A Note on Inextensible Flows of Curves on Oriented Surface

CUBO A Mathematical Journal o Vol.16, N¯ 03, (11–19). October 2014

Onder Gokmen Yildiz Soley Ersoy Melek Masal Department of Mathematics, Department of Mathematics, Department of Mathematics Faculty of Arts and Sciences, Faculty of Arts and Sciences, Teaching, Bilecik Seyh Edebali Sakarya University, Faculty of Education, University, Sakarya/TURKEY Sakarya University, Bilecik/TURKEY [email protected] Sakarya/TURKEY [email protected] [email protected]

ABSTRACT

In this paper, we investigate a general formulation for inextensible flows of curves on an oriented surface in R3. We obtain necessary and sufficient conditions as partial differential equations involving the geodesic curvature and the geodesic torsion for inextensible curve flow lying on an oriented surface. Moreover, some special cases of inextensible curves on oriented surface are given.

RESUMEN

En este art´ıculo investigamos una formulación general para flujos inextensibles de curvas sobre una superficie orientable en R3. Obtenemos condiciones necesarias y suficientes para las ecuaciones diferenciales parciales que involucran la curva geodésica y la torsión geodésica para curvas inextensibles fluyendo sobre superficies orientadas. Más aún, se entregan algunos casos especiales de curvas inextensibles sobre superficies orientadas.

Keywords and Phrases: Curvature ﬂows, inextensible, oriented surface. 2010 AMS Mathematics Subject Classiﬁcation: 53C44, 53A04, 53A05. 12 Onder Gokmen Yildiz, Soley Ersoy & Melek Masal CUBO 16, 3 (2014) 1 Introduction

The flow of the curve is said to be inextensible if its arclength preserved. Curve design using splines is one of the most fundamental topic in CAGD. Inextensible flows of the curves have beautiful shapes preserving connection to their control polygon. On the other hand, physically inextensible curve and surface flows give rise to motion which no strain energy is induced. For example, the swinging motion of a cord of fixed length can be described by inextensible curve and surface flows. Many authors have studied geometric flow problems and applications of inextensible curve flows, [1]–[10]. An evolution equation for inelastic planar curves was derived by [9] and also, the general formulation of inextensible flows of curves and developable surfaces in R3 was exposed by [10]. In this paper, we derive a general formulation for inextensible flows of curves according to Darboux frame in R3. We give the necessary and sufficient conditions for an inextensible curve flow are expressed as a partial differential equations involving the geodesic curvature and geodesic torsion.

2 Preliminaries

Let S be an oriented surface in three-dimensional Euclidean space E3 and α (s) be a curve lying on − − − the surface S. Suppose that the curve α (s) is spatial then there exists the Frenet frame T , N, B − − → → −→ at each points of the curve where T is unit tangent vector, N is principal normal vector and B is binormal vector, respectively. The→ Frenet equation of the curve→ α (s) is given by →

− − T ′ = κN − − − N→′ =−→κ T + τ B − − B→′ =−τN→ → → → where κ and τ are curvature and torsion of the curve α (s), respectively. Since the curve α (s) lies on the surface S there exists another frame of the curve α (s) which is − − − − called Darboux frame and denoted by T , g , n . In this frame T is the unit tangent of the − → − → − − − curve, n is the unit normal of the surface S and→ →g is a unit vector given by g = n × T . Since the − − − − unit tangent→ T is common element of both Frenet→ frame and Darboux frame,→ the→ vectors→ N, B , g − and n lie on→ the same plane. So that the relations between these frames can be given as→ follows→ → → − − T 1 0 0 T − −  →g  =  0 cosϕ sin ϕ   →N  − − →  →n   0 − sin ϕ cos ϕ   B             →  → − − where ϕ is the angle between the vectors g and N. The derivative formulae of the Darboux frame → → CUBO ANoteonInextensibleFlowsofCurvesonOrientedSurface 13 16, 3 (2014)

is −. − T 0 kg kn T . − →−  →g  =  − kg 0 τg   g  −. −  →n   − kn − τg 0   →n             →  where kg, kn and τg are called→ the geodesic curvature, the normal curvature and the geodesic torsions, respectively. Here and in the following, we use ”dot” to denote the derivative with respect to the arc length parameter of a curve. The relations between the geodesic curvature, normal curvature, geodesic torsion and κ, τ are given as follows, [11]

dϕ kg = κ cos ϕ , kn =−κ sin ϕ, τg = τ + ds .

Furthermore, the geodesic curvature kg and geodesic torsion τg of the curve α (s) can be calculated as follows, [11]

→− →− d α d2 α × − kg = ds , ds2 n →− →− Dd α − × d n→E τg = ds , n ds . In the diﬀerential geometry of surfaces, forD any→ curve αE(s) lying on a surface S the following relationships are well-known, [11]

i- α (s) is a geodesic curve if and only if kg = 0,

ii- α (s) is an asymptotic line if and only if kn = 0,

iii- α (s) is a principal line if and only if τg = 0. Through each point on a surface there passes, in general, a geodesic in every direction. A geodesic is uniquely determined by an initial point and tangent at that point. All straight lines on a surface are geodesics. Along all curved geodesics the principal normal coincides with the surface normal. Along asymptotic lines osculating planes and tangent planes coincide, along geodesics they are normal. Through a point of a non-developable surface there are two asymptotic lines which can be real or imaginary.

3 Inextensible Flows of Curve Lying on Oriented Surface

Throughout this paper, we suppose that α : [0, l] × [0, w) M ⊂ E3 is a one parameter family of diﬀerentiable curves on→ orientable surface M in E3, where l is the arclength of the initial curve. Let u be the curve parameterization variable, 0 ≤ u ≤ l. If the speed →− ∂ α of curve α is denoted by v = ∂u then the arclength of α is

14 Onder Gokmen Yildiz, Soley Ersoy & Melek Masal CUBO 16, 3 (2014)

u − u ∂ α S (u)= du = v du. (3.1) Z ∂u Z → 0 0

∂ The operator ∂s is given in terms of u by

∂ 1 ∂ ∂s = v ∂u . (3.2) Thus, the arclength is ds = v du.

Definition 3.1. Let M be an orientable surface and α be a differentiable curve on M in E3. Any − − − flow of the curve α can be expressed with respect to Darboux frame T , g , n in the following form: → → →

→− ∂ α − − − ∂t = f1 T + f2 g + f3 n . (3.3) → → → Here, f1, f2 and f3 are scalar speeds of the curve α. Let the arclength variation be

u S (u, t)= v du. (3.4) R0 In the Euclidean space the requirement that a curve not to be subject to any elongation or com- pression can be expressed by the condition

u ∂ ∂v ∈ ∂t S (u, t)= ∂t du = 0 , u [0, 1]. (3.5) R0 →− ∂ α 3 Deﬁnition 3.2. A curve evolution α (u, t) and its ﬂow ∂t on the oriented surface M in E are said to be inextensible if − ∂ ∂ α = 0. ∂t ∂u →

Now, we research the necessary and suﬃcient condition of a ﬂow to be inextensible. For this reason, we need to the following Lemma. − − − Lemma 3.1. In E3, let M be an orientable surface and T , g , n be a Darboux frame of α on M. There exists following relation between the scalar speed→ fu→nctions→ f1, f2, f3 and the normal curvature kn, geodesic curvature kg of α the curve

∂v ∂f1 ∂t = ∂u − f2vkg − f3vkn. (3.6) CUBO ANoteonInextensibleFlowsofCurvesonOrientedSurface 15 16, 3 (2014)

→− →− ∂ ∂ 2 ∂ α ∂ α Proof. Since ∂u and ∂t commute and v = ∂u , ∂u , we have D E →− →− ∂v ∂ ∂ α ∂ α 2v ∂t = ∂t ∂u , ∂u →− − D∂ α ∂ E − − = 2 ∂u , ∂u f1 T + f2 g + f3 n D ∂f1 → → →E = 2v ∂u − f2vkg − f3vkn .

This completes the proof.

If we consider the conditions of being geodesic and asymptotic of a curve and Lemma 3.1, we can give the following corollary.

Corollary 3.1. If a curve is a geodesic curve or an asymptotic curve, then there are the following equations

∂v ∂f1 ∂t = ∂u − f3vkn or

∂v ∂f1 ∂t = ∂u − f2vkg, respectively. →− − − − ∂ α − − Theorem 3.1. Let T , g , n be the Darboux frame of a curve α on M and ∂t = f1 T + f2 g + − → 3 → f3 n be a differentiable flow→ → of α in R . Then the flow is inextensible if and only if → → ∂f1 ∂s = f2kg + f3kn. (3.7)

Proof. Suppose that the curve ﬂow is inextensible. From the equations (3.4) and (3.6) for u ∈ [0, l] we see that u u ∂ ∂v ∂f S (u, t)= du = 1 − f vk − f vk du = 0. (3.8) ∂t Z ∂t Z ∂u 2 g 3 n 0 0 Thus, it can be seen that ∂f 1 = f vk + f vk . (3.9) ∂u 2 g 3 n Considering the last equation and (3.2), we reach

∂f1 ∂s = f2kg + f3kn.

Conversely, by following a similar way as above, the proof is completed.

From Theorem 3.1, we have following corollary. 16 Onder Gokmen Yildiz, Soley Ersoy & Melek Masal CUBO 16, 3 (2014)

Corollary 3.2. i- Let the curve α is a geodesic curve on M. Then the curve ﬂow is inextensible ∂f1 if and only if ∂s = f3kn. ii- Let the curve α is an asymptotic line on M. Then the curve ﬂow is inextensible if and only ∂f1 if ∂s = f2kg.

Now, we restrict ourselves to the arclength parameterized curves. That is, v = 1 and the local coordinate u corresponds to the curve arclength s. We require the following Lemma. − − − Lemma 3.2. Let M be an orientable surface in E3 and T , g , n be a Darboux frame of the − − − → curve α on M. Then, the diﬀerentiations of T , g , n with respect→ → to t is → →− → → ∂ T ∂f2 − ∂f3 − ∂t = f1kg + ∂s − f3τg g + f1kn + ∂s + f2τg n →− − − ∂ g =− + ∂f2 − + ∂t f1kg ∂s f3τg→T ψ n → →− − − ∂ n =− + ∂f3 + → − ∂t f1kn ∂s f2τg T ψ→g → →− → ∂ g − where ψ = ∂t , n . D →E ∂ ∂ Proof. Since ∂t and ∂s are commutative, it seen that →− →− →− ∂ T ∂ ∂ α ∂ ∂ α ∂ − − − ∂t = ∂t ∂s = ∂s ∂t = ∂s f1 T + f2 g + f3 n − →− →− →− ∂f1 ∂ T ∂f2− ∂ g→ ∂f3 −→ →∂ n = ∂s T + f1 ∂s + ∂s g + f2 ∂s + ∂s n + f3 ∂s . → → → Substituting the equation (3.7) into the last equation and using Theorem 3.1, we have − ∂ T ∂f2 − ∂f3 − = f k + − f τ g + f k + + f τ n . ∂t→ 1 g ∂s 3 g 1 n ∂s 2 g → → Now, let us diﬀerentiate the Darboux frame with respect to t as follows; − − ∂ − − ∂ T − − ∂ g 0 = T , g = , g + T , ∂t * ∂t→ + ∂t→ D→ →E → → − ∂f − ∂ g = f k + 2 − f τ + T , (3.10) 1 g ∂s 3 g ∂t → → − − ∂ − − ∂ T − − ∂ n 0 = T , n = , n + T , ∂t * ∂t→ + ∂t→ D→ →E → → − ∂f − ∂ n = f k + 3 + f τ + T , (3.11) 1 n ∂s 2 g ∂t → → From (3.10) and (3.11), we have obtain − ∂ g ∂f2 − − =− f1kg + − f3τg T + ψ n ∂t→ ∂s → → CUBO ANoteonInextensibleFlowsofCurvesonOrientedSurface 17 16, 3 (2014)

and − ∂ n ∂f3 − − =− f1kn + + f2τg T − ψ g ∂t→ ∂s →− → → ∂ g − respectively, where ψ = ∂t , n . D →E If we take into consideration last Lemma, we have following corollary. Corollary 3.3. Let M be an orientable surface in E3. i- If the curve α is a geodesic curve, then →− ∂ T ∂f2 − ∂f3 − ∂t = ∂s − f3τg g + f1kn + ∂s + f2τg n , →− − − ∂ g =− ∂f2 − + ∂t ∂s f3τg→T ψ n , → →− − − ∂ n =− + ∂f3 +→ − ∂t f1kn ∂s f2τg →T ψ g , →− → → ∂ g − where ψ = ∂t , n . ii- If theD curve→Eα is an asymptotic line, then →− ∂ T ∂f2 − ∂f3 − ∂t = f1kg + ∂s − f3τg g + ∂s + f2τg n , →− − − ∂ g =− + ∂f2 − + ∂t f1kg ∂s f3τg→ T ψ n , → →− − − ∂ n =− ∂f3 + − → ∂t ∂s f2τg T ψ g , → →− → → ∂ g − where ψ = ∂t , n . iii- If theD curve→E is a curvature line, then →− ∂ T ∂f2 − ∂f3 − ∂t = f1kg + ∂s g + f1kn + ∂s n , →− − − ∂ g =− + ∂f2 + ∂t f1kg ∂s →T ψ n , → →− − − ∂ n =− + ∂f3 → − → ∂t f1kn ∂s T ψ g , →− → → ∂ g − where ψ = ∂t , n . → →− D E ∂ α − − − Theorem 3.2. Suppose that the curve ﬂow ∂t = f1 T +f2 g +f3 n is inextensible on the orientable surface on M. In this case, the following partial diﬀerential→ → equations→ are held: 2 ∂kg 2 ∂kg ∂ f2 ∂f3 ∂τg 2 ∂t = f2kg + f3kgkn + f1 ∂s + ∂s2 − 2 ∂s τg − f3 ∂s − f1knτg − f2τg + ψkn, 2 ∂kn 2 ∂kn ∂ f3 ∂f2 ∂τg 2 ∂t = f2kgkn + f3kn + f1 ∂s + ∂s2 + 2 ∂s τg + f2 ∂s + f1kgτg − f3τg − ψkg,

∂τg ∂f2 ∂f3 ∂ψ ∂t = f2kgτg − ∂s kn + ∂s kg + f3knτg + ∂s .

→− →− ∂ ∂ T ∂ ∂ T Proof. Since ∂s ∂t = ∂t ∂s we get →− ∂ ∂ T ∂ ∂f2 − ∂f3 − = f1kg + − f3τg g + f1kn + + f2τg n ∂s ∂t ∂s ∂s ∂s →− ∂k 2 ∂τ − ∂f1 g ∂ f2 →∂f3 g → ∂f2 ∂ g = ∂skg + f1 ∂s + ∂s2 − ∂s τg − f3 ∂s g + f1kg + ∂s − f3τg ∂s →− 2 ∂τ − ∂f1 ∂kn ∂ f3 ∂f2 g → ∂f3 ∂ n + ∂s kn + f1 ∂s + ∂s2 + ∂s τg + f2 ∂s n + f1kn + ∂s + f2τg ∂s → 18 Onder Gokmen Yildiz, Soley Ersoy & Melek Masal CUBO 16, 3 (2014)

i.e., →− 2 − ∂ ∂ T ∂f1 ∂kg ∂ f2 ∂f3 ∂τg − ∂f2 − ∂s ∂t = ∂s kg + f1 ∂s + ∂s2 − ∂s τg − f3 ∂s g + f1kg + ∂s − f3τg −kg T + τg n 2 ∂τ − − − ∂f1 ∂kn ∂ f3 ∂f2 g → ∂f3 → → + ∂s kn + f1 ∂s + ∂s2 + ∂s τg + f2 ∂s n + f1kn + ∂s + f2τg −kg T − τg g → → → while − − − ∂ ∂ T ∂ − − ∂kg − ∂ g ∂kn − ∂ n = kg g + kn n = g + kg + n + kn . ∂t ∂s→ ∂t ∂t ∂t→ ∂t ∂t→ → → → → Thus, from the both of above two equations, we reach

2 ∂kg 2 ∂kg ∂ f2 ∂f3 ∂τg 2 = f2k + f3kgkn + f1 + − 2 τg − f3 − f1knτg − f2τ + ψkn (3.12) ∂t g ∂s ∂s2 ∂s ∂s g and 2 ∂kn 2 ∂kn ∂ f3 ∂f2 ∂τg 2 = f2kgkn + f3k + f1 + + 2 τg + f2 + f1kgτg − f3τ − ψkg. (3.13) ∂t n ∂s ∂s2 ∂s ∂s g →− →− ∂ ∂ g ∂ ∂ g Noting that ∂s ∂t = ∂t ∂s , it is seen that →− − ∂ ∂ g ∂ ∂f2 − ∂s ∂t = ∂s − f1kg + ∂s − f3τg T + ψ n 2 − h∂f1 ∂kg ∂ f2 →∂f3 →i ∂τg = − kg + f1 + 2 − τg − f3 T ∂s ∂s ∂s ∂s ∂s while ∂f2 − − → − f1kg + − f3τg kg g + kn n ∂s − ∂ψ −→ → + ∂s n + ψ −kn T − τg g → → − − − ∂ ∂ g ∂ − − ∂kg − ∂ T ∂τg − ∂ n = −kg T + τg n =− T − kg + n + τg . ∂t ∂s→ ∂t ∂t ∂t→ ∂t ∂t→ → → → → Thus, we obtain ∂τ ∂f ∂f ∂ψ g = f k τ − 2 k + 3 k + f k τ + . (3.14) ∂t 2 g g ∂s n ∂s g 3 n g ∂s →− →− ∂ ∂ n ∂ ∂ n No other new formulas are obtained from the relation ∂s ∂t = ∂t ∂s .

Thus, we give the following corollary from last theorem.

Corollary 3.4. Let M be an orientable surface in E3. i- If the curve α is a geodesic curve on M, then we have

∂k ∂k ∂2f ∂f ∂τ n = f k2 + f n + 3 + 2 2 τ + f g − f τ2 ∂t 3 n 1 ∂s ∂s2 ∂s g 2 ∂s 3 g and ∂τ ∂f ∂ψ g =− 2 k + f k τ + . ∂t ∂s n 3 n g ∂s ii- If the curve α is an asymptotic line, we have

∂k ∂k ∂2f ∂f ∂τ g = f k2 +f g + 2 − 2 3 τ − f g − f τ2 ∂t 2 g 1 ∂s ∂s2 ∂s g 3 ∂s 2 g CUBO ANoteonInextensibleFlowsofCurvesonOrientedSurface 19 16, 3 (2014)

and ∂τ ∂f ∂ψ g = f k τ + 3 k + . ∂t 2 g g ∂s g ∂s iii- If the curve α is a curvature line, then we have

2 ∂kg 2 ∂kg ∂ f2 ∂t = f2kg + f3kgkn + f1 ∂s + ∂s2 + ψkn 2 ∂kn 2 ∂kn ∂ f3 ∂t = f2kgkn + f3kn + f1 ∂s + ∂s2 − ψkg.

Received: September 2013. Accepted: May 2014.

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