A surface is the image of a function with two real variables in three dimensional space. Geometric shapes such as planes, cylinders, cones, and spheres are examples of surfaces. Surfaces are used in such applications as architectural structures, computer graphics, works of art, geometric design, textile and automobile design. Surface theory is an important field of study in differential geometry; the basic theory can be found, for example, [1, 2, 3]. Developable surfaces, in particular, are widely used in industrial applications. Ruled surface have also been widely studied, [4, 10]. Ruled surfaces are called linear surfaces because they are formed by moving a line along a curve, so are represented by an infinite family of straight lines. A generalization of ruled surfaces was introduced by Juza in the 1960s and has since by well studied [5]. The striction point, the striction curve and the distribution parameter (Drall) of a ruled surface with a Frenet frame in 3-dimensional Euclidean space were considered in [6, 7]. Some characteristic properties of a ruled surface with a Frenet frame of a non-cylindrical ruled surface were investigated by Ouarab and Chahdi [8]. On the other hand, Pottmann and Wallner expressed the orthonormal Sannia frame on the striction curve of a ruled surface in 3-dimensional Euclidean space [9].

The aim of this study is to examine a ruled surface with the orthonormal Sannia frame defined on the striction curve of the tangent, normal, binormal and Darboux ruled surfaces.

Let E3 be a 3-dimensional Euclidean space provided with the standard flat metric given by

<,>=dx12+dx22+dx32,

where x=x1,x2,x3 is a rectangular coordinate system of E3. Let α be a space curve with respect to the arclength s in E3, and let T, N and B be the tangent, principal normal and binormal unit vectors at a point α(s) of the curve α, respectively. There exists an orthogonal frame T,N,B which satisfies the Frenet-Serret equations,

(2.1)T′=κN, N′=−κT+τB, B′=−τN,

where κ is the curvature, τ is the torsion of the curve α [2]. The surface obtained by a line r moving along a differentiable curve α is called a ruled surface and its parametric equation is given by

(2.2)Xs,v=α(s)+vrs.

The curve α is called the base curve and the straight line r is called the ruling of the ruled surface [11]. Specifically, if the Frenet vectors of the curve are taken instead of r, the equations of the surfaces are obtained by

XTs,v=α(s)+vTs,XNs,v=α(s)+vNs,XBs,v=α(s)+vBs.

The normal vector field, the components of first and second fundamental forms, the Gaussian curvature and the mean curvature of a surface are computed as

(2.3)uX=Xs×XvXs×Xv, (2.4)E=Xs,Xs, F=Xs,Xv,G=Xv,Xv,l=Xss,uX,m=Xsv,uX, n=Xvv,uX, (2.5)K=ln−m2EG−F2, H=En−2Fm+Gl2EG−F2,

respectively [11]. Frenet vectors of a curve make an instantaneous rotation along the curve and around an axis that is called as the axis of rotation. The Darboux vector W points in the direction of the rotational axis and is calculated by

W=τT+κB.

The unit Darboux vector C, on the other hand, can be computed as following

C=sinφT+cosφB, sinφ=τκ2+τ2, cosφ=κκ2+τ2

where the angle φ is between the Darboux vector W and the binormal vector B of the moving curve [12]. The parametric equation of the ruled surface created by the moving the vector C along the curve α is

 XCs,v=α(s)+vCs.

If there exist a common perpendicular to two consecutive ruling in the ruled surface, then the foot of the common perpendicular on the main ruling is called a striction point and the set of these points is also defined as the striction curve. The equation of the striction curve of the ruled surface given in (2.2) can be written by

(2.6)β(s)=αs−α′,r′ r ′ 2r

[6]. Specifically, if the striction curve is taken to be the base curve on the surface, then the parametric equation of the ruled surface is given as

Xs,v=β(s)+vrs.

Let the curve β be a striction curve of the ruled surface Xs,v. The Sannia orthonormal frame [9] is the orthanormal frame e1,e2,e3 created by unit vectors along the striction curve β such that

(2.7)e1=r, e2= e′1 e′1 , e3=e1∧e2,

where r is the ruling of the ruled surface Xs,v. The Sannia formulae along the striction curve become

e′1=k1e2, e′2=−k1e1+k2e3, e′3=−k2e2,

where k₁ and k₂ are the curvature and the torsion of the striction curve of the ruled surface Xs,v [9].

In this section, we examine the ruled surfaces formed by Sannia frames along the striction curves of ruled surfaces generated by Frenet vectors of a curve. The surfaces obtained are called Sannia ruled surfaces. The relation between the Sannia and Frenet frame, the first and second fundamental forms, and the Gaussian and the mean curvatures of each ruled surface are calculated separately.

3.1. Sannia ruled surfaces associated with tangent ruled surface

Theorem 3.1. Let X_T be a tangent ruled surface and e1,e2,e3 be a Sannia frame on the striction curve of X_T. Then, the relationship between the Sannia frame e1,e2,e3 on striction curve and the Frenet frame T,N,B is as follows:

(3.1)e1=T, e2=N, e3=B.

Proof. Let the curve ζ be a striction curve of the tangent ruled surface X_T. Using (2.6), it can be easily shown that the striction curve ζ is equal to the base curve of X_T, i.e. ζ(s)=αs. Therefore, the equation (3.1) is satisfied.

Definition 3.2. A surface Φ1 is called a e₁ Sannia ruled surface in Euclidean 3-space, if the surface Φ1 is generated by moving the vector e₁ along the striction curve ζ of X_T and its parametric equation is defined as

(3.2)Φ1s,v=ζ(s)+ve1s.

Taking the partial differential of Φ1 with respect to s and v, we get

Φ1s=T+vκN   and   Φ1v=T.

By (2.3), the normal vector field of Φ1, which is denoted by ue1, is found as

ue1s,v=−B.

Theorem 3.3. Let Φ1 be a e₁ Sannia ruled surface in E³. Then, the first and the second fundamental form, the Gaussian curvature and the mean curvature of Φ1 are calculated as

Ie1=1+v2κ2ds2+2dsdv+dv2,IIe1=−vκτds2,Ke1=0, He1=τ2vκ,

κ≠0, respectively.

Proof. Taking the second order partial differentials of the surface Φ1 given by (3.2) with respect to s and v, we get

Φ1ss=κN+v−κ2T+κ′N+κτB,Φ1sv=κN, Φ1vv=0.

Using the equation (2.4), the components of the first and the second fundamental form of Φ1 are obtained as follows:

Ee1=1+v2κ2, Fe1=1, Ge1=1,le1=−vκτ, me1=0, ne1=0.

From here, if the last equations are substituted in the equation (2.5), the proof is complete.

Corollary 3.4. Let X_T and Φ1 be a tangent ruled surface with base curve α and a e₁ Sannia ruled surface with base curve ζ which is striction curve of X_T, respectively. Then, the surfaces X_T and Φ1 are the same surfaces.

Corollary 3.5. Let X_T and Φ1 be the tangent ruled surface with base curve α and e₁ Sannia ruled surface with base curve ζ which is striction curve of X_T, respectively. If the striction curve ζ of X_T is planar curve, the e₁ Sannia ruled surface is developable and the minimal surface.

Definition 3.6. A surface Φ2 is called a e₂ Sannia ruled surface in Euclidean 3-space, if the surface Φ2 is generated by moving the vector e₂ along the striction curve ζ of X_T and its parametrical equation is defined as

(3.3)Φ2s,v=ζ(s)+ve2s.

Taking the first order partial differentials of Φ2 with respect to s and v, we have

Φ2s=1−vκT+vτB   and   Φ2v=N.

So, by (2.3), the normal vector field ue2 of Φ2 is obtained as

ue2s,v=−vτT+1−vκBv2τ2+1−vκ2.

Theorem 3.7. Let Φ2 be a e₂ Sannia ruled surface in E³. Then, the first and the second fundamental forms, the Gaussian curvature and the mean curvature of Φ2 are given as

Ie2=1−vκ2+vτ2ds2+dv2,IIe2=v2τκ′−τ′κ+vτ′v2τ2+1−vκ2ds2+2τv2τ2+1−vκ2dsdv,Ke2=−τ2v2τ2+1−vκ22, He2=v2τκ′−τ′κ+vτ′2v2τ2+1−vκ232,

respectively.

Proof. Taking the second order partial differentials of the surface Φ2 given by (3.3) with respect to s and v, we get

Φ2ss=κN+v−κ′T−κ2+τ2N+τ′B,φ2sv=−κT+τB, φ2vv=0.

From equations (2.4), the components of the first and the second fundamental form of Φ2 are obtained as follows:

Ee2=1−vκ2+vτ2, Fe2=0, Ge2=1,le2=v2τκ′−τ′κ+vτ′v2τ2+1−vκ2, me2=τv2τ2+1−vκ2, ne2=0.

From here, if these equations are substituted in the equation (2.5), the proof is complete.

Corollary 3.8. Let X_T and Φ2 a be tangent ruled surface with base curve α and e₂ Sannia ruled surface with base curve ζ which is striction curve of X_T, respectively. If the striction curve ζ of X_T is planar curve, the ruled surface Φ2 with the Sannia frame is developable and the minimal surface. Also, since Ke2<0, all points of the ruled surface Φ2 are hyperbolic points.

Definition 3.9. A surface Φ3 is called a e₃ Sannia ruled surface in Euclidean 3-space, if the surface Φ3 is generated by moving the vector e₃ along the striction curve ζ of X_T and its parametrical equation is defined as

(3.4)Φ3s,v=ζ(s)+ve3s.

Taking the first order partial differentials of Φ3 with respect to s and v, we have

Φ3s=T−vτN   and   Φ3v=B.

So, by considering (2.3) the normal vector field ue3 of Φ3 is obtained as

ue3s,v=−vτT+N1+vτ2.

Theorem 3.10. Let Φ3 be a e₃ Sannia ruled surface in E³. Then, the first and the second fundamental forms, the Gaussian curvature and the mean curvature of Φ3 are obtained as

Ie31=1+v2κ2ds2+dv2,IIe3=−κ1+v2τ2−vτ′1+vτ2ds2+2τ1+vτ2dsdv,Ke3=−τ21+v2κ22, He3=−κ1+v2τ2−vτ′21+v2κ232,

respectively.

Proof. Taking the second order partial differentials of the surface Φ3 given by (3.2) with respect to s and v, we reach

Φ3ss=vτκT+κ−vτ′N−vτ2B, Φ3sv=−τN, Φ3vv=0.

So, by recalling the equation (2.4), the components of the first and the second fundamental form of Φ3 are given as follows:

Ee3=1+v2τ2, Fe3=0, Ge3=1,le3=−κ1+v2τ2−vτ′1+vτ2, me3=τ1+vτ2, ne3=0.

From here, if these equations are substituted in the equation (2.5), the proof is complete.

Example 3.11. Consider the curve

αs=34coss+cos3s9,sins+sin3s9,−2coss3

with Frenet vectors and the curvatures as follows:

T=34−sins−sin3s3,coss+cos3s3,2sins3,N=−3cos2s2,−3sin2s2,12,B=3coss−cos3s4,sins3,3coss2,κ=3coss,τ=3sins

[10]. Since the striction curve and the base curve of tangent ruled surface are the same curve, the equations of the ruled surfaces with the Sannia frame e1,e2,e3 are

Φ1s,v=34coss+cos3s9,sins+sin3s9,−2coss3+34v−sins−sin3s3,coss+cos3s3,2sins3,Φ2s,v=34coss+cos3s9,sins+sin3s9,−2coss3+v−3cos2s2,−3sin2s2,12,Φ3s,v=34coss+cos3s9,sins+sin3s9,−2coss3+v3coss−cos3s4,sins3,3coss2,

respectively, (Figure.1).

Figure 1. Sannia ruled surfaces associated with tangent ruled surface with s∈−1,3 and v∈−1,1

3.2. Sannia ruled surfaces associated with normal ruled surface

Theorem 3.12. Let XN be a normal ruled surface and f1,f2,f3 be the Sannia frame on the striction curve of XN, denoted by β. Then, the relationship between the Sannia frame f1,f2,f3 on striction curve and the Frenet frame T,N,B is as follows:

f1=N, f2=−κκ2+τ2T+τκ2+τ2B,f3=τκ2+τ2T+κκ2+τ2B

whereκ2+τ2≠0.

Proof. Considering the equation (2.6), we can easily calculate the striction curve of the normal ruled surface by fallowing:

β(s)=αs+κκ2+τ2N.

By the definition of XN, we say f1=N and also, by using the equations (2.1) and (2.7), the vectors f₂ and f₃ are computed as

f2=−κκ2+τ2T+τκ2+τ2B and f3=τκ2+τ2T+κκ2+τ2B.

Definition 3.13. A surface Γ1 is called a f₁ Sannia ruled surface in E³, if the surface Γ1 is generated by moving the vector f₁ along the striction curve β of XN. The parametrical equation of f₁ ruled surface is defined as

Γ1s,v=β(s)+vf1s

where β(s)=αs+κκ2+τ2N   and   f1=N.

Taking the first order partial differentials of Γ1 with respect to s and v, we get

Γ1s=λ1T+λ2N+λ3B, Γ1v=N

such that

λ1=τ2κ2+τ2−vκ,λ2=κκ2 +τ2 ′  and   λ3=τκκ2+τ2+v.

So, by considering (2.3) the normal vector field of Γ1 which is denoted by uf1 is found as

uf1s,v=−λ3λ32+λ12T+λ1λ32+λ12B.

Theorem 3.14. Let Γ1 be a f₁ Sannia ruled surface. Then the Gaussian curvature and the mean curvature of Γ1 are

Kf1=−τ2λ32+λ12  and  Hf1=λ1 λ′ 3−λ2τ− λ′ 1λ32λ12+λ3232 ,

respectively.

Proof. Taking the second order partial differential of Γ1 given by (3.1), we get

Γ1ss=λ′1−λ2κT+λ′2+λ1κ−λ3τN +λ′3+λ2τB,Γ1sv=−κT+τB, Γ1vv=0

By using these equations, the components of the first and the second fundamental form of Γ1 are found as

Ef1=λ12+λ22+λ32, Ff1=λ2, Gf1=1,lf1=λ3− λ′1+λ2κ+λ1 λ′3+λ2τλ12+λ32, mf1=τ, nf1=0.

From the equation (2.5), we reach the desired.

Corollary 3.15. Let XN be a normal ruled surface in E³. if the base curve α of XN is planar curve, then the f₁ Sannia ruled surface is developable and minimal surface.

Definition 3.16. A surface Γ2 is called a f₂ Sannia ruled surface in E³, if the surface Γ2 is generated by moving the vector f₂ along the striction curve β of XN. The parametric equation of f₂ Sannia ruled surface is defined as

(3.2)Γ2s,v=β(s)+vf2s

where β(s)=αs+κκ2+τ2N and f2=−κκ2+τ2T+τκ2+τ2B.

Taking the first order partial differentials of Γ2 with respect to s and v, we get

Γ2s=η1T+η2N+η3B,Γ2v=−κκ2+τ2T+τκ2+τ2B

such that

η1=τ2κ2+τ2−v κ κ 2 + τ 2 ′,η2= κ κ 2 + τ 2 ′−vκ2+τ2,η3=κτκ2+τ2+v τ κ 2 + τ 2 ′.

So, by considering (2.3) the normal vector field of Γ2 which is denoted by uf2 is found as

uf2=η2τT−η1τ+η3κN+η2κBη22κ2+τ2+ η1 τ+η3 κ2.

Theorem 3.17. Let Γ2 be a f₂ Sannia ruled surface in E³, then the Gaussian curvature and the mean curvature of Γ2 are

Kf2=−κ2+τ2 η3 κ+η1 τ η′ 2−η2 η′ 1 τ+ η′ 3 κ2 η3 κ+η1 τ2+η22κ2+τ22,Hf2=κ2+τ2κτη3 2 −η1 2 +η2 τ η′1+κ η′3   −η1η3 κ2 −τ2 − η′2 η3κ+η1τ −2 η1 κ−η3 τκ2+τ2 η3κ+η1τ η′2   −η2 τ η′1+κ η′3 2 η1 τ+η3 κ2+η22κ2+τ232,

respectively.

Proof. Taking the second order partial differential of Γ2, we have

Γ2ss=η′1−η2κT+η′2+η1κ−η3τN+η′3+η2τB,Γ2sv=η′1T+η′2N+η′3B, Γ2vv=0.

From here, the component of the first and the second fundamental forms of Γ2 are computed as

Ef2=η12+η22+η32, Ff2=η3τ−η1κκ2+τ2, Gf2=1,lf2= η2τ η′ 1 −η2 κ+η2κ η′ 3 +η2 τ − η1 τ+η3 κ η′ 2 +η1 κ−η3 τ η22κ2+τ2+ η1τ+η3κ2,mf2=η2 η′ 3κ+ η′ 1τ− η′2η3κ+η1τη22κ2+τ2+ η1τ+η3κ2,nf2=0.

So, substituting these equations into (2.5), the proof is complete.

Definition 3.18. A surface Γ3 is called a f₃ Sannia ruled surface in E³, if the surface Γ3 is generated by moving the vector f₃ along the striction curve β of XN. The parametric equation of f₃ Sannia ruled surface is defined as

(3.3)Γ3s,v=β(s)+vf3s

where β(s)=αs+κκ2+τ2N and f3=τκ2+τ2T+κκ2+τ2B.

Taking the first order partial differentials of Γ3 with respect to s and v, we get

Γ3s=μ1T+μ2N+μ3B,Γ3v=τκ2+τ2T+κκ2+τ2B

such that

μ1=τ2κ2+τ2+v τ κ 2 + τ 2 ′,μ2= κ κ 2 + τ 2 ′,μ3=κτκ2+τ2+v κ κ 2 + τ 2 ′.

So, considering (2.3) the normal vector field of Γ3 which is denoted by uf3 is found as

uf3=μ2κT+μ3τ−μ1κN−μ2τBμ22κ2+τ2+ μ3 τ−μ1 κ2.

Theorem 3.19. Let Γ3 be a f₃ Sannia ruled surface in E³, then the Gaussian curvature and the mean curvature of Γ3 are

Kf3=−κ2+τ2−μ1 μ′ 2κ+μ3 μ′ 2τ+κμ2 μ′ 1−τμ2 μ′ 32μ3τ−μ1κ2+μ22κ2+τ2,Hf3= κ2+τ2 μ1κ−μ3τ 2+τμ2 μ′3 − μ′1 μ2 κ  +μ2 2κ2+μ2 2τ2+μ1 μ ′κ−μ3 μ ′τ +2 μ3κ+μ1τκ2+τ2 μ1 μ′ 2κ−μ3 μ′ 2τ+τμ2 μ′ 3−μ ′ μ2 1κ 2 μ1 κ−μ3 τ2+μ22κ2+τ232,

respectively.

Proof. Taking the second order partial differential of Γ3, we have

Γ3ss=μ′1−μ2κT+μ′2+μ1κ−μ3τN+μ′3+μ2τB, Γ3sv=μ′1T+μ′2N+μ′3B, Γ3vv=0.

From here, the components of the first and the second fundamental forms of Γ3 are computed as

Ef3=μ12+μ22+μ32, Ff3=μ1τ+μ3κκ2+τ2, Gf3=1,lf3= μ2κ μ′ 1 −μ2 κ−μ2τ μ′ 3 +μ2 τ + μ3 τ−μ1 κ μ′ 2 +μ1 κ−μ3 τ μ22κ2+τ2+ μ3τ−μ1κ2,mf3=−μ1κ+μ3τ μ′2+μ2κ μ′ 1−τ μ′ 3μ22κ2+τ2+ μ3τ−μ1κ2, nf3=0.

Substituting these into (2.5) completes the proof.

Example 3.20. Considering the curve α given by example 3.1, the striction curve and Sannia frame vectors of XN are found as

β(s)=−13coss−2+cos2s,2sins33,−coss3,f1=−123cos2s,−123sin2s,12,f2=sin2s,−cos2s,0,f3=12cos2s,cosssins,32.

So, the ruled surfaces with Sannia frame are given by the following forms:

Γ1s,v=−13coss−2+cos2s,2sins33,−coss3 +v−123cos2s,−123sin2s,12,Γ2s,v=−13coss−2+cos2s,2sins33,−coss3 +vsin2s,−cos2s,0,Γ3s,v=−13coss−2+cos2s,2sins33,−coss3 +v12cos2s,cosssins,32.

Figure 2. Sannia ruled surfaces associated with normal ruled surface with s∈−1,3 and v∈−1,1

3.3. Sannia ruled surfaces associated with binormal ruled surface

Theorem 3.21. Let X_B be a binormal ruled surface and g1,g2,g3 be Sannia frame on the striction curve of X_B. Then, the relationship between the Sannia frame and the Frenet frame T,N,B is as follows:

(3.1)g1=B, g2=−N, g3=T.

Proof. Let the curve δ be a striction curve of the binormal ruled surface X_B. By using (2.6), it can be easily shown that the striction curve δ is equal to the base curve of X_B, i.e., δ(s)=αs. From the definition of X_B, we say g1=B and by using the equations (2.1) and (2.7), the vectors we compute g₂ and g₂ as

g2=−N   and   g3=T.

Definition 3.22. The surfaces Ψ1,Ψ2 and Ψ3 are called g₁, g₂ and g₃ Sannia ruled surfaces in E³, if the surfaces Ψ1,Ψ2 and Ψ3 are generated by moving the vectors g₁, g₂ and g₃ along the striction curve δ of X_B, respectively. The parametrical equations of Ψ1,Ψ2 and Ψ3 Sannia ruled surfaces are defined as

Ψ1s,v=δ(s)+vg1s,Ψ2s,v=δ(s)+vg2s,Ψ3s,v=δ(s)+vg3s

where g1=B, g2=−N and g3=T.

Corollary 3.23. Let e₁ and e₃ be Sannia surfaces of the tangent ruled surface and g₁ and g₃ be Sannia ruled surfaces of the binormal ruled surface, then there are the following expressions:

• 1. The g₁ and e₃ Sannia ruled surfaces are the same surfaces.

• 2. The g₃ and e₁ Sannia ruled surfaces are the same surfaces.

Example 3.24. Let us consider the curve α given by example 3.1. As proved above, the striction curve δ and the base curve α of X_B are the same curve and g1=B, g2=−N and g3=T. In that case, The equations of ruled surfaces with Sannia frame g1,g2,g3 of X_B are expressed as

Ψ1s,v=34coss+cos3s9,sins+sin3s9,−2coss3+v3coss−cos3s4,sins3,3coss2,Ψ2s,v=34coss+cos3s9,sins+sin3s9,−2coss3−v−3cos2s2,−3sin2s2,12,Ψ3s,v=34coss+cos3s9,sins+sin3s9,−2coss3+34v−sins−sin3s3,coss+cos3s3,2sins3.

Figure 3. Sannia ruled surfaces associated with binormal ruled surface with s∈−1,3 and v∈−1,1.

3.4. Sannia ruled surfaces associated with Darboux ruled surface

Theorem 3.25. Let  XC be the Darboux ruled surface and q1,q2,q3 be Sannia frame on the striction curve ϖ of  XC in E³. Then the relation between the Sannia frame and the Frenet frame T,N,B is given as

q1=sinφT+cosφB,q2=−cosφT+sinφB, q3=N

where the angle φ is between the Darboux vector W and the binormal vector B.

Proof. By considering the equation (2.6), the striction curve of X_C can be written as

ϖ(s)=αs−α′,C′C′,C′C=αs−cosφφ′C.

By the definition of the surface  XC, the Sannia frame vectors on the striction curve of  XC are computed as

q1=C=sinφT+cosφB,q2=C′C′=−cosφT+sinφB,q3=q1×q2=−N.

Definition 3.26. A surface Δ1 is called q₁ Sannia ruled surfaces in E³, if the surface Δ1 is generated by moving the vector q₁ along the striction curve ϖ of X_C. The parametric equation of q₁ Sannia ruled surface is defined as

(3.1)Δ1s,v=ϖ(s)+vq1s

where ϖ(s)=αs−cosφφ′C  and  q1=sinφT+cosφB.

Theorem 3.27. Let Δ1 be a q₁ Sannia ruled surface, then the normal vector field of Δ1 and the principal normal vector of the curve α are linearly dependent.

Proof. When substituted the equations ϖ(s)=αs−cosφφ′C and q1=sinφT+cosφB into the parametric form of Δ1 given in (3.1), we get

Δ1s,v=αs+vφ′−cosφφ′C.

Taking the first order partial differential of this equation with respect to s and v, and by performing the necessary operation, we can write

Δ1s×Δ1v=−φ′vN.

From here, the normal vector field denoted by uq1 of Δ1 is found as

uq1=±N.

Theorem 3.28. Let Δ1 be a q ₁ Sannia ruled surface, then the Gaussian curvature and the mean curvature of Δ1 are

Kq1=0   and   Hq1=−vφ′.W22cos2φ+2sinφcosφ φ ′′+vφ′ 2−1,

respectively.

Proof. Taking the second order partial differentials of Δ1 results

Δ1ss=ϖ″(s)+vq ″ 1, Δ1sv=q′ 1  and  Δ1vv=0.

By using the equation (2.4), the components of the first and second fundamental forms of Δ1 are computed as

Eq1=1+ cosφ φ′ ′ 2+ vφ′ 2+cos2φ, Fq1=sinφ− cosφ φ ′ ′, Gq1=1,lq1=−vφ′κ−cosφW− vφ′ 2W, mq1=0, nq1=0.

By substituting these equations into (2.5), the proof is complete.

Corollary 3.29. The q₁ Sannia ruled surface is always a developable surface.

Definition 3.30. A surface Δ2 is called q₂ Sannia ruled surfaces in E³, if the surface Δ2 is generated by moving the vector q₂ along the striction curve ϖ of X_C. The parametric equation of q₂ Sannia ruled surface is defined as

(3.2)Δ2s,v=ϖ(s)+vq2s

where ϖ(s)=αs−cosφφ′C and q2=−cosφT+sinφB.

By substituting the latter equations ϖ and q₂ into (3.2), we get

Δ2s,v=αs−cosφφ′C+v−cosφT+sinφB.

Taking the first order partial differentials of this equation with respect to s and v, we simply calculate

Δ2s×Δ2v=sinφ−cosφφ′′N−vφ′N+WC.

So, the normal vector field uq2 of Δ2 is computed as

uq2=sinφ−cosφφ′ ′N−vφ′N+WCsinφ− cosφφ′ ′ 2+v2φ′2+W2.

Theorem 3.31. Let Δ2 be a q₂ Sannia ruled surface, then the Gaussian curvature and the mean curvature of Δ2 are

Kq2=−W2sinφ−cosφφ′ ′−vφ′+vφ″2sinφ−cosφφ′ ′2+v2 φ ′ 2+W2,Hq2=−Wsinφ−cosφ φ ′ ′−vφ′+vφ″.

respectively.

Proof. The second order partial differentials of Δ2 are given as

Δ2ss=ϖ″(s)+vq ″ 2, Δ2sv=q′ 2, Δ2vv=0.

By using (2.4), the components of the first and second fundamental forms of Δ2 are computed as

Eq2=− cosφφ′ ′sinφ+cosφ+ cosφφ′ ′ 2+ vφ ′ 2 +sin2φ+2φ′−sinφ+ cosφ φ ′ ′+v2 φ ′ 2+W2,Fq2=1, Gq2=0,lq2=vW cosφ φ ′ ′′− cosφ φ ′ ′−vκφ′+vτsinφ− cosφ φ ′ ′ 2+v2 φ ′ 2+W 2,mq2=Wsinφ− cosφ φ ′ ′−vφ′+vφ″sinφ− cosφ φ ′ ′ 2+v2 φ ′ 2+W 2,nq2=0.

By substituting these equations into (2.5), the proof is complete.

Definition 3.32. A surface Δ3 is called q₃ Sannia ruled surface in E³, if the surface Δ3 is generated by moving the vector q₃ along the striction curve ϖ of X_C. The parametric equation of q₃ Sannia ruled surface is defined as

Δ3s,v=ϖs+vq3

where ϖs=αs−cosφφ′C   and   q3=−N.

We take derivate of this equation with respect to s and v, it is found that

Δ3s=ϖ′(s)+vN′, Δ3v=−N.

Therefore, the normal vector field of Δ3 can be written as

uq3=cosφcosφφ′′T+1−sinφcosφφ′′B−cosφ−vWC1+ cosφφ′ ′ 2+cos2φ+vW2.

Theorem 3.33. Let Δ3 be a q₃ Sannia ruled surface, then the Gaussian curvature and the mean curvature of Δ3 are

Kq3=0,Hq3=2φ′ cosφ φ ′ ′sinφ− cosφ φ ′ ′ +vW− cosφ φ ′ ′′+φ′cosφ +vκ′sinφ+τ′cosφvW−cosφ −v cosφ φ ′ ′κ′cosφ+τ′sinφ+vτ′ 21+ cosφ φ ′ ′ 2+cos2φ+ vW 2 1+ cosφ φ ′ ′ 2+cos2φ+v2W2

where φ′≠0.

Proof. Taking the second order partial differential of Δ3, it follows that

Δ3ss=ϖ″(s)−vN″, Δ3sv=κT−τB, Δ3vv=0.

By using the equation (2.4), the components of the first and second fundamental forms of Δ3 are computed as

Eq3=1+ cosφ φ′′2+cos2φ+ vW2,Fq3=0, Gq3=1,lq3= 2φ′sinφ cosφ φ ′ ′−2φ′ cosφ φ′ ′ 2 +v W − cosφ φ ′ ′′ +φ′ cosφ+vτ′ +v κ′sinφ+τ′ cosφ v W −cosφ −v cosφ φ ′ ′ κ′cosφ+τ′sinφ 1+ cosφ φ ′ ′ 2 +cos2 φ+ v W 2 , mq3=0, nq3=0.

When substituted these into (2.5), the proof is complete.

Example 3.34. Considering the curve α given in example 3.1, the striction curve and the Sannia frame vectors of X_C are found as

ϖ(s)=2coss−cosscos2s3,2Sins33,−3Coss,q1=12cos2s,12sin2s,32,q2=sin2s,−cos2s,0,q3=3cos2s2,3sin2s2,−12.

So, the q₁, q₂ and q₃ Sannia ruled surfaces are given by the following forms:

Δ1s,v= 3coss+3vcos2s−cos3s 6, v3sin2s+4sin s 36,3 v−2coss 2 ,Δ2s,v= 2coss−cosscos2s+3vsin2s 3, 2sin s 3−3vcos2s 3,−3coss ,Δ3s,v= 3coss+33vcos2s−cos3s 6, 4sin s 3+33vsin2s 6,−v+23coss 2 .

Figure 4. Sannia ruled surfaces associated with Darboux ruled surface with s∈−1,3 and v∈−1,1.