopes of the parallel curves and evolutes. The envelope of parallel curves of a one-parameter family of Legendre curves is equal to the parallel curve of the envelope of the one-parameter family of Legendre curves. Under a condition, the envelope of the evolute of a one-parameter family of Legendre curves is equal to the evolute of the envelope of the one-parameter family of Legendre curves. In Section 4, we give relationships among one-parameter families of Legendre curves in the unit spherical bundle over the unit sphere and one-parameter families of Legendre curves in the unit tangent bundle over the Euclidean plane. In Section 5, we give two examples and some pictures to illustrate our results. All maps and manifolds considered here are differential of class C∞. 2. Legendre curves in the unit spherical bundle over the unit sphere We first recall some definitions and theorems of Legendre curves in the unit spherical bundle over the unit sphere. For more detailed descriptions, see [10]. Let R3 be the 3-dimensional Euclidean space equipped with the inner product a·b=a1b1+a2b2+a3b3, where a=(a1,a2,a3) and b=(b1,b2,b3)∈R3. The vector product is given by a×b=∣e1e2e3a1a2a3b1b2b3∣, where e1,e2,e3 are the canonical basis on R3. Let S2={(x,y,z)∈R3∣x2+y2+z2=1} be the unit sphere. We denote the set {(a,b)∈S2×S2∣a·b=0} by Δ (cf. [10]). Then Δ is a 3-dimensional smooth manifold. We say that (γ,ν):I→Δ⊂S2×S2 is a Legendre curve (or, spherical Legendre curve) if γ˙(t)·ν(t)=0 for all t∈I, that is, (γ,ν) is an integrable curve with respect to the canonical contact 1-form on Δ. We call γ a frontal and ν a dual of γ. Moreover, if (γ,ν) is a Legendre immersion, we call γ a front. We define μ(t)=γ(t)×ν(t). By definition, μ(t)∈S2, γ(t)·μ(t)=0 and ν(t)·μ(t)=0 for all t∈I. It follows that {γ(t),ν(t),μ(t)} is a moving frame along the frontal γ(t). Let (γ,ν):I→Δ be a Legendre curve. We have the Frenet type formula. (γ˙(t)ν˙(t)μ˙(t))=(00m(t)00n(t)−m(t)−n(t)0)(γ(t)ν(t)μ(t)), where m(t)=γ˙(t)·μ(t) and n(t)=ν˙(t)·μ(t). We say that the pair of the functions (m,n) is the curvature of the Legendre curve (γ,ν):I→Δ. Definition 2.1. Let (γ,ν),(γ˜,ν˜):I→Δ be Legendre curves. We say that (γ,ν) and (γ˜,ν˜) are congruent as Legendre curves if there exists a special orthogonal matrix A∈SO(3) such that γ˜(t)=A(γ(t)), ν˜(t)=A(ν(t)) for all t∈I. Then we have the following existence and uniqueness theorems in terms of the curvature of the Legendre curve [10]. Theorem 2.2. (The Existence Theorem of spherical Legendre curves). Let (m,n):I→R×Rbe a smooth mapping. There exists a Legendre curve (γ,ν):I→Δ, whose associated curvature is (m,n). Theorem 2.3. (The Uniqueness Theorem of spherical Legendre curves). Let (γ,ν)and (γ˜,ν˜):I→Δbe Legendre curves whose curvatures (m,n)and (m˜,n˜), respectively. Then (γ,ν)and (γ˜,ν˜)are congruent as Legendre curves if and only if (m,n)and (m˜,n˜)coincide. We consider one-parameter families of Legendre curves in the unit spherical bundle Δ⊂S2×S2. Let I and Λ be intervals of R. Definition 2.4. Let (γ,ν):I×Λ→Δ be a smooth mapping. We say that (γ,ν) is a one-parameter family of spherical Legendre curves if γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. By definition, (γ(·,λ),ν(·,λ)):I→Δ is a Legendre curve for each fixed parameter λ∈Λ. We define μ(t,λ)=γ(t,λ)×ν(t,λ). Then {γ(t,λ),ν(t,λ),μ(t,λ)} is a moving frame along the frontal γ(t,λ) on S2. We have the Frenet type formula: (γt(t,λ)νt(t,λ)μt(t,λ))=(00m(t,λ)00n(t,λ)−m(t,λ)−n(t,λ)0)(γ(t,λ)ν(t,λ)μ(t,λ)), (γλ(t,λ)νλ(t,λ)μλ(t,λ))=(0L(t,λ)M(t,λ)−L(t,λ)0N(t,λ)−M(t,λ)−N(t,λ)0)(γ(t,λ)ν(t,λ)μ(t,λ)), where m(t,λ)=γt(t,λ)·μ(t,λ),n(t,λ)=νt(t,λ)·μ(t,λ),L(t,λ)=γλ(t,λ)·ν(t,λ),M(t,λ)=γλ(t,λ)·μ(t,λ),N(t,λ)=νλ(t,λ)·μ(t,λ). We denote the matrices A(t,λ)=(00m(t,λ)00n(t,λ)−m(t,λ)−n(t,λ)0),B(t,λ)=(0L(t,λ)M(t,λ)−L(t,λ)0N(t,λ)−M(t,λ)−N(t,λ)0). By γtλ(t,λ)=γλt(t,λ),νtλ(t,λ)=νλt(t,λ) and μtλ(t,λ)=μλt(t,λ), we have the integrability condition Aλ(t,λ)+A(t,λ)B(t,λ)=Bt(t,λ)+B(t,λ)A(t,λ), that is, (2.1) for all (t,λ)∈I×Λ. We call the tuple (m,n,L,M,N) with the integrability condition (2.1) the curvature of the one-parameter family of Legendre curves. Remark 2.5. Let (γ,ν):I×Λ→Δ be a one-parameter family of Legendre curves with the curvature (m,n,L,M,N). Then, it is easy to check that (γ,−ν), (−γ,ν) and (ν,γ) are also one-parameter family of Legendre curves. The curvatures are (−m,n,−L,−M,N), (m,−n,−L,M,−N) and (−n,−m,−L,−N,−M), respectively. Definition 2.6. Let (γ,ν) and (γ˜,ν˜):I×Λ→Δ be one-parameter families of Legendre curves. We say that (γ,ν) and (γ˜,ν˜) are congruent as one-parameter families of Legendre curves if there exists a special orthogonal matrix A∈SO(3) such that γ˜(t,λ)=A(γ(t,λ)) and ν˜(t,λ)=A(ν(t,λ)) for all (t,λ)∈I×Λ. Then we have the following existence and uniqueness theorems for one-parameter families of Legendre curves. Theorem 2.7. (The Existence Theorem for one-parameter families of spherical Legendre curves). Let (m,n,L,M,N):I×Λ→R5be a smooth mapping with the integrability condition. There exists a one-parameter family of Legendre curves (γ,ν):I×Λ→Δ, whose associated curvature is (m,n,L,M,N). Proof Choose any fixed value t=t0,λ=λ0 of the parameter. We consider an initial value problem, Ft(t,λ)=A(t,λ)F(t,λ),Fλ(t,λ)=B(t,λ)F(t,λ),F(t0,λ0)=I3, where F(t,λ)∈M(3), A(t,λ),B(t,λ) as the above, M(3) is the set of 3×3 matrices and I3 is the identity matrix. Then we consider Ftλ=AλF+AFλ=AλF+ABF=(Aλ+AB)F,Fλt=BtF+BFt=BtF+BAF=(Bt+BA)F. By the integrability condition Aλ+AB=Bt+BA, we have Ftλ=Fλt. Since I×Λ is simply connected, there exists a solution F(t,λ). Therefore, there exists a one-parameter family of Legendre curves (γ,ν):I×Λ→Δ whose associated curvature is (m,n,L,M,N).□ Lemma 2.8. Let (γ,ν)and (γ˜,ν˜):I×Λ→Δbe one-parameter families of Legendre curves having equal curvature, that is, (m(t,λ),n(t,λ),L(t,λ),M(t,λ),N(t,λ))=(m˜(t,λ),n˜(t,λ),L˜(t,λ),M˜(t,λ),N˜(t,λ))for all (t,λ)∈I×Λ. If there exist two parameters t=t0,λ=λ0for which (γ(t0,λ0),ν(t0,λ0))=(γ˜(t0,λ0),ν˜(t0,λ0)), then (γ,ν)and (γ˜,ν˜)coincide. Proof Define a smooth function f:I×Λ→R by f(t,λ)=γ(t,λ)·γ˜(t,λ)+ν(t,λ)·ν˜(t,λ)+μ(t,λ)·μ˜(t,λ). Since (m(t,λ),n(t,λ),L(t,λ),M(t,λ),N(t,λ))=(m˜(t,λ),n˜(t,λ),L˜(t,λ),M˜(t,λ),N˜(t,λ)), we have ft(t,λ)=(γt·γ˜+γ·γ˜t+νt·ν˜+ν·ν˜t+μt·μ˜+μ·μ˜t)(t,λ)=((mμ)·γ˜+γ·(m˜μ˜)+(nμ)·ν˜+ν·(n˜μ˜)+(−mγ−nν))·μ˜+μ·(−m˜γ˜−n˜ν˜)(t,λ)=((m−m˜)μ·γ˜+(m˜−m)γ·μ˜+(n−n˜)μ·ν˜+(n˜−n)ν·μ˜)(t,λ)=0,fλ(t,λ)=(γλ·γ˜+γ·γ˜λ+νλ·ν˜+ν·ν˜λ+μλ·μ˜+μ·μ˜λ)(t,λ)=((Lν+Mμ)·γ˜+γ·(L˜ν˜+M˜μ˜)+(−Lγ+Nμ)·ν˜+ν·(−L˜γ˜+N˜μ˜))+(−Mγ−Nν)·μ˜+μ·(−M˜γ˜−N˜ν˜)(t,λ)=((L−L˜)ν·γ˜+(L˜−L)γ·ν˜+(M−M˜)μ·γ˜+(M˜−M)γ·μ˜)+(N−N˜)μ·ν˜+(N˜−N)ν·μ˜(t,λ)=0 for all (t,λ)∈I×Λ. It follows that f is constant. By γ(t0,λ0)=γ˜(t0,λ0) and ν(t0,λ0)=ν˜(t0,λ0), we have f(t0,λ0)=3 and the function f is constant with value 3. By the Cauchy–Schwarz inequality, we have γ(t,λ)·γ˜(t,λ)≤∣γ(t,λ)∣∣γ˜(t,λ)∣=1,ν(t,λ)·ν˜(t,λ)≤∣ν(t,λ)∣∣ν˜(t,λ)∣=1,μ(t,λ)·μ˜(t,λ)≤∣μ(t,λ)∣∣μ˜(t,λ)∣=1. If one of these inequalities is strict, the value of f(t,λ) would be <3. It follows that these inequalities are equalities, and we have γ(t,λ)·γ˜(t,λ)=1,ν(t,λ)·ν˜(t,λ)=1,μ(t,λ)·μ˜(t,λ)=1 for all (t,λ)∈I×Λ. Then we have ∣γ(t,λ)−γ˜(t,λ)∣2=∣ν(t,λ)−ν˜(t,λ)∣2=∣μ(t,λ)−μ˜(t,λ)∣2=0. It follows that γ(t,λ)=γ˜(t,λ),ν(t,λ)=ν˜(t,λ),μ(t,λ)=μ˜(t,λ)forall(t,λ)∈I×Λ. □ Theorem 2.9. (The Uniqueness Theorem for one-parameter families of Legendre curves). Let (γ,ν)and (γ˜,ν˜):I×Λ→Δbe one-parameter families of Legendre curves with the curvatures (m,n,L,M,N)and (m˜,n˜,L˜,M˜,N˜), respectively. Then (γ,ν)and (γ˜,ν˜)are congruent as one-parameter family of Legendre curves if and only if (m,n,L,M,N)and (m˜,n˜,L˜,M˜,N˜)coincide. Proof Suppose that (γ,ν) and (γ˜,ν˜) are congruent as one-parameter families of Legendre curves. By a direct calculation, we have γ˜t(t,λ)=∂∂t(A(γ(t,λ)))=A(γt(t,λ))=m(t,λ)A(μ(t,λ))=m(t,λ)μ˜(t,λ),ν˜t(t,λ)=∂∂t(A(ν(t,λ)))=A(νt(t,λ))=n(t,λ)A(μ(t,λ))=n(t,λ)μ˜(t,λ),γ˜λ(t,λ)=∂∂λ(A(γ(t,λ)))=A(γλ(t,λ))=L(t,λ)A(ν(t,λ))+M(t,λ)A(μ(t,λ))=L(t,λ)ν˜(t,λ)+M(t,λ)μ˜(t,λ),ν˜λ(t,λ)=∂∂λ(A(ν(t,λ)))=A(νλ(t,λ))=−L(t,λ)A(γ(t,λ))+N(t,λ)A(μ(t,λ))=−L(t,λ)γ˜(t,λ)+N(t,λ)μ˜(t,λ). Therefore, the curvatures (m,n,L,M,N) and (m˜,n˜,L˜,M˜,N˜) coincide. Conversely, suppose that (m,n,L,M,N) and (m˜,n˜,L˜,M˜,N˜) coincide. Let (t0,λ0)∈I×Λ be fixed. By using a congruence as one-parameter family of Legendre curves, we may assume γ(t0,λ0)=γ˜(t0,λ0) and ν(t0,λ0)=ν˜(t0,λ0). By Lemma 2.8, we have γ(t,λ)=γ˜(t,λ) and ν(t,λ)=ν˜(t,λ) for all (t,λ)∈I×Λ.□ 3. Envelopes of one-parameter families of Legendre curves in the unit spherical bundle Let (γ,ν):I×Λ→Δ be a one-parameter family of Legendre curves with the curvature (m,n,L,M,N) and let e:U→I×Λ, e(u)=(t(u),λ(u)) be a smooth curve, where U is an interval of R. We denote Eγ=γ◦e:U→S2, Eγ(u)=γ◦e(u) and Eν=ν◦e:U→S2, Eν(u)=ν◦e(u). Definition 3.1. We call Eγ an envelope (and e a pre-envelope) for the one-parameter family of Legendre curves (γ,ν), when the following conditions are satisfied: The function λ is non-constant on any non-trivial subinterval of U. (The Variability Condition.) For all u, the curve Eγ is tangent at u to the curve γ(t,λ) at the parameter (t(u),λ(u)), meaning that the tangent vectors Eγ′(u)=(dE/du)(u) and μ(t(u),λ(u)) are linearly dependent. (The Tangency Condition.) Note that the tangency condition is equivalent to the condition Eγ′(u)·ν(t(u),λ(u))=Eγ′(u)·Eν(u)=0 for all u∈U. Therefore, we have the following Proposition. Proposition 3.2. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N). Suppose that e:U→I×Λ,e(u)=(t(u),λ(u))is a pre-envelope and Eγ=γ◦e:U→S2is an envelope of (γ,ν). Then Eγis a frontal. More precisely, (Eγ,Eν):U→Δis a Legendre curve with the curvature mEγ(u)=t′(u)m(e(u))+λ′(u)M(e(u)),nEγ(u)=t′(u)n(e(u))+λ′(u)N(e(u)). Proof By definition, Eγ(u)·Eν(u)=γ(e(u))·ν(e(u))=0 for all u∈U. Since Eγ is an envelope, Eγ′(u)·Eν(u)=0 for all u∈U. It follows that (Eγ,Eν):U→Δ is a Legendre curve. Then mEγ(u)=Eγ′(u)·μ(e(u))=(t′(u)γt(e(u))+λ′(u)γλ(e(u)))·μ(e(u))=t′(u)m(e(u))+λ′(u)M(e(u)),nEγ(u)=Eν′(u)·μ(e(u))=(t′(u)νt(e(u))+λ′(u)νλ(e(u)))·μ(e(u))=t′(u)n(e(u))+λ′(u)N(e(u)). □ By Proposition 3.2, the envelope of the dual is the dual of the envelope of the one-parameter family of Legendre curves. We have the envelope theorem as follows: Theorem 3.3. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves and let e:U→I×Λbe a smooth curve satisfying the variability condition. Then eis a pre-envelope of (γ,ν) (and Eγis an envelope) if and only if γλ(e(u))·ν(e(u))=0for all u∈U. Proof Suppose that e is a pre-envelope of (γ,ν). By the tangency condition, there exists a function c(u)∈R such that Eγ′(u)=c(u)μ(e(u)). By differentiating Eγ(u)=γ◦e(u), we have Eγ′(u)=t′(u)γt(e(u))+λ′(u)γλ(e(u)). It follows from γt(t,λ)=m(t,λ)μ(t,λ) that (t′(u)m(e(u))−c(u))μ(e(u))+λ′(u)γλ(e(u))=0. Then we have λ′(u)γλ(e(u))·ν(e(u))=0. By the variability condition, we have γλ(e(u))·ν(e(u))=0 for all u∈U. Conversely, suppose that γλ(e(u))·ν(e(u))=0 for all u∈U. Since Eγ′(u)·ν(e(u))=(t′(u)γt(e(u))+λ′(u)γλ(e(u)))·ν(e(u))=0, e is a pre-envelope of (γ,ν).□ By using the curvature of the one-parameter family of Legendre curves, we have the corollary of Theorem 3.3. Corollary 3.4. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N)and let e:U→I×Λbe a smooth curve satisfying the variability condition. Then e:U→I×Λis a pre-envelope of (γ,ν) (and Eγis an envelope) if and only if L(e(u))=0for all u∈U. Proposition 3.5. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves. Suppose that e:U→I×Λis a pre-envelope and Eγis an envelope of (γ,ν). Then e:U→I×Λis also a pre-envelope of (−γ,ν), (γ,−ν)and (ν,γ). Moreover, −Eγis an envelope of (−γ,ν), Eγis an envelope of (γ,−ν)and Eνis an envelope of (ν,γ). Proof Since e:U→I×Λ is a pre-envelope, we have γλ(e(u))·ν(e(u))=0 for all u∈U. It follows that −γλ(e(u))·ν(e(u))=0, γλ(e(u))·(−ν(e(u)))=0 and νλ(e(u))·γ(e(u))=0 for all u∈U. Thus e:U→I×Λ is also a pre-envelope of (−γ,ν), (γ,−ν) and (ν,γ). It follows that −Eγ=−γ◦e, Eγ=γ◦e and Eν=ν◦e are envelopes of (−γ,ν), (γ,−ν) and (ν,γ), respectively.□ Definition 3.6. We say that a map Φ:I˜×Λ˜→I×Λ is a one-parameter family of parameter change if Φ is a diffeomorphism and given by the form Φ(s,k)=(ϕ(s,k),φ(k)). Proposition 3.7. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N). Suppose that Φ:I˜×Λ˜→I×Λis a one-parameter family of parameter change. Then (γ˜,ν˜)=(γ◦Φ,ν◦Φ):I˜×Λ˜→Δis also a one-parameter family of Legendre curves with the curvature m˜(s,k)=m(Φ(s,k))ϕs(s,k),n˜(s,k)=n(Φ(s,k))ϕs(s,k),L˜(s,k)=L(Φ(s,k))φ′(k),M˜(s,k)=m(Φ(s,k))ϕk(s,k)+M(Φ(s,k))φ′(k),N˜(s,k)=n(Φ(s,k))ϕk(s,k)+N(Φ(s,k))φ′(k).If e:U→I×Λis a pre-envelope, Eγis an envelope, then Φ−1◦e:U→I˜×Λ˜is a pre-envelope and Eγis also an envelope of (γ˜,ν˜). Proof Since γ˜s(s,k)=γt(Φ(s,k))ϕs(s,k) and γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ, we have γ˜s(s,k)·ν˜(s,k)=0forall(s,k)∈I˜×Λ˜. Therefore, (γ˜,ν˜) is a one-parameter family of Legendre curves. Then we have m˜(s,k)=γ˜s(s,k)·μ˜(s,k)=γt(Φ(s,k))ϕs(s,k)·μ(Φ(s,k))=m(Φ(s,k))ϕs(s,k),n˜(s,k)=ν˜s(s,k)·μ˜(s,k)=νt(Φ(s,k))ϕs(s,k)·μ(Φ(s,k))=n(Φ(s,k))ϕs(s,k),L˜(s,k)=γ˜k(s,k)·ν˜(s,k)=(γt(Φ(s,k))ϕk(s,k)+γλ(Φ(s,k))φ′(k))·ν(Φ(s,k))=L(Φ(s,k))φ′(k),M˜(s,k)=γ˜k(s,k)·μ˜(s,k)=(γt(Φ(s,k))ϕk(s,k)+γλ(Φ(s,k))φ′(k))·μ(Φ(s,k))=m(Φ(s,k))ϕk(s,k)+M(Φ(s,k))φ′(k),N˜(s,k)=ν˜k(s,k)·μ˜(s,k)=(νt(Φ(s,k))ϕk(s,k)+νλ(Φ(s,k))φ′(k))·μ(Φ(s,k))=n(Φ(s,k))ϕk(s,k)+N(Φ(s,k))φ′(k). By the form of the diffeomorphism Φ(s,k)=(ϕ(s,k),φ(k)),Φ−1:I×Λ→I˜×Λ˜ is given by the form Φ−1(t,λ)=(ψ(t,λ),φ−1(λ)). It follows that Φ−1◦e(u)=(ψ(t(u),λ(u)),φ−1(λ(u))). Since (d/du)φ−1(λ(u))=φλ−1(λ(u))λ′(u), the variability condition holds. Moreover, we have γ˜k(s,k)·ν˜(s,k)=(γt(Φ(s,k))ϕk(s,k)+γλ(Φ(s,k))φ′(k))·ν(Φ(s,k))=φ′(k)γλ(Φ(s,k))·ν(Φ(s,k)). It follows that γ˜k(φ−1◦e(u))·ν˜(φ−1◦e(u))=φ′(φ−1(λ(u)))γλ(e(u))·ν(e(u))=0forallu∈U. By Theorem 3.3, Φ−1◦e is a pre-envelope of (γ˜,ν˜). Therefore, γ˜◦Φ−1◦e=γ◦Φ◦Φ−1◦e=γ◦e=Eγ is also an envelope of (γ˜,ν˜).□ Definition 3.8. Let (γ,ν):I×Λ→Δ be a one-parameter family of Legendre curves, we define the parallel curves of the one-parameter family of Legendre curves by γθ(t,λ)=cosθγ(t,λ)−sinθν(t,λ),νθ(t,λ)=sinθγ(t,λ)+cosθν(t,λ). (3.1) Proposition 3.9. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N), then (γθ,νθ):I×Λ→Δis also a one-parameter family of Legendre curves with the curvature mθ(t,λ)=cosθm(t,λ)−sinθn(t,λ),nθ(t,λ)=sinθm(t,λ)+cosθn(t,λ),Lθ(t,λ)=L(t,λ),Mθ(t,λ)=cosθM(t,λ)−sinθN(t,λ),Nθ(t,λ)=sinθM(t,λ)+cosθN(t,λ).If e:U→I×Λis a pre-envelope of (γ,ν), then e:U→I×Λis also a pre-envelope of (γθ,νθ). Moreover, we have (Eγθ(u),Eνθ(u))=(Eγθ(u),Eνθ(u))for all u∈U, where (Eγθ,Eνθ)is parallel curve of (Eγ,Eν)and (Eγθ,Eγθ)=(γθ◦e,νθ◦e). Proof Since (γ,ν):I×Λ→Δ is a one-parameter family of Legendre curves, γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. It follows that γtθ(t,λ)·νθ(t,λ)=0 for all (t,λ)∈I×Λ. Thus (γθ,νθ):I×Λ→Δ is also a one-parameter family of Legendre curves. By definition, μθ(t,λ)=γθ(t,λ)×νθ(t,λ)=(cosθγ(t,λ)−sinθν(t,λ))×(sinθγ(t,λ)+cosθν(t,λ))=μ(t,λ). Therefore, we have mθ(t,λ)=γtθ(t,λ)·μθ(t,λ)=cosθm(t,λ)−sinθn(t,λ),nθ(t,λ)=νtθ(t,λ)·μθ(t,λ)=sinθm(t,λ)+cosθn(t,λ),Lθ(t,λ)=γλθ(t,λ)·νθ(t,λ)=L(t,λ),Mθ(t,λ)=γλθ(t,λ)·μθ(t,λ)=cosθM(t,λ)−sinθN(t,λ),Nθ(t,λ)=γλθ(t,λ)·μθ(t,λ)=sinθM(t,λ)+cosθN(t,λ). Since e:U→I×Λ is a pre-envelope of (γ,ν), we have γλ(e(u))·ν(e(u))=0 for all u∈U. It follows that γλθ(e(u))·νθ(e(u))=0 for all u∈U. Thus, e:U→I×Λ is a pre-envelope of (γθ,νθ) by Theorem 3.3. Moreover, Eγθ(u)=cosθEγ(u)−sinθEν(u)=cosθγ◦e(u)−sinθν◦e(u),Eνθ(u)=sinθEγ(u)+cosθEν(u)=sinθγ◦e(u)+cosθν◦e(u),Eγθ(u)=γθ◦e(u)=(cosθγ−sinθν)◦e(u)=cosθγ◦e(u)−sinθν◦e(u),Eνθ(u)=νθ◦e(u)=(sinθγ+cosθν)◦e(u)=sinθγ◦e(u)+cosθν◦e(u). Thus, we have (Eγθ(u),Eνθ(u))=(Eγθ(u),Eνθ(u)) for all u∈U.□ In [10], the evolute of the spherical Legendre curve is defined. Now, we define the evolute of a one-parameter family of Legendre curves in the unit sphere bundle. Definition 3.10. Let (γ,ν):I×Λ→Δ be a one-parameter family of Legendre curves with the curvature (m,n,L,M,N). Suppose that (m(t,λ),n(t,λ))≠(0,0) for all (t,λ)∈I×Λ. We define the evolute of the one-parameter family of Legendre curves (γ,ν) by E(γ)(t,λ)=±n(t,λ)m2(t,λ)+n2(t,λ)γ(t,λ)∓m(t,λ)m2(t,λ)+n2(t,λ)ν(t,λ). (3.2) Proposition 3.11. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N). Suppose that (m(t,λ),n(t,λ))≠(0,0), for all (t,λ)∈I×Λ. Then the evolute (E(γ),μ):I×Λ→Δof (γ,ν)is also a one-parameter family of Legendre curves with the curvature (mE,nE,LE,ME,NE), where mE(t,λ)=mtn−mntm2+n2(t,λ),nE(t,λ)=±m2+n2(t,λ),LE(t,λ)=±nM−mNm2+n2(t,λ),ME(t,λ)=mλn−mnλ−L(m2+n2)m2+n2(t,λ),NE(t,λ)=±mM+nNm2+n2(t,λ).If e:U→I×Λis a pre-envelope of (γ,ν)and (nM−mN)◦(e(u))=0for all u∈U, then e:U→I×Λis also a pre-envelope of (E(γ),μ). Moreover, we have EE(γ)(u)=EEγ(u)for all u∈U, where EE(γ)is the envelope of E(γ), EEγis the evolute of Eγ. Proof Since (γ,ν):I×Λ→Δ is a one-parameter family of Legendre curves and {γ(t,λ),ν(t,λ),μ(t,λ)} is a moving frame along the frontal γ(t,λ), we have E(γ)(t,λ)·μ(t,λ)=0,Et(γ)(t,λ)·μ(t,λ)=0 for all (t,λ)∈I×Λ. Therefore, (E(γ),μ):I×Λ→Δ is a one-parameter family of Legendre curves. We denote (γE,νE)=(E(γ),μ). By definition μE(t,λ)=γE(t,λ)×νE(t,λ)=∓m(t,λ)m2(t,λ)+n2(t,λ)γ(t,λ)∓n(t,λ)m2(t,λ)+n2(t,λ)ν(t,λ). Thus, mE(t,λ)=γEt(t,λ)·μE(t,λ)=mtn−mntm2+n2(t,λ),nE(t,λ)=νEt(t,λ)·μE(t,λ)=±m2+n2(t,λ),LE(t,λ)=γEλ(t,λ)·νE(t,λ)=±nM−mNm2+n2(t,λ),ME(t,λ)=γEλ(t,λ)·μE(t,λ)=mλn−mnλ−L(m2+n2)m2+n2(t,λ),NE(t,λ)=νEλ(t,λ)·μE(t,λ)=±mM+nNm2+n2(t,λ). Since (nM−mN)◦(e(u))=0 for all u∈U, we have E(γ)λ(e(u))·μ(e(u))=±nM−mNm2+n2◦e(u)=0. By Theorem 3.3, e:U→I×Λ is a pre-envelope of (E(γ),μ). The envelope of E(γ) is given by EE(γ)(u)=E(γ)◦e(u)=(±mm2+n2γ∓nm2+n2ν)◦e(u)=±mm2+n2(e(u))γ(e(u))∓nm2+n2(e(u))ν(e(u)). On the other hand, by Proposition 3.2, the evolute of Eγ is given by EEγ(u)=±mEγ(u)mEγ2(u)+nEγ2(u)Eγ(u)∓nEγ(u)mEγ2(u)+nEγ2(u)Eν(u)=±t′m+λ′M(t′m+λ′M)2+(t′n+λ′N)2(e(u))γ(e(u))∓t′n+λ′N(t′m+λ′M)2+(t′n+λ′N)2(e(u))ν(e(u)). Since (nM−mN)(e(u))=0 for all u∈U, we have (t′m+λ′M)2(m2+n2)(e(u))=m2((t′m+λ′M)2+(t′n+λ′N)2)(e(u)),(t′n+λ′N)2(m2+n2)(e(u))=n2((t′m+λ′M)2+(t′n+λ′N)2)(e(u)). Then t′m+λ′M(t′m+λ′M)2+(t′n+λ′N)2(e(u))=mm2+n2(e(u)),t′n+λ′N(t′m+λ′M)2+(t′n+λ′N)2(e(u))=nm2+n2(e(u)). Thus, we have EE(γ)(u)=EEγ(u) for all u∈U.□ 4. Relationships among envelopes of Legendre curves in the spherical bundle over the unit sphere and the unit tangent bundle over the Euclidean plane We first recall the definition of the envelope of a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane. For more detailed descriptions, see [11]. Let (γ,ν):I×Λ→R2×S1 be a smooth mapping. We say that (γ,ν) is a one-parameter family of Legendre curves if γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. We denote J(a)=(−a2,a1) the anticlockwise rotation by π/2 of a vector a=(a1,a2). We define μ(t,λ)=J(ν(t,λ)). Since {ν(t,λ),μ(t,λ)} is a moving frame along γ(t,λ) on R2, we have the Frenet type formula: (νt(t,λ)μt(t,λ))=(0ℓ(t,λ)−ℓ(t,λ)0)(ν(t,λ)μ(t,λ)),(νλ(t,λ)μλ(t,λ))=(0m(t,λ)−m(t,λ)0)(ν(t,λ)μ(t,λ)),γt(t,λ)=β(t,λ)μ(t,λ), where ℓ(t,λ)=νt(t,λ)·μ(t,λ), m(t,λ)=νλ(t,λ)·μ(t,λ) and β(t,λ)=γt(t,λ)·μ(t,λ). By the integrability condition νtλ(t,λ)=νλt(t,λ), ℓ and m satisfy the condition ℓλ(t,λ)=mt(t,λ) for all (t,λ)∈I×Λ. We call the triple (ℓ,m,β) are the curvature of the one-parameter family of Legendre curves (γ,ν). Let (γ,ν):I×Λ→R2×S1 be a one-parameter family of Legendre curves with the curvature (ℓ,m,β) and let e:U→I×Λ, e(u)=(t(u),λ(u)) be a smooth curve, where U is an interval of R. We denote Eγ=γ◦e:U→R2, Eγ(u)=γ◦e(u) and Eν=ν◦e:U→R2, Eν(u)=ν◦e(u). We call Eγ an envelope (and e a pre-envelope) for the one-parameter family of Legendre curves (γ,ν):I×Λ→R2×S1, when the following conditions satisfy: The function λ is non-constant on any non-trivial subinterval of U. (The Variability Condition.) For all u, the curve Eγ is tangent at u to the curve γ(t,λ) at the parameter (t(u),λ(u)), meaning that the tangent vectors Eγ′(u)=(dE/du)(u) and μ(t(u),λ(u)) are linearly dependent. (The Tangency Condition.) We consider relationships among envelopes of Legendre curves in the spherical bundle over the unit sphere and the unit tangent bundle over the Euclidean plane. We denote a hemisphere S+={(x,y,z)∈S2∣z>0}. Now we consider the central projection ϕ:S+→R2 by ϕ(x,y,z)=(xz,yz). Proposition 4.1. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves with the curvature (m,n,L,M,N)and γ(I×Λ)⊂S+. We denote γ(t,λ)=(x(t,λ),y(t,λ),z(t,λ)), ν(t,λ)=(a(t,λ),b(t,λ),c(t,λ)). Suppose that (a(t,λ),b(t,λ))≠(0,0)for all (t,λ)∈I×Λ. Then (γ˜,ν˜):I×Λ→R2×S1is a one-parameter family of Legendre curves with the curvature (ℓ˜,m˜,β˜), where γ˜(t,λ)=ϕ◦γ(t,λ)=(x(t,λ)z(t,λ),y(t,λ)z(t,λ)),ν˜(t,λ)=1a2(t,λ)+b2(t,λ)(a(t,λ),b(t,λ)),ℓ˜(t,λ)=nza2+b2(t,λ),m˜(t,λ)=Nza2+b2(t,λ),β˜(t,λ)=mz2+(xb−ya)ztz2a2+b2(t,λ). Proof Since (γ,ν):I×Λ→Δ is a one-parameter family of Legendre curves in the unit spherical bundle, we have γ(t,λ)·ν(t,λ)=0,γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. It follows that x(t,λ)at(t,λ)+y(t,λ)bt(t,λ)+z(t,λ)ct(t,λ)=0. By definition, we have μ(t,λ)=γ(t,λ)×ν(t,λ)=(y(t,λ)c(t,λ)−z(t,λ)b(t,λ),z(t,λ)a(t,λ)−x(t,λ)c(t,λ),x(t,λ)b(t,λ)−y(t,λ)a(t,λ)). By a direct calculation, we have m(t,λ)=γt(t,λ)·μ(t,λ)=−xtb+ytaz(t,λ),n(t,λ)=νt(t,λ)·μ(t,λ)=−atb+btaz(t,λ),N(t,λ)=νλ(t,λ)·μ(t,λ)=−aλb+bλaz(t,λ). By the assumption (a(t,λ),b(t,λ))≠(0,0), ν˜:I×Λ→S1 is a smooth mapping. Moreover, we have γ˜t(t,λ)=(xt(t,λ)z(t,λ)−x(t,λ)zt(t,λ),yt(t,λ)z(t,λ)−y(t,λ)zt(t,λ))/z2(t,λ)andγ˜t(t,λ)·ν˜(t,λ)=0. Therefore, (γ˜,ν˜):I×Λ→R2×S1 is a one-parameter family of Legendre curves. By definition, we have μ˜(t,λ)=J(ν˜(t,λ))=(−b(t,λ),a(t,λ))/a2(t,λ)+b2(t,λ)and the curvature ℓ˜(t,λ)=ν˜t(t,λ)·μ˜(t,λ)=−atb+btaa2+b2(t,λ)=nza2+b2(t,λ),m˜(t,λ)=ν˜λ(t,λ)·μ˜(t,λ)=−aλb+bλaa2+b2(t,λ)=Nza2+b2(t,λ),β˜(t,λ)=γ˜t(t,λ)·μ˜(t,λ)=(−xtb+yta)z+(xb−ya)ztz2a2+b2(t,λ)=mz2+(xb−ya)ztz2a2+b2(t,λ). □ Proposition 4.2. Under the same assumptions in Proposition4.1, suppose that e:U→I×Λis a pre-envelope of (γ,ν)and Eγ:U→S2is an envelope. Then e:U→I×Λis also a pre-envelope of (γ˜,ν˜):I×Λ→R2×S1. Moreover, we have Eγ˜(u)=E˜γ(u)for all u∈U, where Eγ˜=γ˜◦eand E˜γ=ϕ◦Eγ. Proof Since (γ,ν):I×Λ→Δ is a one-parameter family of Legendre curves in the unit spherical bundle and e:U→I×Λ is a pre-envelope of (γ,ν), we have γ(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ, γλ(e(u))·ν(e(u))=0 for all u∈U. It follows that (a(t,λ)(xλ(t,λ)z(t,λ)−x(t,λ)zλ(t,λ))+b(t,λ)(yλ(t,λ)z(t,λ)−y(t,λ)zλ(t,λ)))◦e(u)=0. Then we have γ˜λ(e(u))·ν˜(e(u))=0 for all u∈U. Therefore, e:U→I×Λ is a pre-envelope of (γ˜,ν˜) (cf. [11]). Moreover, we have Eγ˜(u)=γ˜◦e(u)=ϕ◦γ◦e(u)=ϕ(Eγ(u))=E˜γ(u) for all u∈U.□ Conversely, we have the following results. Proposition 4.3. Let (γ˜,ν˜):I×Λ→R2×S1be a one-parameter family of Legendre curves with the curvature (ℓ˜,m˜,β˜). We denote γ˜(t,λ)=(x(t,λ),y(t,λ)), ν˜(t,λ)=(a(t,λ),b(t,λ)). Then (γ,ν):I×Λ→Δ⊂S+×S2is a one-parameter family of Legendre curves in the unit spherical bundle with the curvature (m,n,L,M,N), where γ(t,λ)=ϕ−1◦γ˜(t,λ)=(x,y,1)1+x2+y2(t,λ),ν(t,λ)=(a,b,−xa−yb)1+(xa+yb)2(t,λ),m(t,λ)=β˜+(ytx−xty)(xa+yb)(1+x2+y2)1+(xa+yb)2(t,λ),n(t,λ)=ℓ˜1+x2+y21+(xa+yb)2(t,λ),L(t,λ)=xλa+yλb1+x2+y21+(xa+yb)2(t,λ),M(t,λ)=(yλx−xλy)(xa+yb)+yλa−xλb(1+x2+y2)1+(xa+yb)2(t,λ),N(t,λ)=m˜(1+x2+y2)+(xλa+yλb)(xb−ya)1+x2+y2(1+(xa+yb)2)(t,λ). Proof Since (γ˜,ν˜):I×Λ→R2×S1 is a one-parameter family of Legendre curves, then we have γ˜t(t,λ)·ν˜(t,λ)=(xta+ytb)(t,λ)=0forall(t,λ)∈I×Λ. By the definition, μ˜(t,λ)=J(ν˜(t,λ))=(−b(t,λ),a(t,λ)). It follows that ℓ˜(t,λ)=(−atb+abt)(t,λ),β˜(t,λ)=(−xtb+yta)(t,λ),m˜(t,λ)=(−aλb+abλ)(t,λ). By a direct calculation, we have γt(t,λ)=11+x2+y2((1+y2)xt−xyyt,(1+x2)yt−xxty,−xxt−yyt)(t,λ). Then γ(t,λ)·ν(t,λ)=0 and γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. Therefore, (γ,ν):I×Λ→Δ is a one-parameter family of Legendre curves in the unit spherical bundle. By definition, μ(t,λ) is given by μ(t,λ)=γ(t,λ)×ν(t,λ)=(−xya−(1+y2)b,(1+x2)a+xyb,xb−ya)(1+x2+y2)(1+(xa+yb)2)(t,λ). By a direct calculation, we have the curvature (m,n,L,M,N) of (γ,ν).□ Proposition 4.4. Under the same assumptions in Proposition4.3, suppose that e:U→I×Λis a pre-envelope of (γ˜,ν˜)and Eγ˜:U→R2is an envelope. Then e:U→I×Λis also a pre-envelope of (γ,ν):I×Λ→Δ⊂S+×S2. Moreover, we have ϕ−1◦Eγ˜(u)=Eγ(u)for all u∈U. Proof Since e:U→I×Λ is a pre-envelope of (γ˜,ν˜), we have γ˜λ(e(u))·ν˜(e(u))=0 for all u∈U (cf. [11]). It follows that (xλ(t,λ)·a(t,λ)+yλ(t,λ)·b(t,λ))◦e(u)=0. By a direct calculation, we have γλ(e(u))·ν(e(u))=0 for all u∈U. By Theorem 3.3, e:U→I×Λ is a pre-envelope of (γ,ν). Moreover, we have ϕ−1◦Eγ˜(u)=ϕ−1◦γ˜◦e(u)=γ◦e(u)=Eγ(u) for all u∈U.□ Also, we consider the canonical projection π:S+→D2⊂R2 by π(x,y,z)=(x,y), where D2={(x,y)∈R2∣x2+y2<1}. Proposition 4.5. Let (γ,ν):I×Λ→Δbe a one-parameter family of Legendre curves in the unit spherical bundle with the curvature (m,n,L,M,N)and γ(I×Λ)⊂S+. We denote γ(t,λ)=(x(t,λ),y(t,λ),z(t,λ))and ν(t,λ)=(a(t,λ),b(t,λ),c(t,λ)). Then (γ˜,ν˜):I×Λ→D2×S1is a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane with the curvature (ℓ˜,m˜,β˜), where γ˜(t,λ)=π◦γ(t,λ)=(x(t,λ),y(t,λ)),ν˜(t,λ)=(za−xc,zb−yc)(za−xc)2+(zb−yc)2(t,λ),ℓ˜(t,λ)=nz+xyt−xty(za−xc)2+(zb−yc)2(t,λ),m˜(t,λ)=Nz+xyλ−xλy(za−xc)2+(zb−yc)2(t,λ),β˜(t,λ)=m−(xb−ya)zt(za−xc)2+(zb−yc)2(t,λ). Proof If z(t,λ)a(t,λ)−x(t,λ)c(t,λ)=0 and z(t,λ)b(t,λ)−y(t,λ)c(t,λ)=0, then a(t,λ)=x(t,λ)c(t,λ)/z(t,λ)andb(t,λ)=y(t,λ)c(t,λ)/z(t,λ). Since ν(t,λ)∈S2, we have c2(t,λ)=z2(t,λ) and hence c(t,λ)=±z(t,λ). It follows that a(t,λ)=±x(t,λ)andb(t,λ)=±y(t,λ). This contradicts the fact that γ(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. Hence, ν˜:I×Λ→S1 is a smooth mapping. By (xta+ytb+ztc)(t,λ)=0 and (xtx+yty+ztz)(t,λ)=0, we have γ˜t(t,λ)·ν˜(t,λ)=0forall(t,λ)∈I×Λ. Therefore, (γ˜,ν˜):I×Λ→D2×S1 is a one-parameter family of Legendre curves. By a similar calculation as in Proposition 4.1, we have the curvature (ℓ˜,m˜,β˜) of (γ˜,ν˜).□ Proposition 4.6. Under the same assumptions in Proposition4.5, suppose that e:U→I×Λis a pre-envelope of (γ,ν)and Eγ:U→S2is an envelope. Then e:U→I×Λis also a pre-envelope of (γ˜,ν˜):I×Λ→D2×S1. Moreover, we have Eγ˜(u)=E˜γ(u)for all u∈U, where Eγ˜=γ˜◦eand E˜γ=π◦Eγ. Proof Since e:U→I×Λ is a pre-envelope of (γ,ν), we have γλ(e(u))·ν(e(u))=0 for all u∈U. It follows that (xλ(t,λ)(a(t,λ)z(t,λ)−x(t,λ)c(t,λ))+yλ(t,λ)(b(t,λ)z(t,λ)−y(t,λ)c(t,λ)))◦e(u)=0. By a direct calculation, we have γ˜λ(e(u))·ν˜(e(u))=0 for all u∈U. Therefore, e:U→I×Λ is a pre-envelope of (γ˜,ν˜) (cf. [11]). Moreover, we have Eγ˜(u)=γ˜◦e(u)=π◦γ◦e(u)=π(Eγ(u))=E˜γ(u) for all u∈U.□ Conversely, we have the following results. Proposition 4.7. Let (γ˜,ν˜):I×Λ→D2×S1be a one-parameter family of Legendre curves with the curvature (ℓ˜,m˜,β˜). We denote γ˜(t,λ)=(x(t,λ),y(t,λ)),ν˜(t,λ)=(a(t,λ),b(t,λ)).Then we have (γ,ν):I×Λ→Δ⊂S+×S2is a one-parameter family of Legendre curves, where γ(t,λ)=π−1◦γ˜(t,λ)=(x(t,λ),y(t,λ)z(t,λ)),ν(t,λ)=11−(xa+yb)2(a−x(xa−yb),b−y(xa−yb),−z(xa+yb))(t,λ).Here we put z(t,λ)=1−x(t,λ)2−y(t,λ)2. Proof Since γ˜(t,λ)·γ˜(t,λ)<1 and ν˜(t,λ)·ν˜(t,λ)=1, we have x(t,λ)a(t,λ)+y(t,λ)b(t,λ)<1 for all (t,λ)∈I×Λ. Therefore, ν:I×Λ→S2 is a smooth mapping. By the same argument as in Proposition 4.5, we have ℓ˜(t,λ)=−at(t,λ)b(t,λ)+a(t,λ)bt(t,λ),β˜(t,λ)=−xt(t,λ)b(t,λ)+yt(t,λ)a(t,λ),m˜(t,λ)=−aλ(t,λ)b(t,λ)+a(t,λ)bλ(t,λ). Since γt(t,λ)=(xt(t,λ),yt(t,λ),−x(t,λ)xt(t,λ)−y(t,λ)yt(t,λ)z(t,λ)),x2(t,λ)+y2(t,λ)+z2(t,λ)=1, we have γ(t,λ)·ν(t,λ)=0 and γt(t,λ)·ν(t,λ)=0 for all (t,λ)∈I×Λ. Therefore, (γ,ν):I×Λ→Δ⊂S+×S2 is a one-parameter family of Legendre curves.□ Remark 4.8. We can detect to the curvature (m,n,L,M,N) of (γ,ν) in Proposition 4.7. However, the description is a little bit long, we omit it. Proposition 4.9. Under the same assumptions in Proposition4.7, suppose that e:U→I×Λis a pre-envelope of (γ˜,ν˜)and Eγ˜:U→R2is an envelope. Then e:U→I×Λis also a pre-envelope of (γ,ν):I×Λ→Δ⊂S+×S2. Moreover, we have π−1◦Eγ˜(u)=Eγ(u) for all u∈U. Proof Since e:U→I×Λ is a pre-envelope of (γ˜,ν˜), we have γ˜λ(e(u))·ν˜(e(u))=0 for all u∈U (cf. [11]). It follows that (xλ(t,λ)·a(t,λ)+yλ(t,λ)·b(t,λ))◦e(u)=0. By a direct calculation, we have γλ(e(u))·ν(e(u))=0 for all u∈U. By Theorem 3.3, e:U→I×Λ is a pre-envelope of (γ,ν). Moreover, we have π−1◦Eγ˜(u)=π−1◦γ˜◦e(u)=γ◦e(u)=Eγ(u) for all u∈U.□ 5. Examples Example 5.1. Let (γ,ν):[0,2π)×[0,2π)→Δ, γ(t,θ)=(cosθ(34cost−14cos3t)−32sinθcost,34sint−14sin3t,32cosθcost+sinθ(34cost−14cos3t)),ν(t,θ)=(cosθ(−34sint−14sin3t)−32sinθsint,34cost+14cos3t,32cosθsint−sinθ(34sint+14sin3t)). Then (γ,ν) is a one-parameter family of Legendre curves. By definition μ(t,θ)=γ(t,θ)×ν(t,θ)=(−334cosθcos2t−38cosθcos2t+338cosθ−12sinθ,−3sintcost,−334sinθcos2t−38sinθcos2t+338sinθ+12cosθ). Then the curvature is given by m(t,θ)=332(96sintcos4t−54sintcos2t+9sintcostcos3t−33sin3tcos2t−20sint+12sin3t),n(t,θ)=38(6costcos2t+5cost−3cos3t),L(t,θ)=3sintcost,M(t,θ)=98(2cos2t−1)−18cos3t,N(t,θ)=34sintcos2t+116sin3t−1516sint. If we take e:[0,2π)→[0,2π)×[0,2π),e(u)=(0,u),(π/2,u),(π,u),(3π/2,u), then we got L(e(u))=0 for all u∈[0,2π). Thus, e are pre-envelopes of (γ,ν). Hence, the envelopes Eγ:[0,2π)→S2 are given by Eγ=(1/2cosu−3/2sinu,0,1/2sinu+3/2cosu),(0,1,0),(−1/2cosu+3/2sinu,0,−1/2sinu−3/2cosu),(0,−1,0), see Fig. 1. Example 5.2. Let n,m and k be natural numbers with m=k+n. We give a mapping (γ,ν):R×[0,2π)→Δ by γ(t,θ)=1t2m+t2n+1(cosθ−tnsinθ,sinθ+tncosθ,tm),ν(t,θ)=1k2t2m+m2t2k+n2(ktmcosθ+mtksinθ,ktmsinθ−mtkcosθ,n). Then (γ,ν):R×[0,2π)→Δ is a one-parameter family of Legendre curves. By definition, μ(t,θ)=γ(t,θ)×ν(t,θ)=1t2m+t2n+1k2t2m+m2t2k+n2(tn(mt2k+n)cosθ+(−kt2m+n)sinθ,tn(mt2k+n)sinθ+(kt2m−n)cosθ,−kt2n+k−mtk). Then the curvature is given by m(t,θ)=−tn−1k2t2m+m2t2k+n2t2m+t2n+1,n(t,θ)=mnktk−1t2m+t2n+1k2t2m+m2t2k+n2,L(t,θ)=−mtk+ktm+nt2m+t2n+1k2t2m+m2t2k+n2,M(t,θ)=−nk2t2m+m2t2k+n2,N(t,θ)=tmt2m+t2n+1. If we take e:[0,2π)→R×[0,2π), e(u)=(0,u), then we got L(e(u))=0 for all u∈[0,2π). Thus, e is a pre-envelope of (γ,ν). Hence, the envelope Eγ:[0,2π)→S2 is given by Eγ(u)=(cosu,sinu,0). For example, when n=2,m=3,k=1, then we have γ(t,θ)=1t6+t4+1(cosθ−t2sinθ,sinθ+t2cosθ,t3),ν(t,θ)=1t6+9t2+4(t3cosθ+3tsinθ,t3sinθ−3tcosθ,2). By definition, μ(t,θ)=γ(t,θ)×ν(t,θ)=1t6+t4+1t6+9t2+4(t2(3t2+2)cosθ+(−t6+2)sinθ,t2(3t2+2)sinθ+(t6−2)cosθ,−t5−3t). Then the curvature is given by m(t,θ)=−tt6+9t2+4t6+t4+1,n(t,θ)=6t6+t4+1t6+9t2+4,L(t,θ)=−t5−3tt6+9t2+4t6+t4+1,M(t,θ)=−2t6+9t2+4,N(t,θ)=t3t6+t4+1. The envelope Eγ:[0,2π)→S2 is given by Eγ(u)=(cosu,sinu,0), see Fig. 2. The parallel curve of the envelope Eγ(u) for λ=π/6 is Eγπ6(u)=(3/2cosu,3/2sinu,−1/2). The parallel curves of (γ,ν) is γπ6(t,θ)=cosπ6γ(t,θ)−sinπ6ν(t,θ)=(3(cosθ−t2sinθ)2t6+t4+1−t3cosθ+3tsinθ2t6+9t2+4,3(sinθ+t2cosθ)2t6+t4+1−t3sinθ−3tcosθ2t6+9t2+4,3t32t6+t4+1−1t6+9t2+4). Then the envelope of these parallel curves is given by Eγπ6(u)=(3/2cosu,3/2sinu,−1/2). Thus, we have Eγπ6(u)=Eγπ6(u) for all u∈U. Figure 3 put these curves in the one sphere. Figure 1. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelopes. Figure 1. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelopes. Figure 2. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelope. Figure 2. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelope. Figure 3. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelope, parallel curves and the envelope of these parallel curves. Figure 3. View largeDownload slide These curves are one-parameter family of Legendre curves and their envelope, parallel curves and the envelope of these parallel curves. Funding This work was supported by the National Natural Science Foundation of China (D.P.) (Grant no. 11671070), China Scholarship Council (Y.L.) (Grant no. 201606620072) and JSPS KAKENHI (M.T.) (Grant no. JP 17K05238). Conflict of interests The authors declare that there is no conflicts of interests in this work. Acknowledgements The authors would like to thank the referee for helpful comments to improve the original manuscript. References 1 V. I. Arnol’d , Singularities of Caustics and Wave Fronts. Mathematics and Its Applications Vol. 62 , Kluwer Academic Publishers , Dordrecht , 1990 . Google Scholar CrossRef Search ADS 2 V. I. Arnol’d , S. M. Gusein-Zade and A. N. Varchenko , Singularities of Differentiable Maps, 1986 . 3 J. W. Bruce and P. J. Giblin , What is an envelope? Math. Gaz. 65 ( 1981 ), 186 – 192 . Google Scholar CrossRef Search ADS 4 J. W. Bruce and P. J. Giblin , Curves and Singularities. A geometrical introduction to singularity theory , 2nd edn , Cambridge University Press , Cambridge , 1992 . Google Scholar CrossRef Search ADS 5 C. G. Gibson , Elementary Geometry of Differentiable Curves. An undergraduate introduction , Cambridge University Press , Cambridge , 2001 . Google Scholar CrossRef Search ADS 6 A. Gray , E. Abbena and S. Salamon , Modern differential geometry of curves and surfaces with Mathematica. Studies in Advanced Mathematics , 3rd edn , Chapman and Hall/CRC , Boca Raton, FL , 2006 . 7 G. Ishikawa , Singularities of Curves and Surfaces in Various Geometric Problems, CAS Lecture Notes 10, Exact Sciences. 2015 . 8 S. Izumiya , M. C. Romero-Fuster , M. A. S. Ruas and F. Tari , Differential Geometry from a Singularity Theory Viewpoint , World Scientific Pub. Co Inc , Hackensack , 2015 . Google Scholar CrossRef Search ADS 9 J. W. Rutter , Geometry of Curves, Chapman & Hall/CRC Mathematics , Chapman & Hall/CRC , Boca Raton, FL , 2000 . 10 M. Takahashi , Legendre Curves in the Unit Spherical Bundle over the Unit Sphere and Evolutes, Real and Complex Singularities, 337–355, Contemp. Math., 675, Amer. Math. Soc., Providence, RI, 2016 . 11 M. Takahashi , Envelopes of Legendre curves in the unit tangent bundle over the Euclidean plane , Results Math. 71 ( 2017 ), 1473 – 1489 . DOI:10.1007/s00025-016-0619-7 . Google Scholar CrossRef Search ADS © The Author 2017. Published by Oxford University Press. All rights reserved. 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