Contact between Framed Rectifying Curves

Generic Properties of Framed Rectifying Curves

7. Contact between Framed Rectifying Curves

In this section, the contact between framed rectifying curves is considered. We now introduce the notion of circular rectifying curves as follows.

Definition 5. Letγ(s)be a framed rectifying curve and:

γ(s) =ρ(tan²( |g(s)|ds+C) +1)¹²g(s),

whereρis a positive number and C is a constant. We callγa circular rectifying curve ifg(s)is a circle on S². Let(γ,μ1,μ2):I→S²×Δ2be a framed spherical curve. We chooseμ1=γ, thenν=γ×μ2and γ(s) =α(s)ν(s). We show that the spherical Frenet–Serret formula ofγis as follows:

⎧⎪

⎨

⎪⎩

γ(s) =α(s)ν(s) μ₂(s) =l(s)ν(s)

ν(s) =−α(s)γ(s)−l(s)μ2(s),

whereμ₂(s),ν(s)=l(s). By the curvature functionsα(s)andl(s), we show the following proposition for framed spherical curves:

Mathematics2019,7, 37

Proposition 2. Let(γ,γ,μ2): I→S²×Δ2be a framed spherical curve, thenγis a circle if and only if α(s) =0and l(s)/α(s) =constant.

Proof. Ifα(s) =0 and(l/α)(s) =k, wherekis a constant, then we consider a normal vector field N(s) =_k2^k+1² γ(s)−_k2^k+1μ2(s). By taking the derivative ofN(s), we haveN(s) = _k2^k+1² (α(s)ν(s)− α(s)ν(s)) =0. This means thatN(s)is a constant vector. Moreover, we have:

N(s),γ(s)−N(s)= k²

k²+1γ(s)− k

k²+1μ2(s), 1

k²+1γ(s) + k

k²+1μ2(s)=0.

This means thatγis the intersection of a plane andS², soγis a circle.

Letγbe a circle onS². Obviously,γis a plane curve andα(s) =0, so thatγ(s),γ(s)×γ(s)= 0. Then, we can calculate thatγ(s),γ(s)×γ(s)=α⁴(s)l(s)−α³(s)α(s)l(s). Sinceα(s) =0, we haveα(s)l(s)−α(s)l(s) =0. This is equivalent to(l/α)(s) =0. Thus,l(s)/α(s) =constant.

As a corollary of Proposition2, we have the following result:

Corollary 1. Let(γ,γ,μ2):I→S²×Δ2be a framed spherical curve, thenγis a great circle on S²if and only ifα(s) =0and l(s) =0.

Now, we review the notions of contact between framed curves [14]. Let(γ,μ1,μ2):I→R³×Δ2; s→(γ(s),μ1(s),μ2(s))and(γ,1μ11,1μ2): 1I →R³×Δ2;u→(γ(u),1 μ11(u),1μ2(u))be framed curves.

We say that(γ,μ1,μ2)and(1γ,μ11,1μ2)havek^thorder contact ats=s₀,u=u₀if:

(γ,μ1,μ2)(s0) = (1γ,1μ1,μ12)(u0), d

ds(γ,μ1,μ2)(s0) = d

du(1γ,μ11,μ12)(u0), . . . , d^k−1

ds^k−1(γ,μ1,μ2)(s₀) = d^k−1

du^k−1(γ,11μ1,μ12)(u₀), d^k

ds^k(γ,μ1,μ2)(s₀) = d^k

du^k(1γ,μ11,1μ2)(u₀).

In addition, we say that(γ,μ1,μ2)and(1γ,1μ1,μ12)have at leastk^thorder contact ats=s₀,u=u₀ if:

(γ,μ1,μ2)(s0) = (1γ,1μ1,μ12)(u0), d

ds(γ,μ1,μ2)(s0) = d

du(1γ,μ11,μ12)(u0), . . . , d^k−1

ds^k−1(γ,μ1,μ2)(s0) = d^k−1

du^k−1(1γ,1μ1,μ12)(u0).

We generally say that(γ,μ1,μ2)and(1γ,1μ1,μ12)have at least first order contact at any points=s₀, u=u₀, up to congruence as framed curves. As a conclusion of Theorem 3.7 in [14], we show the following proposition:

Proposition 3. Let(γ,γ,μ2) : I → S²×Δ2, s → (γ(s),γ(s),μ2(s))and(1γ,γ,1μ12) : 1I → S²×Δ2, u→(γ(u),1 γ(u),1 μ12(u))be framed spherical curves. If(γ,γ,μ2)and(1γ,γ,1μ12)have at least(k+1)^thorder contact at s=s₀, u=u₀, we have:

α(s₀) =1α(u₀), d

dsα(s₀) = d

du1α(u₀), . . . , d^k−1

ds^k−1α(s₀) = d^k−1

du^k−11α(u₀), (14)

l(s0) =1l(u0), d

dsl(s0) = d

du1l(u0), . . . , d^k−1

ds^k−1l(s0) = d^k−1

du^k−11l(u0). (15) Conversely, if the conditions (14) and (15) hold, then(γ,γ,μ2)and(γ,1γ,1μ12)have at least(k+1)^thorder contact at s=s₀, u=u₀, up to congruence as framed spherical curves.

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Now, we consider the contact between circles and framed spherical curves. We have a corollary of Propositions2and3as follows:

Corollary 2. Let(γ,γ,μ2):I→S²×Δ2be a framed spherical curve.γand a circle have at least(k+1)^th order contact at s=s₀if and only if there exists a constantσsuch that:

l(s₀) =σα(s₀), d

dsl(s₀) =σd

dsα(s₀), . . . , d^k−1

ds^k−1l(s₀) =σd^k−1 ds^k−1α(s₀).

For the construction of the framed rectifying curve in Theorem4, we fix positive numberρand constantC. Letgi:I→S²(i=1, 2)be framed spherical curves. We knowγ1,γ2havek^thorder contact ats₀if and only ifg1,g2havek^thorder contact ats₀. By Corollary2, we have the following theorem, which can describe the contact between framed rectifying curves and circular rectifying curves.

Theorem 9. Letγbe a framed rectifying curve andα(s)and l(s)be curvature functions of the corresponding framed spherical curve. Then,γand a circular rectifying curve have at least k^thorder(k≥2)contact at s₀if and only if there exists a constantσsuch that:

l(s0) =σα(s0), d

dsl(s0) =σd

dsα(s0), . . . , d^k−2

ds^k−2l(s0) =σd^k−2 ds^k−2α(s0).

Author Contributions:Conceptualization, Y.W.; Writing—Original Draft Preparation, Y.W.; Calculations, D.P.;

Manuscript Correction, D.P.; Giving the Examples, R.G.; Drawing the Pictures, R.G.

Funding:This research was funded by National Natural Science Foundation of China grant numbers 11271063 and 11671070, the Fundamental Research Funds for the Central Universities grant number 3132018220, and the Project of Science and Technology of Jilin Provincial Education Department grant number JJKH2090547KJ.

Acknowledgments:This work was partially supported by the National Natural Science Foundation of China Nos. 11271063 and 11671070, the Fundamental Research Funds for the Central Universities No. 3132018220, and The Project of Science and Technology of Jilin Provincial Education Department No. JJKH2090547KJ.

Conflicts of Interest:The authors declare no conflict of interest.

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