On Characteristic Curves of Developable Surfaces in Euclidean

Home , Tangent developable

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In section 3, both the coeﬃcients of the ﬁrst and second fundamental forms of developable surfaces and the Gaussian curvatures of normal and binormal surfaces are obtained. Finally, in section 4, the main theorems for particular curves on developable surfaces are given.

2. Preliminaries We shortly give some basic concepts concerning theory of curves and surfaces in E3 that will use in the subsequent section. Let α : I R E3 be a curve with ′ ⊂ −→ ′ dα α (s) = 1, where s is the arc-length parameter of α and α (s) = (s). The ds ′′ function κ : I R, κ(s)= α (s) is deﬁned as the curvature of α. For κ> 0, the −→ Frenet frame along the curve α and the corresponding derivative formulas (Frenet formulas) are as follows. ′ T (s)= α (s), ′′ α (s) N(s)= ′′ , α (s) k k B(s)= T (s) N(s), × where T , N, and B are the tangent, the principal normal, and the binormal of α, respectively. ′ T (s)= κ(s)N(s), ′ N (s)= κ(s)T (s)+ τ(s)B(s), ′ − B (s)= τ(s)N(s), − ′ ′′ ′′′ α (s) α (s), α (s) where the function τ : I R, τ(s)= D × E is the torsion of α; −→ κ2(s) ” , ” is the standart inner product, and ” ” is the cross product on R3. h Leti M be a regular surface and α : I ×R M be a unit-speed curve. Some particular curves lying on M are characterized⊂ −→ as follows. a) A curve α lying on M is a geodesic curve if and only if the acceleration vector ′′ α is normal to M, in other words, ′′ U α =0. × b) A curve α lying on M is an asymptotic curve if and only if the acceleration vector ′′ α is tangent to M, that is, ′′ U, α =0. D E c) A curve α lying on M is a line of curvature if and only if S(T ) and T are linearly dependent, where T is the tangent of α, U is the unit normal, and S is the shape operator of M. A ruled surface in R3 is (locally) the map

F(γ,δ)(t,u)= γ(t)+ uδ(t), where γ : I R3, δ : I R3 0 are smooth mappings and I is an open interval or a unit circle−→ S1 [5], where−→ γ \{and}δ are called the base and generator (director) ON CHARACTERISTIC CURVES 3 curves, respectively. For δ(t) = 1, the Gaussian curvature of F is [4] k k (γ,δ) ′ ′ [det(γ (t),δ(t),δ (t))]2 (2.1) K = , − (EG F 2)2 − ′ ′ ′ where E = E(t,u) = γ (t)+ uδ (t) , F = F (t,u) = γ (t),δ(t) , and G = D E G(t,u) = 1 are the coefficients of the first fundamental form of F . (γ,δ) Theorem 1 ([1]). A necessary and sufficient condition for the parameter curves of a surface to be lines of curvature in a neighborhood of a nonumbilical point is that F = f =0, where F and f are the respective the first and second fundamental coefficients. Theorem 2 (The Lancret Theorem). Let α be a unit-speed space curve with κ(s) = τ 6 0. Then α is a general helix if and only if ( )(s) is a constant function. κ Theorem 3 ([5]). A unit-speed curve α : I R E3 with κ(s) = 0 is a slant helix if and only if ⊂ −→ 6 2 κ τ ′ σ(s)= ( ) (s) ∓ (κ2 + τ 2)3/2 κ is a constant function.

3. Developable, Normal, and Binormal Surfaces In this section, we introduce developable, normal and binormal surfaces associ- ated to a space curve and after that obtain the coefficients of the first and second fundamental forms of them, respectively. The non-singular ruled surfaces whose the Gaussian curvature vanish are called developable surfaces. Now, we firstly get the unit normal U and the acceleration vectors of the parameter curves of developable surfaces. Let K(s, v)= α(s)+ vδ(s) be a ruled surface with δ(s) = 1. Then, we have k k ′ Ks = T + vδ ,

Kv = δ,

K K 1 ′ U = s × v = [(T + vδ ) δ], K K ′ × k s × vk (T + vδ ) δ ×

′ Kvs = δ , ′′ Kss = κN + vδ ,

Kvv = 0, (3.1) where T is the tangent, N is the principal normal, and κ is the curvature of α. From Eq.(2.1), the ruled surface K(s, v) is developable if and only if

′ ′ det(T,δ,δ )= T δ,δ =0. D × E 4 FATIH DOGAN˘

By diﬀerentiating the last equation with respect to s, we get

′ ′ ′ ′ ′′ T δ,δ + T δ ,δ + T δ,δ =0 D × E D × E D × E ′′ ′ (3.2) T δ,δ = κ N δ,δ . D × E − D × E Secondly, we obtain the coeﬃcients of the ﬁrst and second fundamental forms of special developable surfaces. Let α : I R E3 be a unit-speed curve with κ = 0 and let T,N,B,κ,τ be Frenet apparatus⊂ −→ of α. 6 { } 1) The ruled surface K(s, v)= α(s)+ vN(s) is called the principal normal surface of α. The partial derivatives of K(s, v) with respect to s and v are as follows.

K = (1 vκ)T + vτB, s − Kv = N, ′ ′ K = vκ T + [κ v(κ2 + τ 2)]N + vτ B, ss − − K = κT + τB, vs − Kvv =0.

Thus, the unit normal U and the Gaussian curvature K of K(s, v) are obtained as follows. K K 1 U = s × v = [ vτT + (1 vκ)B], Ks Kv (1 vκ)2 + v2τ 2 − − k × k − p

E = K ,K = (1 vκ)2 + v2τ 2, h s si − F = K ,K =0, h s vi G = K ,K =1, h v vi

′ κ ′ vτ + v2τ 2( ) τ e = U,Kss = , (3.3) h i (1 vκ)2 + v2τ 2 − p τ f = U,Kvs = , h i (1 vκ)2 + v2τ 2 − g = U,K =0p, h vvi

eg f 2 τ 2 K = − = , EG F 2 −[(1 vκ)2 + v2τ 2]2 − − 2) The ruled surface K(s, v)= α(s)+ vB(s) ON CHARACTERISTIC CURVES 5 is called the binormal surface of α. The partial derivatives of K(s, v) with respect to s and v are as follows. K = T vτN, s − Kv = B, ′ K = vκτT + (κ vτ )N vτ 2B, ss − − K = τN, vs − Kvv =0. Thus, the unit normal U and the Gaussian curvature K of K(s, v) are obtained as follows. K K 1 U = s × v = [ vτT N], Ks Kv √1+ v2τ 2 − − k × k E = K ,K =1+ v2τ 2, h s si F = K ,K =0, h s vi G = K ,K =1, h v vi ′ v2κτ 2 κ + vτ e = U,Kss = − − , (3.4) h i √1+ v2τ 2 τ f = U,Kvs = , h i √1+ v2τ 2 g = U,K =0, h vvi eg f 2 τ 2 K = − = . EG F 2 −[1 + v2τ 2]2 − As we can see above, the normal and binormal surfaces of α are developable if and only if the base curve α is a plane curve. 3) The ruled surface K(s, v)= α(s)+ vT (s) is called the tangent developable surface of α. The partial derivatives of K(s, v) with respect to s and v are as follows.

Ks = T + vκN,

Kv = T, ′ K = vκ2T + (κ + vκ )N + vκτB, ss − Kvs = κN,

Kvv =0. Thus, the unit normal U and the coeﬃcients of the ﬁrst and second fundamental forms of K(s, v) are obtained as follows. K K U = s × v = B, K K ± k s × vk E = K ,K =1+ v2κ2, h s si F = K ,K =1, h s vi G = K ,K =1, h v vi 6 FATIH DOGAN˘

e = U,K = vκτ, (3.5) h ssi ± f = U,K =0, h vsi g = U,K =0. h vvi 4) The ruled surface K(s, v)= B(s)+ vT (s) is called the Darboux developable surface of α. The partial derivatives of K(s, v) with respect to s and v are as follows. K = (vκ τ)N, s − Kv = T, ′ K = κ(vκ τ)T + (vκ τ) N + τ(vκ τ)B, ss − − − − Kvs = κN,

Kvv =0. Thus, the unit normal U and the coefficients of the first and second fundamental forms of K(s, v) are obtained as follows. K K U = s × v = B, K K ± k s × vk E = K ,K = (vκ τ)2, h s si − F = K ,K =0, h s vi G = K ,K =1, h v vi e = U,K = τ(vκ τ), (3.6) h ssi ± − f = U,K =0, h vsi g = U,K =0. h vvi 5) The ruled surface ∼ K(s, v)= α(s)+ vD(s) is called the rectifying developable surface of α, where ∼ τ D(s) = ( )(s)T (s)+ B(s) κ is the modified Darboux vector field of α. The partial derivatives of K(s, v) with respect to s and v are as follows. τ ′ K =(1+ v( ) )T, s κ τ K = T + B, v κ τ ′′ τ ′ K = v( ) T + κ(1 + v( ) )N, ss κ κ τ ′ K = ( ) T, vs κ Kvv =0. ON CHARACTERISTIC CURVES 7

Thus, the unit normal U and the coeﬃcients of the ﬁrst and second fundamental forms of K(s, v) are obtained as follows. K K U = s × v = N, K K ± k s × vk τ ′ E = K ,K =(1+ v( ) )2, h s si κ τ τ ′ F = K ,K = (1 + v( ) ), h s vi κ κ τ 2 G = K ,K =1+ , h v vi κ2

τ ′ e = U,K = κ(1 + v( ) ), (3.7) h ssi ± κ f = U,K =0, h vsi g = U,K =0. h vvi 6) The ruled surface − K(s, v)= D(s)+ vN(s) is called the tangential Darboux developable surface of α, where − 1 D(s)= (τ(s)T (s)+ κ(s)B(s)) κ2(s)+ τ 2(s) is the unit Darboux vectorp ﬁeld of α. The partial derivatives of K(s, v) with respect to s and v are as follows. ′ K = (v σ(s))N , s − Kv = N, ′ ′ ′′ Kss = σ (s)N + (v σ(s))N , − ′ − Kvs = N ,

Kvv =0,

′′ ′ ′ where N = κ T (κ2 +τ 2)N +τ B. Thus, the unit normal U and the coeﬃcients of the ﬁrst and− second− fundamental forms of K(s, v) are obtained as follows. K K − U = s × v = D K K ± k s × vk E = K ,K = (v σ(s))2(κ2 + τ 2), h s si − F = K ,K =0, h s vi G = K ,K =1, h v vi (3.8) e = U,K = (κ2 + τ 2)σ(s)(v σ(s)), h ssi ± − f = U,K =0, h vsi g = U,K =0. h vvi From now on, we will investigate the relationship among characteristic curves of developable surfaces. 8 FATIH DOGAN˘

4. Main Results We give main theorems that characterize the parameter curves are also particular curves on K(s, v) such as geodesic curves, asymptotic curves or lines of curvature. Theorem 4. Let K(s, v)= α(s)+ vδ(s) be a developable surface with δ(s) =1. Then, the following expressions are satisfied for the parameter curves ofk K(s,k v). i) s parameter curves are also asymptotic curves if and only if − ′ ′′ det(δ,δ ,δ ) κ = , det(T,δ,N) v2 where T is the tangent, N is the principal normal, and κ is the curvature of α. ii) s parameter curves cannot also be geodesic curves. iii) v− parameter curves are straight lines. iv) The− parameter curves are also lines of curvature if and only if the tangent of the base curve α is perpendicular to the director curve δ. Proof. i) If we use Eq.(3.1) and Eq.(3.2), and then take inner product of U and Kss, we obtain ′ ′′ ′ ′′ 2 U,Kss = κ T δ,N + κv δ δ,N + v T δ,δ + v δ δ,δ =0 h i h × i D × E D × E D × E ′ ′′ det(δ,δ ,δ ) κ = . det(T,δ,N) v2 ii) If we take cross product of U and Kss, we get ′′ ′′ ′ ′′ ′ 2 U Kss = [κ δ,N + v δ,δ ]T + [κv δ ,N + v T,δ + v δ ,δ ]δ × − h i D E D E D E ′′ ′ [κv δ,N + v2 δ,δ ]δ . − h i ′ D E Since T , δ, and δ are linearly dependent, the coefficients of them cannot be zero at the same time, i.e, s parameter curves cannot also be geodesic curves. − iii) Since U Kvv = 0 and U,Kvv = 0, v parameter curves are both geodesic curves and asymptotic× curves,h namely,i v parameter− curves are straight lines. iv) Since K(s, v) is the developable surface,− from Eq.(3.1), we obtain ′ ′ ′ f = U,K = T δ,δ + v δ δ,δ =0, h vsi × × D E ′ D E F = Ks,Kv = T,δ + v δ,δ . h i h i D E By the last equation, the parameter curves are also lines of curvature if and only if T,δ = 0, that is, the tangent of the base curve α is perpendicular to the director hcurvei δ. Theorem 5. Let K(s, v) be the normal surface of α. Then, the following expressions are satisfied for the parameter curves of K(s, v). i) s parameter curves are also asymptotic curves if and only if the exactly one s0 parameter− curve is the base curve, i.e., K(s, 0) = α(s) or the base curve α is− a circular helix. ii) s parameter curves are also geodesic curves if and only if the base curve α is a circular− helix. iii) v parameter curves are straight lines. − ON CHARACTERISTIC CURVES 9 iv) The parameter curves are also lines of curvature if and only if the base curve α is a plane curve. Proof. i) From Eq.(3.3), we have ′ κ ′ U,K =0 vτ + v2τ 2( ) =0 h ssi ⇐⇒ τ ′ κ ′ vτ + v2τ 2( ) =0 τ

1 ′ κ ′ v = 0 or ( ) = v( ) τ τ 1 cτ v = 0 or κ = − , v where c is a constant. Then, the s0 parameter curve is the base curve K(s, 0) = α(s) or κ and τ are constants. As− a result, the base curve α becomes a circular helix. ii) s parameter curves are also geodesic curves if and only if − (κ v(κ2 + τ 2))(1 vκ)T − − 1 ′ 2 ′ v 2 2 U Kss =  +(vκ (κ + τ ) )N  =0. × (1 vκ)2 + v2τ 2 − 2 −  +vτ(κ v(κ2 + τ 2))B  p  −  Since T , N, and B are linearly independent, we have (κ v(κ2 + τ 2))(1 vκ) = 0, − 2 − ′ v ′ vκ (κ2 + τ 2) = 0, − 2 vτ(κ v(κ2 + τ 2)) = 0. − From the expression of U, we see that both 1 vκ and vτ cannot be zero at the same time. Hence, by the ﬁrst and last equations− above, we obtain κ = v(κ2 + τ 2). 2 v ′ If we substitute this in the second equation, we get (κ2 + τ 2) = 0. As κ = 0, 2 6 v = 0. Then we have κ2 + τ 2 = constant. Since v and κ2 + τ 2 are constants, from κ 6= v(κ2 + τ 2), it follows that κ is a constant and thus τ is a constant. In other words, the base curve α is a circular helix. iii) Since U Kvv = 0 and U,Kvv = 0, v parameter curves are both geodesic curves and asymptotic× curves,h namely,i v parameter− curves are straight lines. iv) We have F = 0. Moreover, − τ f = =0 τ =0. (1 vκ)2 + v2τ 2 ⇐⇒ − Then, the parameter curvesp are also lines of curvature if and only if the base curve α is a plane curve.

Theorem 6. Let K(s, v) be the binormal surface of α. Then, the following expressions are satisﬁed for the parameter curves of K(s, v). i) s parameter curves are also asymptotic curves if and only the curvatures of α hold− the diﬀerential equation ′ vτ v2κτ 2 κ =0. − − 10 FATIH DOGAN˘ ii) s parameter curves are also geodesic curves if and only if the exactly one s0 parameter− curve is the base curve, i.e., K(s, 0) = α(s) or the base curve α is− a plane curve. iii) v parameter curves are straight lines. iv) The− parameter curves are also lines of curvature if and only if the base curve α is a plane curve. Proof. i) From Eq.(3.4), we have ′ U,K =0 v2κτ 2 κ + vτ =0. h ssi ⇐⇒ − − ii) s parameter curves are also geodesic curves if and only if − ′ 1 2 2 3 2 U Kss = [vτ T v τ N + v ττ B]=0. × √1+ v2τ 2 − Since T , N, and B are linearly independent, we have vτ 2 = 0, v2τ 3 = 0, ′ v2ττ = 0.

Then, it follows that v =0or τ = 0, i.e., the s0 parameter curve is the base curve K(s, 0) = α(s) or the base curve α is a plane curve.− iii) Since U Kvv = 0 and U,Kvv = 0, v parameter curves are both geodesic curves and asymptotic× curves,h namely,i v parameter− curves are straight lines. iv) We have F = 0. Moreover, − τ f = =0 τ =0. √1+ v2τ 2 ⇐⇒ Then, the parameter curves are also lines of curvature if and only if the base curve α is a plane curve. Theorem 7. Let K(s, v) be the tangent developable surface of α. Then, the following expressions are satisﬁed for the parameter curves of K(s, v). i) s parameter curves are also asymptotic curves if and only if the exactly one s0 parameter− curve is the base curve, i.e., K(s, 0) = α(s) or the base curve α is− a plane curve. ii) s parameter curves are also geodesic curves if and only if the exactly one s0 parameter− curve is the base curve, i.e., K(s, 0) = α(s). − iii) v parameter curves are straight lines. iv) The− parameter curves cannot also be lines of curvature. Proof. i) From Eq.(3.5), we have U,K =0 vκτ =0. h ssi ⇐⇒ ± Since κ = 0, v = 0 or τ = 0, i.e., the s0 parameter curve is the base curve K(s, 0) =6 α(s) or the base curve α is a plane− curve. ii) s parameter curves are also geodesic curves if and only if − ′ U K = (κ + vκ )T vκ2N =0. × ss ∓ ∓ Since T and N are linearly independent, we have ′ κ + vκ = 0, vκ2 = 0. ON CHARACTERISTIC CURVES 11

Since κ = 0, it follows that v = 0, i.e., the s0 parameter curve is the base curve K(s, 0) =6 α(s). − iii) Since U Kvv = 0 and U,Kvv = 0, v parameter curves are both geodesic curves and asymptotic× curves,h namely,i v parameter− curves are straight lines. iv) We have F = 0 and f = 1. Therefore,− the parameter curves cannot also be lines of curvature.

Theorem 8. Let K(s, v) be the Darboux developable surface of α. Then, the following expressions are satisﬁed for the parameter curves of K(s, v). i) s parameter curves are also asymptotic curves if and only if the base curve α is a plane− curve or a general helix. ii) s parameter curves are also geodesic curves if and only if the base curve α is a general− helix. iii) v parameter curves are straight lines. iv) The− parameter curves are also lines of curvature. Proof. i) From Eq.(3.6), we have U,K =0 τ(vκ τ)=0. h ssi ⇐⇒ ± − τ Then, τ = 0 or = v = constant, i.e., the base curve α is a plane curve or a κ general helix. ii) s parameter curves are also geodesic curves if and only if − ′ U K = (vκ τ) T κ(vκ τ)N =0. × ss ∓ − ∓ − Since T and N are linearly independent, we have ′ (vκ τ) = 0, − κ(vκ τ) = 0. − τ Then, it follows that = v = constant, i.e., the base curve α is a general helix. κ iii) Since U Kvv = 0 and U,Kvv = 0, v parameter curves are both geodesic curves and asymptotic× curves,h namely,i v parameter− curves are straight lines. iv) We have F = f = 0. Consequently,− the parameter curves are also lines of curvature.

Theorem 9. Let K(s, v) be the rectifying developable surface of α. Then, the following expressions are satisﬁed for the parameter curves of K(s, v). τ i) s parameter curves are also asymptotic curves if and only if is a linear func- − κ tion. ii) s parameter curves are also geodesic curves if and only if the exactly one s0 − τ − parameter curve is the base curve, i.e., K(s, 0) = α(s) or is a linear function. κ iii) v parameter curves are straight lines. iv) The− parameter curves are also lines of curvature if and only if the base curve α τ is a plane curve or is a linear function. κ Proof. i) From Eq.(3.7), we have τ ′ U,K =0 κ(1 + v( ) )=0. h ssi ⇐⇒ ± κ 12 FATIH DOGAN˘

τ 1 Since κ = 0, we get = s + c, where c is a constant. 6 κ −v ii) s parameter curves are also geodesic curves if and only if − τ ′′ U K = v( ) B =0. × ss ± κ τ ′′ Then, it follows that v =0or( ) = 0, i.e., the s parameter curve is the base κ 0 τ − curve K(s, 0) = α(s) or = as + c, where a and c are constants. κ − iii) Since U Kvv = 0 and U,Kvv = 0, v parameter curves are both geodesic curves and asymptotic× curves,h namely,i v parameter− curves are straight lines. iv) We have f = 0. Furthermore, −

τ τ ′ τ 1 F = (1 + v( ) )=0 τ =0 or = s + c, κ κ ⇐⇒ κ −v τ where c is a constant. Then, the base curve α is a plane curve or is a linear κ function.

Theorem 10. Let K(s, v) be the tangential Darboux developable surface of α. Then, the following expressions are satisﬁed for the parameter curves of K(s, v). i) s parameter curves are also asymptotic curves if and only if the base curve α is a general− helix or a slant helix. ii) s parameter curves are also geodesic curves if and only if the base curve α is a plane− curve or a slant helix. iii) v parameter curves are straight lines. iv) The− parameter curves are also lines of curvature. Proof. i) From Eq.(3.8), we have U,K =0 (κ2 + τ 2)σ(s)(v σ(s))=0. h ssi ⇐⇒ ± − In the last equation, κ and τ cannot be zero at the same time. Hence, it concludes that σ(s)=0or σ(s)= v = constant in other words the base curve α is a general helix or a slant helix. ii) s parameter curves are also geodesic curves if and only if − ′ 2 2 2 2 σ(s) v 2 2 U Kss = κ(v σ(s)) κ + τ T (κ +τ )( − ) N τ(v σ(s)) κ + τ B =0. 2 2 × ∓ − p ± √κ + τ ± − p Since T , N, and B are linearly independent, we obtain κ(v σ(s)) κ2 + τ 2 = 0, − σp(s) v ′ (κ2 + τ 2)( − ) = 0, √κ2 + τ 2 τ(v σ(s)) κ2 + τ 2 = 0. − p From this, we get τ =0 or σ(s) = v = constant, i.e., the base curve α is a plane curve or a slant helix. iii) Since U Kvv = 0 and U,Kvv = 0, v parameter curves are both geodesic curves and asymptotic× curves,h namely,i v parameter− curves are straight lines. iv) We have F = f = 0. Consequently,− the parameter curves are also lines of curvature. ON CHARACTERISTIC CURVES 13

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