Processing math: 81%
Research article Topical Sections

Construction and analysis of the quasi-ruled surfaces based on the quasi-focal curves in R3

  • This paper presents the concept of a quasi-ruled surface, which is a ruled surface generated by a base curve and a ruling, both of which are defined by the quasi-frame (q-frame). This study begins with the original curve defined by the q-frame, and then we focus on the focal curve of the original curve, which serves as the base curve of the ruled surface. We define the focal curve by the q-frame, so the terminology quasi-focal curve is used in this paper. This paper investigates the formation and properties of the quasi-ruled surface (QRS) using a quasi-focal curve (QFC) as the base curve (directrix). The ruling of the surface is expressed in terms of the q-frame associated with the QFC. A variety of QRS types are discussed in this study, including the osculating, normal, and rectifying types. In addition, the types of a quasi-tangent developable surface, a quasi-principal normal surface, and a quasi-binormal ruled surface will also be discussed. The geometric properties of these surfaces, such as the first and second fundamental quantities, Gaussian curvature, mean curvature, second Gaussian curvature, and second mean curvature, are described. The conditions for their developability and minimality are derived. Moreover, we provide an example that includes the study of geometric properties and clear visualizations of these novel types of QRS.

    Citation: Samah Gaber, Asmahan Essa Alajyan, Adel H. Sorour. Construction and analysis of the quasi-ruled surfaces based on the quasi-focal curves in R3[J]. AIMS Mathematics, 2025, 10(3): 5583-5611. doi: 10.3934/math.2025258

    Related Papers:

    [1] Ayman Elsharkawy, Clemente Cesarano, Abdelrhman Tawfiq, Abdul Aziz Ismail . The non-linear Schrödinger equation associated with the soliton surfaces in Minkowski 3-space. AIMS Mathematics, 2022, 7(10): 17879-17893. doi: 10.3934/math.2022985
    [2] Yanlin Li, Kemal Eren, Kebire Hilal Ayvacı, Soley Ersoy . The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space. AIMS Mathematics, 2023, 8(1): 2226-2239. doi: 10.3934/math.2023115
    [3] Kemal Eren, Hidayet Huda Kosal . Evolution of space curves and the special ruled surfaces with modified orthogonal frame. AIMS Mathematics, 2020, 5(3): 2027-2039. doi: 10.3934/math.2020134
    [4] Emad Solouma, Ibrahim Al-Dayel, Meraj Ali Khan, Youssef A. A. Lazer . Characterization of imbricate-ruled surfaces via rotation-minimizing Darboux frame in Minkowski 3-space E31. AIMS Mathematics, 2024, 9(5): 13028-13042. doi: 10.3934/math.2024635
    [5] Emel Karaca . Non-null slant ruled surfaces and tangent bundle of pseudo-sphere. AIMS Mathematics, 2024, 9(8): 22842-22858. doi: 10.3934/math.20241111
    [6] Kemal Eren, Soley Ersoy, Mohammad Nazrul Islam Khan . Simultaneous characterizations of alternative partner-ruled surfaces. AIMS Mathematics, 2025, 10(4): 8891-8906. doi: 10.3934/math.2025407
    [7] Nural Yüksel, Burçin Saltık . On inextensible ruled surfaces generated via a curve derived from a curve with constant torsion. AIMS Mathematics, 2023, 8(5): 11312-11324. doi: 10.3934/math.2023573
    [8] Chang Sun, Kaixin Yao, Donghe Pei . Special non-lightlike ruled surfaces in Minkowski 3-space. AIMS Mathematics, 2023, 8(11): 26600-26613. doi: 10.3934/math.20231360
    [9] Yanlin Li, Kemal Eren, Soley Ersoy . On simultaneous characterizations of partner-ruled surfaces in Minkowski 3-space. AIMS Mathematics, 2023, 8(9): 22256-22273. doi: 10.3934/math.20231135
    [10] Yanlin Li, Kemal Eren, Kebire Hilal Ayvacı, Soley Ersoy . Simultaneous characterizations of partner ruled surfaces using Flc frame. AIMS Mathematics, 2022, 7(11): 20213-20229. doi: 10.3934/math.20221106
  • This paper presents the concept of a quasi-ruled surface, which is a ruled surface generated by a base curve and a ruling, both of which are defined by the quasi-frame (q-frame). This study begins with the original curve defined by the q-frame, and then we focus on the focal curve of the original curve, which serves as the base curve of the ruled surface. We define the focal curve by the q-frame, so the terminology quasi-focal curve is used in this paper. This paper investigates the formation and properties of the quasi-ruled surface (QRS) using a quasi-focal curve (QFC) as the base curve (directrix). The ruling of the surface is expressed in terms of the q-frame associated with the QFC. A variety of QRS types are discussed in this study, including the osculating, normal, and rectifying types. In addition, the types of a quasi-tangent developable surface, a quasi-principal normal surface, and a quasi-binormal ruled surface will also be discussed. The geometric properties of these surfaces, such as the first and second fundamental quantities, Gaussian curvature, mean curvature, second Gaussian curvature, and second mean curvature, are described. The conditions for their developability and minimality are derived. Moreover, we provide an example that includes the study of geometric properties and clear visualizations of these novel types of QRS.



    Ruled surfaces play a crucial role in differential geometry. They are characterized by the movement of generators, which are straight lines that produce the surface. Furthermore, a directrix (base curve) is any curve that crosses all of the generators (rulings). Understanding this concept is crucial for grasping the characteristics and applications of ruled surfaces in various geometric contexts.

    Many researchers are interested in studying ruled surfaces according to different frames. Tuncer [1] used a novel technique to study ruled surfaces in R3. These surfaces were defined in terms of their rulings, base curve curvatures, shape operators, and Gauss curvatures.

    In [2], the pitch, angle of pitch, and dual angle of pitch of the ruled surface in R3, corresponding to a closed curve on the dual unit sphere, were examined. The vectors of the Frenet and Bishop frames of the closed curve were also analyzed, resulting in a relationship between the dual angle of pitch and the pitch angle. In [3], a fundamental method was adopted to analyze the ruled surfaces, focusing on the most basic foliated submanifolds in R3. The structural functions of the ruled surfaces were specified. The geometric properties and kinematical characterizations of the non-developable ruled surfaces in R3 were investigated.

    In [4], the ruled surfaces in R3 were studied using the base curves with the Bishop frame. These surfaces were characterized by their directrices, Bishop curvatures, shape operators, and Gauss curvatures. Masal [5] developed ruled surfaces created by type-2 Bishop vectors, distinguishing Gaussian curvature (GC) and mean curvature (MC), as well as integral invariants. The fundamental forms, geodesic curvatures, normal curvatures, and geodesic torsions were determined.

    In [6], the Darboux frame was used to define the ruled surface and study its properties, including geodesic curvature, normal curvature, and geodesic torsion. In [7], parallel ruled surfaces with the Darboux frame in R3 were introduced, highlighting aspects such as developability, striction points, and distribution parameters. The Steiner rotation vector for such a kind of surface was determined, and the pitch length and angle of the parallel ruled surfaces associated with the Darboux frame were computed. In [8], a necessary and sufficient condition was established for a ruled surface to be the principal normal ruled surface of a space curve using the theories of ruled invariants in R3.

    In [9], the ruled surfaces created by normal and binormal vectors throughout a timelike space curve utilizing a q-frame were explored in three-dimensional Minkowski space. The directional evolutions of quasi-principal normal and quasi-binormal ruled surfaces were investigated, employing their directrices. The geometric properties of the ruled surfaces were examined, including their inextensibility, minimality, and developability. In [10], the striction curve of a non-cylindrical ruled surface is considered to be the base curve, with its ruling represented as linear combinations of Frenet-Serret frame (FSF) vectors from the first ruled surface.

    In [11], a novel family of ruled surfaces was constructed and studied via q-frame vectors, known as quasi-vectors. The features of these governed surfaces, such as the first and second fundamental forms, GC and MC, were determined. Furthermore, several geometric properties such as developability, minimality, striction curve, and distribution parameters were investigated. Senyurt et al. [12] introduced a new type of special ruled surface, where the construction of each surface is based on a Smarandache curve and a specified curve according to the FSF. The generator (ruling) is selected as the unit Darboux vector. The properties of those ruled surfaces were investigated using the first and second fundamental forms, as well as their corresponding curvatures.

    The q-frames of the rational and polynomial Bezier curves were computed algorithmically in [13]. The frame was constructed even at singular points based on the curve's second derivative. This study provides an important improvement to computer-aided geometric design research.

    Kaymanli et al. [14] derived ruled surfaces using a quasi-principal normal, and a quasi-binormal vectors along a spacelike curve in three-dimensional Minkowski space, leading to the formulation of the time evolution equations based on quasi-curvatures. Pal et al. [15] introduced a new type of ruled surfaces in R3, called ruled-like surfaces, which are generated by a base curve and a director curve. In addition, the properties of these surfaces, such as GC, MC, and the existence of Bertrand mates, were investigated.

    Using the FSF in R3, Gaber et al. [16] investigated a family of ruled surfaces formed of circular helices (W-curves). The second mean curvature (SMC), and the second Gaussian curvature (SGC) formulas were obtained, the properties of the constructed ruled surfaces were described, and the conditions for minimal, flat, Ⅱ-minimal, and Ⅱ-flat surfaces were determined. In addition, the conditions for the base curves of these surfaces were classified as a geodesic curve, an asymptotic line, and a principal line.

    In this work, we introduce a specified concept of QRS, which refers to a ruled surface whose base curve is defined by a q-frame, and the q-frame vectors of the base curve describe the ruling.

    This study focuses on a directrix, which is the focal curve of the original curve. In [17], the focal curve given by the q-frame is defined as QFC.

    The structure of this work is as follows: Section 2 provides background information on the fundamental ideas of curves and ruled surfaces in three-dimensional Euclidean space. Section 3 covers the construction of QRS from the QFC with specific geometric features. Section 4 provides techniques for constructing several innovative types of QRS, using a QFC as the base curve and influencing its ruling vector. Section 5 presents and visualizes novel types of QRS. Finally, we give a conclusion.

    In this section, we present some geometric concepts on curves in R3, defining the FSF, the q-frame, and their relationship. The construction of the QRS is based on specific concepts of the q-frame of the original curve, the quasi-focal curve (QFC). Therefore, it is important to highlight these concepts.

    Consider a unit speed curve α:IRR3 with an arc length parameter s. Let F={T,N,B} be an orthogonal FSF at the point p0 on the open curve, where T,N, and B are the unit tangent, unit principal normal, and unit binomial vectors, respectively. The FSF has the following characteristics [18]:

    T,N=N,B=T,B=0, T,T=N,N=B,B=1.

    TN=B, NB=T, and BT=N.

    Let κ=κ(s) and τ=τ(s) be the curvature and torsion of the open curve. Then, the Frenet equations are given by

    T=κN,N=κT+τB,B=τN,()=dds(). (2.1)

    The Frenet-Serret frame FSF loses effectiveness when the curvature of a curve is zero. To solve this issue, we use an alternative frame known as a q-frame, which is related to the equations of the Frenet-Serret frame. The q-frame offers several advantages, including the ability to be defined even in the absence of a tangent line. Additionally, the formation of the q-frame does not require the space curve to have a unit speed. Finally, the q-frame is easy to calculate.

    Definition 1. Let s represent the arc length along the curve α:IRR3 within the interval I. Assume that α(s) is a unit speed curve. Assume that {Tqα,Nqα,Bqα} is the q-frame, where Tqα is the unit quasi-tangent vector, Nqα is the unit quasi-principal normal vector, and Bqα is the unit quasi-binormal vector. The q-frame is defined as follows [19,20]:

    Tqα=α(s),Nqα=TqαuTqαu,Bqα=TqαNqα, (2.2)

    where () refers to the cross product and u represents the projection vector; for convenience, we select u=(0,0,1).

    Definition 2. Consider the q-frame {Tqα,Nqα,Bqα} of the curve α(s) at a point p, alongside the FSF {T,N,B} at the same point p on the curve. Let θ represent the Euclidean angle between the principal normal vector N and the quasi-principal normal vector Nqα. The relation between the directional q-frames and the FSF is provided by [19,20] as follows:

    [TqαNqαBqα]=[1000cosθsinθ0sinθcosθ][TNB]. (2.3)

    Definition 3. [19] The q-frame {Tqα,Nqα,Bqα}, where Tqα=T, possesses the characteristics outlined below:

    Tqα,Tqα=Nqα,Nqα=Bqα,Bqα=1,Tqα,Nqα=Tqα,Bqα=Nqα,Bqα=0,Nqα,N=Bqα,B=cosθ,Nqα,B=Bqα,N=sinθ. (2.4)

    Definition 4. [20] The relation between the curvatures κ and τ of the curve α described by the FSF and the curvatures κ1,κ2, and κ3 of the curve α described by the q-frame is established as follows:

    κ1=κcosθ,κ2=κsinθ,κ3=dθ+τ. (2.5)

    This paper uses quasi-curvatures, referred to κ1,κ2, and κ3, which are defined in the following manner [13]:

    κ1=Tqα,Nqα=det[α,α,u]αu,()=dds(),κ2=Tqα,Bqα=α,uα,αα,uαu,κ3=Bqα,Nqα=α,udet[α,α,u]αu2. (2.6)

    Lemma 1. [19] Let s represent the arc length along the curve α:IRR3 within the interval I. Assume that α(s) is a unit speed curve. The derivatives of the q-frame {T,Nqα,Bqα} with respect to the arc length s satisfy the following equations:

    [αTqα(s)Nqα(s)Bqα(s)]=[010000κ1(s)κ2(s)0κ1(s)0κ3(s)0κ2(s)κ3(s)0][αTqα(s)Nqα(s)Bqα(s)]. (2.7)

    In this paper, we focus on studying focal curves by employing a q-frame, and we refer to them as quasi-focal curves QFC.

    Definition 5. [17] Let s represent the arc length along the curve α:IRR3 within the interval I. Assume that α(s) is a unit speed curve. Consider {Tqα,Nqα,Bqα} be a quasi-frame for the original curve, and assume that Fα is its unit speed QFC, which is defined by

    Fα(s)=α(s)+φ1(s)Nqα+φ2(s)Bqα, (2.8)

    where the smooth functions φ1,φ2 are the quasi-focal curvatures. Here, we call the curve α the original curve.

    Theorem 2. Let s represent the parameter of the arc length along the curve α:IRR3 within the interval I. Assume that α(s) is a unit speed curve defined by the q-frame {Tqα,Nqα,Bqα}. Let Fα(sF(s)) be a QFC for the curve α and assume that Fα is a unit speed curve defined by the q-frame {TFq,NFq,BFq}. Let sF(s) be the QFC arc length parameter and assume that sF(s) is measured on the focal curve Fα(sF(s)) in the direction of increasing s on the curve α. The relation between the q-frame for the QFC Fα(s) and the q-frame for the original curve α is given by

    [TFqNFqBFq]=[0010ε0ε00][TqαNqαBqα],ε=±1. (2.9)

    Proof. Taking the s-derivative of (2.8) with respect to s, we have

    dFαds=α+φ1Nqα+φ1Nqα+φ2Bqα+φ2Bqα. (2.10)

    Substituting from (2.7) into (2.10), we have

    dFαds=dFαdsFdsFds=(1κ1φ1κ2φ2)Tqα+(φ1κ3φ2)Nqα+(φ2+κ3φ1)Bqα. (2.11)

    Since the QFC represents the centers of the tangential oscillating spheres, the components of Tqα and Nqα vanish. Then,

    1κ1φ1κ2φ2=0,φ1κ3φ2=0. (2.12)

    Hence, we have

    dFαdsFdsFds=(φ2+κ3φ1)Bqα. (2.13)

    Since Fα is a unit speed curve, then dFαdsF=1. Define TFq=dFαdsF as the unit quasi-tangent vector of Fα. Then,

    TFqdsFds=(φ2+κ3φ1)Bqα. (2.14)

    Taking the norm of the two sides of (2.14), then

    dsFds=|φ2+κ3φ1|.

    Since sF is measured on Fα(sF(s)) in the direction of increasing s on the curve α(s), then sF is an increasing function of s. So, dsFds>0, and then dsFds=φ2+κ3φ1. Hence, we obtain the quasi-binormal vector for the QFC:

    TFq=Bqα. (2.15)

    Let NFq be the quasi-principal normal vector to Fα, where

    NFq=TFquTFqu=BqαuBqαu,u=(1,0,0). (2.16)

    Assume that the quasi-tangent and the quasi-principal normal vectors for the curve α are defined by the following components:

    Tqα=(t1,t2,t3),Nqα=(n1,n2,n3). (2.17)

    Then,

    Bqαu=TqαNqαu=t3Nqα. (2.18)

    Substituting from (2.18) into (2.16), we have

    NFq=t3|t3|Nqα. (2.19)

    Hence,

    NFq=εNqα,ε=±1. (2.20)

    Since BFq=TFqNFq, then, by using (2.15) and (2.20), we obtain the quasi-binormal vector of the QFC as

    BFq=εTqα.

    Hence, the theorem holds.

    Remark 1. Throughout this paper, we assume ε=1. Therefore,

    TFq=Bqα,NFq=Nqα,BFq=Tqα. (2.21)

    Lemma 3. Consider a unit speed curve, α:IR3 defined by the q-frame {Tqα,Nqα,Bqα} with arc length s. Let Fα(sF(s)) be a QFC for the original curve α. Let sF(s) be the arc length parameter of the QFC and assume that sF(s) is measured on the focal curve Fα(sF(s)) in the direction of increasing s on the curve α. Let {TFq,NFq,BFq} be the q-frame for Fα. The q-frame of the quasi-focal curve Fα is constructed similarly to the q-frame of any curve by the following equations:

    ddsF[TFqNFqBFq]=[0κF1κF2κF10κF3κF2κF30][TFqNFqBFq]. (2.22)

    where κF1,κF2, and κF3 are the quasi-curvatures for the quasi-focal curve Fα, and they have the following relations with the quasi-curvatures of the original curve κ1,κ2, and κ3:

    κF1=dTFqdsF,NFq=εκ3|φ2+κ3φ1|,κF2=dTFqdsF,BFq=εκ2|φ2+κ3φ1|,κF3=dBFqdsF,NFq=κ1|φ2+κ3φ1|. (2.23)

    Theorem 4. [17] Consider a unit speed curve α:IR3 with its QFC Fα. Then, the quasi-focal curvatures φ1 and φ2 are given by

    φ1=eκ1κ3κ2ds(eκ1κ3κ2dsκ3κ2ds+C),φ2=1κ2κ1κ2eκ1κ3κ2ds(eκ1κ3κ2dsκ3κ2ds+C), (2.24)

    where C is a constant of integration.

    Definition 6. [21] Let γ(s):IRR3 be a unit speed curve with an arc length parameter s. A ruled surface is a surface constructed by straight lines parametrized by γ(s) and η(s). It has the following parametrization:

    Ψ(s,v)=γ(s)+vη(s),

    where γ=γ(s):IRR3 is the directrix or base curve, and η(s) represents a unit vector in the direction of the ruling of the ruled surface.

    Definition 7. [22] Let γ(s):IRR3 be a unit speed curve with the arc length parameter s, and let {T,N,B} be the Frenet frame of the curve at a point q. The ruled surface Ψ(s,v)=I×RR3 defined by

    Ψ(s,v)=γ(s)+vη(s),η(s)=μ1(s)T(s)+μ2(s)N(s),

    is called the generalized osculating type ruled surface, where μ1(s) and μ2(s) are smooth functions (μ21+μ22=1). The following cases can be given:

    1. If μ1(s)=0 and μ2(s)=±1, then the surface Ψ(s,v) is a principal normal surface along the base curve.

    2. If μ2(s)=0 and μ1(s)=±1, then the surface Ψ(s,(v) is a tangent developable surface along the base curve.

    Definition 8. [23] Let γ(s):IRR3 be a unit speed curve with the arc length parameter s with Frenet frame {T,N,B} at a point q on the base curve. The ruled surface Ψ(s,v):I×RR3 defined by

    Ψ(s,v)=γ(s)+vη(s),η(s)=μ2(s)N(s)+μ3(s)B(s),

    is called the generalized normal ruled surface, where μ2,μ3 are smooth functions of the arc length parameter s, and μ22+μ23=1. The following cases can be given:

    1. If μ2(s)=±1 and μ3(s)=0, then the ruled surface Ψ(s,v) is called the principal normal surface along the base curve γ(s).

    2. If μ2(s)=0 and μ3(s)=±1, then the ruled surface Ψ(s,v) is called the binormal surface along the base curve γ(s).

    Definition 9. [24] Let γ(s):IRR3 be a unit speed curve with the arc length parameter s, with FSF {T,N,B} at a point q on the base cuve, and assume that κ and τ are the curvature and torsion of the curve. The ruled surface Ψ(s,v):I×RR3 is determined as

    Ψ(s,v)=γ(s)+vη(s),η(s)=μ1(s)T(s)+μ3(s)B,

    is called the generalized rectifying ruled surface, where μ1(s), and μ3(s) are smooth functions and μ21+μ23=1.

    Lemma 5. [24] Let Ψ(s,v) be a generalized rectifying ruled surface of the base curve γ(s). Then:

    1. If μ1(s)=τ(s)κ2(s)+τ2(s) and μ3(s)=κ(s)κ2(s)+τ2(s), then the surface Ψ(s,v) is a rectifying developable surface along the base curve γ(s).

    2. If μ1(s)=0 and μ3(s)=±1, then the surface Ψ(s,v) is a binormal surface along the base curve γ(s).

    3. If μ1(s)=±1 and μ3(s)=0, then Ψ(s,v) is the tangent developable surface along γ(s).

    In this paper, we define a quasi-ruled surface QRS as a ruled surface generated by a base curve, which is described by the q-frame, and the ruling is defined by the q-frame of the base curve. We focus on the QFC of the original curve as a base curve (directrix) of the constructed QRS.

    The QRS has the following parametrization:

    ψ(s,v)=Fα(s)+vη(s), (3.1)

    where Fα is a quasi-focal curve and it serves as the base curve (directrix), and the line passing through Fα is called the ruling of the surface ψ(s,v) at Fα. The surface ψ(s,v) has singular points at (s,v) if ψsψv=0. Substituting from (2.8) into (3.1), the QRS can be expressed in terms of the original curve α(s) as

    ψ(s,v)=α(s)+φ1Nqα+φ2Bqα+vη(s), (3.2)

    where φ1 and φ2 are quasi-focal curvatures of α satisfying (2.12) and are given explicitly by Eq (2.24).

    Definition 10. [21] Consider the QRS that is defined by (3.2). It has a unit normal vector field nψ which is defined by

    nψ=ψsψvψsψv. (3.3)

    where ψs=ψ(s,v)s and ψv=ψ(s,v)v.

    Definition 11. [21] The geodesic curvature κg, normal curvature κn, and geodesic torsion τg of the QFC Fα on the surface ψ are defined as follows:

    κg=nψTFq,(TFq),κn=(TFq),nψ,τg=nψns,(TFq),()=dds(). (3.4)

    Definition 12. [21] The curve Fα lying on the surface ψ is a geodesic curve, an asymptotic line, and a principal line if and only if κg=0, κn=0, and τg=0, respectively.

    Definition 13. [21] The coefficients of the first fundamental form (CFFF) are defined as follows:

    g11=ψs,ψs,g12=ψs,ψv,g22=ψv,ψv. (3.5)

    Also, the coefficients of the second fundamental form (CSFF) are defined as follows:

    L11=ψss,nψ,L12=ψsv,nψ,L22=ψvv,nψ. (3.6)

    Definition 14. [21] The Gaussian curvature GC, the mean curvature MC, and the distribution parameter are denoted, respectively, by K, H, and λ, where they are given by

    K=L11L22L212g11g22g212, (3.7)
    H=g11L222g12L12+g22L112(g11g22g212), (3.8)
    λ=det(Fα,η,η)η2,()=dds(). (3.9)

    Definition 15. [25] The second mean curvature (SMC), denoted as HII, is defined for the QRS in three-dimensional Euclidean space R3 by

    HII=H+14ΔIIlog(|K|), (3.10)

    where ΔII stands for the Laplacian function. In explicit terms, we have

    HII=H+12|det(II)|i,jxi(|det(II)|Lijxj(ln|K|)), (3.11)

    where Lij is the inverse of Lij, and the indices i,j belong to {1,2}. Let the parameters {x1,x2} correspond to the coordinates {s,v}.

    Definition 16. [26] Let KII stand for the SGC of the QRS in R3. It is defined by using Brioschi's formula by replacing the curvature tensor L11,L12, and L22 with the metric tensor components g11,g12, and g22, respectively:

    KII=1(det(II))2(|12L11,vv+L12,sv12L22,ss12L11,sL12,s12L11,vL12,v12L22,sL11L1212L22,vL12L22||012L11,v12L22,s12L11,vL11L1212L22,sL12L22|), (3.12)

    where (),v=v,(),vv=2v2,(),s=s,(),ss=2s2, and (),sv=2vs. While the minimal surfaces are characterized by a vanishing SGC, KII=0, the converse is not true: A surface with KII=0 is not necessarily minimal.

    Definition 17. [21] A developable surface in R3 has a vanishing GC (K=0), while a minimal surface has a vanishing MC (H=0).

    Definition 18. [27] A non-developable surface in R3 is called Ⅱ-flat if the SGC, (KII=0), and Ⅱ-minimal if the SMC, (HII=0).

    Let s represent the arc length along the curve α:IRR3 within the interval I. Assume that α(s) is a unit speed curve defined by the q-frame {Tqα,Nqα,Bqα}. Consider Fα(sF(s)) to be a unit speed QFC for the original curve α, with arc length sF(s), and described by the q-frame {TFq,NFq,BFq}. The QFC Fα(sF(s)) is defined by (2.8), and the relation between the q-frame for the QFC and the q-frame of the original curve α is obtained by (2.21). In this section, we present some novel types of QRS constructed by the QFC as a base curve (directrix), and with the ruling that is given by the q-frame of Fα. We define the following novel types of QRS as follows:

    1. The osculating type of quasi-ruled surface whose ruling lies in the osculating plane {TFq,NFq} of the base curve Fα.

    ψ1(s,v)=Fα(s)+v(μ1TFq+μ2NFq),μ21+μ22=1.

    2. The normal type of quasi-ruled surface whose ruling lies in the normal plane {NFq,BFq} of the base curve Fα.

    ψ2(s,v)=Fα(s)+v(μ2NFq+μ3BFq),μ22+μ23=1.

    3. The rectifying type of quasi-ruled surface whose ruling lies in the rectifying plane {TFq,BFq} of the base curve Fα.

    ψ3(s,v)=Fα(s)+v(μ1TFq+μ3BFq),μ21+μ23=1.

    4. The quasi-tangent developable surface whose ruling parallels the quasi-tangent vector of Fα.

    ψ4(s,v)=Fα(s)+vTFq.

    5. The quasi-principal normal ruled surface whose ruling parallels the quasi-principal normal vector of Fα.

    ψ5(s,v)=Fα(s)+vNFq.

    6. The quasi-binormal ruled surface whose ruling parallels the quasi-binormal vector of Fα.

    ψ6(s,v)=Fα(s)+vBFq.

    Let Fα(sF(s)) be the QFC of the original curve α. Assume that the ruling lies in the osculating plane {TFq,NFq} of the base curve Fα. In this case, the constructed surface is called the osculating type of quasi-ruled surface, where

    η(s)=μ1TFq+μ2NFq,μ21+μ22=1. (4.1)

    Substituting from (2.21) into (4.1), then

    η(s)=μ2Nqα+μ1Bqα. (4.2)

    Substituting from (4.2) into (3.2), then we obtain the osculating type QRS:

    ψ1(s,v)=α(s)+(φ1+vμ2)Nqα+(φ2+vμ1)Bqα. (4.3)

    Taking the first derivative of (4.3) with respect to s, we have

    ψ1,s=α+(φ1+vμ2)Nqα+(φ1+vμ2)Nqα+(φ2+vμ1)Bqα+(φ2+vμ1)Bqα. (4.4)

    Substituting from (2.7) into (4.4), then

    ψ1,s=(1κ1(φ1+vμ2)κ2(φ2+vμ1))Tqα+(φ1+vμ2κ3(φ2+vμ1))Nqα+(φ2+vμ1+κ3(φ1+vμ2))Bqα. (4.5)

    Using relation (2.12), we obtain

    ψ1,s=v(κ1μ2+κ2μ1)Tqα+v(μ2κ3μ1)Nqα+(v(μ1+κ3μ2)+φ2+κ3φ1)Bqα.

    Choose

    ξ1=(κ1μ2+κ2μ1),ξ2=μ2κ3μ1,ξ3=μ1+κ3μ2,μ1ξ3+μ2ξ2=0,ξ4=φ2+κ3φ1. (4.6)

    Then,

    ψ1,s=vξ1Tqα+vξ2Nqα+(vξ3+ξ4)Bqα. (4.7)

    Taking the first derivative of (4.3) with respect to v, we have

    ψ1,v=μ2Nqα+μ1Bqα. (4.8)

    By substituting from (4.7) and (4.8) into (3.5), we obtain the following lemma.

    Lemma 6. The CFFF of the osculating type of QRS are given by

    g11=v2(ξ21+ξ22+ξ23)+2vξ3ξ4+ξ24,g12=μ1ξ4,g22=1. (4.9)

    Lemma 7. The normal vector nψ1 to the osculating type of QRS is given by

    nψ1=1ϵ1(((μ1ξ2μ2ξ3)vμ2ξ4)Tqαvμ1ξ1Nqα+vμ2ξ1Bqα),ϵ1=(v2(ξ21+ξ22+ξ23)+2vξ3ξ4+μ22ξ24)1/2. (4.10)

    Lemma 8. Consider the osculating type of QRS that is defined by (4.3). Then, the second partial derivatives with respect to s and v are given by

    ψ1,ss=(λ1vκ2ξ4)Tqα+(λ2vκ3ξ4)Nqα+(λ3v+ξ4)Bqα,ψ1,sv=ξ1Tqα+ξ2Nqα+ξ3Bqα,ψ1,vv=0, (4.11)

    where

    λ1=ξ1κ1ξ2κ2ξ3,λ2=ξ2+κ1ξ1κ3ξ3,λ3=ξ3+κ2ξ1+κ3ξ2. (4.12)

    Lemma 9. The CSFF of the osculating type of QRS are given as

    L11=1ϵ1(A1v2+A2v+κ2μ2ξ24),L12=μ2ξ1ξ4ϵ1,L22=0, (4.13)

    where

    A1=λ1(μ1ξ2μ2ξ3)λ2μ1ξ1+λ3μ2ξ1,A2=λ1μ2ξ4κ2ξ4(μ1ξ2μ2ξ3)+μ1κ3ξ1ξ4+μ2ξ1ξ4. (4.14)

    Lemma 10. The MC and GC for the osculating type of QRS are given directly by substituting from (4.9) and (4.13) into (3.7) and (3.8):

    H=12ϵ31(A1v2+A2v+μ2ξ24(κ2+2μ1ξ1)),K=(μ2ξ1ξ4)2ϵ41. (4.15)

    Let Fα(sF(s)) be the QFC of the original curve α. Assume that the ruling lies in the normal plane {NFq,BFq} of the base curve Fα. In this case, the constructed surface is called the normal type of quasi-ruled surface QRS, where

    η(s)=μ2NFq+μ3BFq,μ22+μ23=1. (4.16)

    Substituting from (2.21) into (4.16), then

    η(s)=μ2Nqαμ3Tqα. (4.17)

    Substituting from (4.17) into (3.2), we obtain the normal type of QRS, which has the following parametrization:

    ψ2(s,v)=α(s)vμ3Tqα+(φ1+vμ2)Nqα+φ2Bqα. (4.18)

    Taking the first derivative of (4.18) with respect to s, then

    ψ2,s=(1κ1(φ1+vμ2)κ2φ2vμ3)Tqα+(φ1+vμ2κ3φ2vκ1μ3)Nqα+(φ2+κ3φ1+v(μ2κ3μ3κ2))Bqα.

    Using relation (2.12), we obtain

    ψ2,s=v(μ3+μ2κ1)Tqα+v(μ2μ3κ1)Nqα+(φ2+κ3φ1+v(μ2κ3μ3κ2))Bqα.

    Choose

    ˜ξ1=(μ3+μ2κ1),˜ξ2=μ2μ3κ1,μ2˜ξ2μ3˜ξ1=0,˜ξ3=μ2κ3μ3κ2,ξ4=φ2+κ3φ1. (4.19)

    Then,

    ψ2,s=v˜ξ1Tqα+v˜ξ2Nqα+(v˜ξ3+ξ4)Bqα. (4.20)

    Taking the first derivative of (4.18) with respect to v, we have

    ψ2,v=μ3Tqα+μ2Nqα. (4.21)

    By substituting from (4.20) and (4.21) into (3.5), we obtain the following lemma.

    Lemma 11. The CFFF of the normal type of QRS are given as

    g11=v2(˜ξ21+˜ξ22+˜ξ23)+2v˜ξ3ξ4+ξ24,g12=0,g22=1. (4.22)

    Lemma 12. The normal vector nψ2 to the normal type of QRS is given by

    nψ2=1ϵ2(μ2(v˜ξ3+ξ4)Tqαμ3(v˜ξ3+ξ4)Nqα+v(μ2˜ξ1+μ3˜ξ2)Bqα),ϵ2=(v2(˜ξ21+˜ξ22+˜ξ23)+2v˜ξ3ξ4+ξ24)1/2. (4.23)

    Lemma 13. Consider the normal type of QRS that is defined by (4.18). Then, the second partial derivatives with respect to s and v are given by

    ψ2,ss=(˜λ1vκ2ξ4)Tqα+(˜λ2vκ3ξ4)Nqα+(˜λ3v+ξ4)Bqα,ψ2,sv=˜ξ1Tqα+˜ξ2Nqα+˜ξ3Bqα,ψ2,vv=0, (4.24)

    where

    ˜λ1=˜ξ1κ1˜ξ2κ2˜ξ3,˜λ2=˜ξ2+κ1˜ξ1κ3˜ξ3,˜λ3=˜ξ3+κ2˜ξ1+κ3˜ξ2. (4.25)

    Lemma 14. The CSFF of the normal type of QRS are given as

    L11=1ϵ2(˜A1v2+˜A2v+(κ2μ2+κ3μ3)ξ24),L12=ξ4ϵ2(μ2˜ξ1+μ3˜ξ2),L22=0, (4.26)

    where

    ˜A1=(˜λ1μ2+˜λ2μ3)˜ξ3+˜λ3(μ2˜ξ1+μ3˜ξ2),˜A2=(˜λ1μ2+˜λ2μ3)ξ4+(μ2κ2+μ3κ3)˜ξ3ξ4+(μ2˜ξ1+μ3˜ξ2)ξ4. (4.27)

    Lemma 15. The MC and GC for the normal type of QRS are given directly by substituting from (4.22) and (4.26) into (3.7) and (3.8).

    H=12ϵ32(˜A1v2+˜A2v+(μ2κ2+μ3κ3)ξ24),K=ξ24ϵ42(μ2˜ξ1+μ3˜ξ2)2. (4.28)

    Let Fα(sF(s)) be the QFC of the original curve α. Assume that the ruling lies in the rectifying plane {TFq,BFq} of the base curve Fα. In this case, the constructed surface is called the rectifying type of quasi-ruled surface QRS, where

    η(s)=μ1TFq+μ3BFq,μ21+μ23=1. (4.29)

    Substituting from (2.21) into (4.29), then

    η(s)=μ3Tqα+μ1Bqα. (4.30)

    Substituting from (4.30) into (3.2), we obtain the rectifying type of QRS with the following parametrization:

    ψ3(s,v)=α(s)vμ3Tqα+φ1Nqα+(φ2+vμ1)Bqα. (4.31)

    Taking the first derivative of (4.31) with respect to s, then

    ψ3,s=(1κ1φ1κ2φ2v(μ3+μ1κ2))Tqα+(φ1κ3φ2v(μ3κ1+μ1κ3))Nqα+(φ2+κ3φ1+v(μ1μ3κ2))Bqα.

    Using relation (2.12), we obtain

    ψ3,s=v(μ3+μ1κ2)Tqαv(μ3κ1+μ1κ3)Nqα+(φ2+κ3φ1+v(μ1μ3κ2))Bqα.

    Choose

    ˆξ1=(μ3+μ1κ2),ˆξ2=(μ3κ1+μ1κ3),ˆξ3=μ1μ3κ2,μ1ˆξ3μ3ˆξ1=0,ξ4=φ2+κ3φ1. (4.32)

    Then,

    ψ3,s=vˆξ1Tqα+vˆξ2Nqα+(vˆξ3+ξ4)Bqα. (4.33)

    Taking the first derivative of (4.31) with respect to v, we have

    ψ3,v=μ3Tqα+μ1Bqα. (4.34)

    By substituting from (4.33) and (4.34) into (3.5), we obtain the following lemma.

    Lemma 16. The CFFF of the rectifying type of QRS are given as

    g11=v2(ˆξ21+ˆξ22+ˆξ23)+2vˆξ3ξ4+ξ24,g12=μ1ξ4,g22=1. (4.35)

    Lemma 17. The normal vector nψ3 to the rectifying type of QRS is given by

    nψ3=1ϵ3(vμ1ˆξ2Tqα(v(μ1ˆξ1+μ3ˆξ3)+μ3ξ4)Nqα+vμ3ˆξ2Bqα),ϵ3=(v2(ˆξ21+ˆξ22+ˆξ23)+2vˆξ3ξ4+μ23ξ24)1/2. (4.36)

    Lemma 18. Consider the rectifying type of QRS that is defined by (4.31). Then, the second partial derivatives with respect to s and v are given by

    ψ3,ss=(ˆλ1vκ2ξ4)Tqα+(ˆλ2vκ3ξ4)Nqα+(ˆλ3v+ξ4)Bqα,ψ3,sv=ˆξ1Tqα+ˆξ2Nqα+ˆξ3Bqα,ψ3,vv=0, (4.37)

    where

    ˆλ1=ˆξ1κ1ˆξ2κ2ˆξ3,ˆλ2=ˆξ2+κ1ˆξ1κ3ˆξ3,ˆλ3=ˆξ3+κ2ˆξ1+κ3ˆξ2. (4.38)

    Lemma 19. The CSFF of the rectifying type of QRS are given as

    L11=1ϵ3(ˆA1v2+ˆA2v+κ3μ3ξ24),L12=μ3ˆξ2ξ4ϵ3,L22=0, (4.39)

    where

    ˆA1=(ˆλ1μ1+ˆλ3μ3)ˆξ2ˆλ2(μ1ˆξ1+μ3ˆξ3),ˆA2=(μ3ˆλ2+μ1κ2ˆξ2)ξ4+μ3ˆξ2ξ4+κ3ξ4(μ1ˆξ1+μ3ˆξ3). (4.40)

    Lemma 20. The MC and GC for the rectifying type of QRS are given directly by substituting from (4.35) and (4.39) into (3.7) and (3.8):

    H=12ϵ33(ˆA1v2+ˆA2v+μ3ξ24(κ3+2μ1ˆξ2)),K=(μ3ˆξ2ξ4)2ϵ43. (4.41)

    Let Fα(sF(s)) be the QFC of the original curve α. Assume that the ruling parallels the quasi-tangent vector TFq of the base curve Fα, so

    η(s)=TFq=Bqα. (4.42)

    Substituting from (4.42) into (3.2), we obtain the quasi-tangent developable surface as follows:

    ψ4(s,v)=α(s)+φ1Nqα+(φ2+v)Bqα. (4.43)

    Taking the first derivative of (4.43) with respect to s, we have

    ψ4,s=α+φ1Nqα+φ1Nqα+φ2Bqα+(φ2+v)Bqα. (4.44)

    Substituting from (2.7) into (4.44), then

    ψ4,s=(1κ1φ1κ2(φ2+v))Tqα+(φ1κ3(φ2+v))Nqα+(φ2+κ3φ1)Bqα. (4.45)

    Using relation (2.12), we obtain

    ψ4,s=vκ2Tqαvκ3Nqα+(φ2+κ3φ1)Bqα.

    Choose

    ξ4=φ2+κ3φ1,whereφ2=(1κ1φ1κ2). (4.46)

    Then, we have

    ψ4,s=vκ2Tqαvκ3Nqα+ξ4Bqα. (4.47)

    Taking the first partial derivative of (4.43) with respect to v, we have

    ψ4,v=Bqα. (4.48)

    Lemma 21. The CFFF for the quasi-tangent developable surface are given by

    g11=ξ24+v2(κ22+κ23),g12=ξ4,g22=1. (4.49)

    Lemma 22. The normal vector nψ4 to the quasi-tangent developable surface is given by

    nψ4=κ3Tqα+κ2Nqακ22+κ23. (4.50)

    Lemma 23. The CSFF of the quasi-tangent developable surface are given as

    L11=vκ22+κ23(κ2κ3κ2κ3κ1(κ22+κ23)),L12=0,L22=0. (4.51)

    Proof. Taking the second partial derivatives of (4.47) and (4.48) with respect to s and v, we obtain

    ψ4,ss=(ξ4κ2+v(κ2κ1κ3))Tqα(ξ4κ3+v(κ3+κ1κ2))Nqα+(ξ4v(κ22+κ23))Bqα,ψ4,sv=κ2Tqακ3Nqα,ψ4,vv=0. (4.52)

    Taking the inner product of (4.50) and (4.52) and substituting into (3.6), the lemma holds.

    Lemma 24. The MC and GC for the quasi-tangent developable surface are given directly by substituting from (4.49) and (4.51) into (3.7) and (3.8).

    H=κ2κ3κ2κ3κ1(κ22+κ23)2v(κ22+κ23)3/2,K=0. (4.53)

    Let Fα(sF(s)) be the QFC of the original curve α. Consider the case where the ruling parallels the quasi-principal normal vector NFq of the base curve Fα. Then,

    η(s)=NFq. (4.54)

    Substituting from (2.21) into (4.54), then

    η(s)=Nqα. (4.55)

    Substituting from (4.55) into (3.2), we obtain the quasi-principal normal ruled surface, which is given by

    ψ5(s,v)=α(s)+(φ1+v)Nqα+φ2Bqα. (4.56)

    Taking the first derivative of (4.56) with respect to s and using (2.7), then

    ψ5,s=(1κ1(φ1+v)κ2φ2)Tqα+(φ1κ3φ2)Nqα+(φ2+κ3φ1+vκ3)Bqα. (4.57)

    Using relation (2.12), we obtain

    ψ5,s=vκ1Tqα+(vκ3+ξ4)Bqα,ξ4=φ2+κ3φ1. (4.58)

    Taking the first derivative of (4.56) with respect to v, we have

    ψ5,v=Nqα. (4.59)

    By substituting from (4.58) and (4.59) into (3.5), we obtain the following lemma.

    Lemma 25. The CFFF of the quasi-principal normal ruled surface are given by

    g11=(κ21+κ23)v2+2vκ3ξ4+ξ24,g12=0,g22=1. (4.60)

    Lemma 26. The normal vector nψ5 to the quasi-principal normal ruled surface is given by

    nψ5=(vκ3+ξ4)Tqαvκ1Bqαv2(κ21+κ23)+2vκ3ξ4+ξ24. (4.61)

    Lemma 27. Consider the quasi-principal normal ruled surface that is defined by (4.56). Then, the second partial derivatives with respect to s and v are given by

    ψ5,ss=(v(κ1+κ2κ3)κ2ξ4)Tqα(v(κ21+κ23)+κ3ξ4)Nqα+(v(κ3κ1κ2)+ξ4)Bqα,ψ5,sv=κ1Tqα+κ3Bqα,ψ5,vv=0. (4.62)

    Lemma 28. The CSFF of the quasi-principal normal ruled surface are given as

    L11=1v2(κ21+κ23)+2vκ3ξ4+ξ24(v2(κ1κ3κ1κ3+κ2(κ21+κ23))+v(ξ4(κ1+2κ2κ3)κ1ξ4)+κ2ξ24),L12=κ1ξ4v2(κ21+κ23)+2vκ3ξ4+ξ24,L22=0. (4.63)

    Lemma 29. The MC and GC for the quasi-principal normal ruled surface are given directly by substituting from (4.60) and (4.63) into (3.7) and (3.8).

    H=v2(κ1κ3κ1κ3+κ2(κ21+κ23))+v(ξ4(κ1+2κ2κ3)κ1ξ4)+κ2ξ242(v2(κ21+κ23)+2vκ3ξ4+ξ24)3/2,K=(κ1ξ4)2(v2(κ21+κ23)+2vκ3ξ4+ξ24)2. (4.64)

    Let Fα(sF(s)) be the QFC of the original curve α. Consider the case where the ruling parallels the quasi-binormal vector BFq of the base curve Fα. Then,

    η(s)=BFq. (4.65)

    Substituting from (2.21) into (4.65), then

    η(s)=Tqα. (4.66)

    Substituting from (4.66) into (3.2), we obtain the quasi-binormal ruled surface as

    ψ6(s,v)=α(s)vTqα+φ1Nqα+φ2Bqα. (4.67)

    Taking the first derivative of (4.67) with respect to s and using (2.7), then

    ψ6,s=(1κ1φ1κ2φ2)Tqα+(φ1κ3φ2vκ1)Nqα+(φ2+κ3φ1vκ2)Bqα. (4.68)

    Using relation (2.12), we obtain

    ψ6,s=vκ1Nqα+(ξ4vκ2)Bqα,ξ4=φ2+κ3φ1. (4.69)

    Taking the first derivative of (4.67) with respect to v, we have

    ψ6,v=Tqα. (4.70)

    By substituting from (4.69) and (4.70) into (3.5), we obtain the following lemma.

    Lemma 30. The CFFF of the quasi-binormal ruled surface are given by

    g11=(κ21+κ22)v22vκ2ξ4+ξ24,g12=0,g22=1. (4.71)

    Lemma 31. The normal vector nψ6 to the quasi-binormal ruled surface is given by

    nψ6=(ξ4vκ2)Nqα+vκ1Bqαv2(κ21+κ22)+2vκ2ξ4+ξ24. (4.72)

    Lemma 32. Consider the quasi-binormal ruled surface that is defined by (4.67). Then, the second partial derivatives with respect to s and v are given by

    ψ6,ss=(v(κ21+κ22)κ2ξ4)Tqα(v(κ1κ2κ3)+κ3ξ4)Nqα+(ξ4v(κ2+κ1κ3))Bqα,ψ6,sv=κ1Nqακ2Bqα,ψ6,vv=0, (4.73)

    Lemma 33. The CSFF of the quasi-binormal ruled surface are given as

    L11=1v2(κ21+κ22)+2vκ2ξ4+ξ24(v2(κ1κ2κ1κ2+κ3(κ21+κ22))+v(ξ4(κ12κ2κ3)κ1ξ4)+κ3ξ24),L12=κ1ξ4v2(κ21+κ22)+2vκ2ξ4+ξ24,L22=0. (4.74)

    Lemma 34. The MC and GC for the quasi-binormal ruled surface are given directly by substituting from (4.71) and (4.74) into (3.7) and (3.8).

    H=v2(κ1κ2κ1κ2+κ3(κ21+κ22))+v(ξ4(κ12κ2κ3)κ1ξ4)+κ3ξ242(v2(κ21+κ22)+2vκ2ξ4+ξ24)3/2,K=(κ1ξ4)2(v2(κ21+κ22)+2vκ2ξ4+ξ24)2. (4.75)

    Remark 2. For the previous types of QRS, we obtained L22=0. So, the SMC and SGC for these types of QRS are given by

    HII=H+12L12(2s(vln|K|)v(L11L12vln|K|)),KII=12(L12)3(L12(2L12,svL11,vv)+L12,v(L11,v2L12,s)). (4.76)

    Ruled surfaces can be created in different ways, depending on the type of base curve, ruling, or modification to the base curve frame. Methods like using the Frenet frame, Bishop frame, and q-frame can be used. The choice of the process depends on the practical application.

    Example 1. Consider a unit speed curve α(s) given as the original curve with the following parametrization:

    α=(23(coss1),23sins,53s), (5.1)

    where s represents the arc length along the curve α with FSF given by

    T=(23sins,23coss,53),N=(coss,sins,0),B=(53sins,53coss,23). (5.2)

    The curvature and torsion κ,τ are given by

    κ=23,τ=53.

    The q-frame of the original curve α is given by

    Tqα=(23sins,23coss,53),Nqα=(coss,sins,0),Bqα=(53sins,53coss,23), (5.3)

    with quasi-curvatures κ1,κ2,κ3 given by

    \kappa_1 = \frac{-2}{3}, \; \kappa_2 = 0 , \;\kappa_3 = \frac{\sqrt{5}}{3}.

    Using (2.12), then \varphi_1 = \frac{-3}{2}\; , \; \varphi_2 = 0 , and the QFC associated with the original curve is given by

    \begin{equation} F_\alpha = \left(-\frac{1}{6} (4+5\cos s ), -\frac{5}{6}\sin s , \frac{\sqrt{5}}{3} s \right), \end{equation} (5.4)

    Using (2.21), the q-frame of the QFC F_\alpha is given by

    \begin{equation} \boldsymbol{T}_{q}^{F} = (\frac{-\sqrt{5}}{3} \sin s , \frac{\sqrt{5}}{3}\cos s , \frac{-2}{3} ), \; \boldsymbol{N}_{q}^{F} = (\cos s , \sin s , 0), \; \boldsymbol{B}_{q}^{F} = (\frac{2}{3} \sin s , \frac{-2}{3}\cos s , \frac{-\sqrt{5}}{3}). \end{equation} (5.5)

    Using (2.23), the quasi-curvatures for the QFC are given as

    \kappa_1^F = \frac{-2}{3}\;, \;\kappa_2^F = 0\;, \;\kappa_3^F = \frac{-4}{3\sqrt{5}}.

    Now, we can construct new types of QRS as follows:

    1. The osculating type of QRS:

    The osculating type of QRS has the following parametrization:

    \begin{equation*} \psi_{1}( s , v ) = \frac{1}{6}\left(-\sqrt{10} v \sin s -(3 \sqrt{2} v +5) \cos s -4, \sqrt{10} v \cos s -(3 \sqrt{2} v +5) \sin s , 2 (\sqrt{5} s -\sqrt{2} v )\right), \end{equation*}

    for \eta( s ) = \frac{1}{\sqrt{2}}({\bf T}_{q}^{F} -{\bf N}_{q}^{F}), \; \; \mu_1 = -\mu_2 = \frac{1}{\sqrt{2}} . This surface is illustrated with Figure 1(a). The normal vector to the surface is

    n_{\psi_{1}} = \frac{\left(2 \sqrt{5} \left(\sqrt{2} v +3\right) \sin s +6 \sqrt{2} v \cos s , 6 \sqrt{2} v \sin s -2 \sqrt{5} \left(\sqrt{2} v +3\right) \cos s , -14 \sqrt{2} v -15\right)}{3 \sqrt{56 v ^2+60 \sqrt{2} v +45}}.
    Figure 1.  Visualization of the osculating type, normal type, and rectifying type of QRS. The green curve represents the original curve, and the blue curve represents the quasi-focal curve for s \in [0, 6\pi] and v \in [0, 5] .

    Lemma 35. The CFFF and CSFF for the osculating type of QRS are given, respectively, by

    \begin{equation*} \begin{split} \boldsymbol{g}_{11} & = \frac{7 }{9} v ^2+\frac{5 }{3 \sqrt{2}} v +\frac{5}{4}, \quad \boldsymbol{g}_{12} = -\frac{\sqrt{5}}{2\sqrt{2}}, \quad \boldsymbol{g}_{22} = 1, \\ \boldsymbol{L}_{11} & = \frac{2 v \left(14 v +15 \sqrt{2}\right)}{9 \sqrt{56 v ^2+60 \sqrt{2} v +45}}, \quad \boldsymbol{L}_{12} = \frac{\sqrt{10}}{\sqrt{56 v ^2+60 \sqrt{2} v +45}}, \quad \boldsymbol{L}_{22} = 0. \end{split} \end{equation*}

    Lemma 36. The MC, GC, SMC, and SGC for the osculating type of QRS are given by

    \begin{equation*} \boldsymbol{H} = \frac{4 \left(28 v ^2+30 \sqrt{2} v +45\right)}{\left(56 v ^2+60 \sqrt{2} v +45\right)^{3/2}}, \quad \boldsymbol{K} = -\frac{720}{\left(56 v ^2+60 \sqrt{2} v +45\right)^2}. \end{equation*}
    \begin{equation*} \begin{split} \boldsymbol{H}_{II}& = \frac{8\bigg(5488 v ^4+11760 \sqrt{2} v ^3+25844 v ^2+14190 \sqrt{2} v +5085\bigg) }{45\left(56 v ^2+60 \sqrt{2} v +45\right)^{3/2}}, \\ \boldsymbol{K}_{II} & = \frac{28 v \left(784 v ^3+1680 \sqrt{2} v ^2+2430 v +675 \sqrt{2}\right)+8100}{45 \left(56 v ^2+60 \sqrt{2} v +45\right)^{3/2}}. \end{split} \end{equation*}

    Lemma 37. The geodesic curvature \kappa_g , normal curvature \kappa_n , and geodesic torsion \tau_g of the QFC F_\alpha on the surface \psi_1 are given, respectively, according to Eq (3.4), as follows:

    \begin{equation*} \kappa_g = \frac{-5 \left(2 \sqrt{2} v +3\right)}{3 \sqrt{56 v ^2+60 \sqrt{2} v +45}}, \; \kappa_n = \frac{-2 \sqrt{10} v }{3 \sqrt{56 v ^2+60 \sqrt{2} v +45}}, \; \tau_g = \frac{-2 \sqrt{5} v \left(28 v +15 \sqrt{2}\right)}{9 \left(56 v ^2+60 \sqrt{2} v +45\right)}. \end{equation*}

    2. The normal type of quasi-ruled surfaces:

    The normal type of quasi-ruled surface has the following parametrization:

    \begin{equation*} \psi_{2}( s , v ) = \frac{1}{6}\left(\sqrt{2} v (3 \cos s -2 \sin s )-5 \cos s -4, \sqrt{2} v (3 \sin s +2 \cos s )-5 \sin s , \sqrt{5} \left(2 s +\sqrt{2} v \right)\right), \end{equation*}

    for \eta( s ) = \frac{1}{\sqrt{2}}(\boldsymbol{N}_{q}^{F} -\boldsymbol{B}_{q}^{F}), \; \; \mu_2 = -\mu_3 = \frac{1}{\sqrt{2}} . This surface is illustrated with Figure 1(b).

    The normal vector to the surface is

    n_{\psi_{2}} = \frac{\sqrt{5}\left(-(4 v +6 \sqrt{2}) \sin s +\left(6 v -9 \sqrt{2}\right) \cos s , (6 v -9 \sqrt{2}) \sin s +(4 v +6 \sqrt{2}) \cos s , \frac{6} {\sqrt{5}}(\frac{5}{\sqrt{2}}-\frac{13 v }{3})\right)}{6 \sqrt{26 v ^2-30 \sqrt{2} v +45}}.

    Lemma 38. The CFFF and CSFF for the normal type of QRS are given, respectively, by

    \begin{equation*} \begin{split} \boldsymbol{g}_{11} & = \frac{1}{36} \left(26 v ^2-30 \sqrt{2} v +45\right), \quad \boldsymbol{g}_{12} = 0, \quad \boldsymbol{g}_{22} = 1, \\ \boldsymbol{L}_{11} & = -\frac{1}{18} \sqrt{65 v ^2-75 \sqrt{2} v +\frac{225}{2}}, \quad \boldsymbol{L}_{12} = \frac{2 \sqrt{5}}{\sqrt{26 v ^2-30 \sqrt{2} v +45}}, \quad \boldsymbol{L}_{22} = 0. \end{split} \end{equation*}

    Lemma 39. The MC, GC, SMC, and SGC for the normal type of QRS are given by

    \begin{equation*} \boldsymbol{H} = -\frac{\sqrt{10}}{2 \sqrt{26 v ^2-30 \sqrt{2} v +45}}, \quad \boldsymbol{K} = -\frac{720}{\left(26 v ^2-30 \sqrt{2} v +45\right)^2}, \end{equation*}
    \begin{equation*} \boldsymbol{H}_{II} = -\frac{13 \left(26 v ^2-30 \sqrt{2} v +45\right)+180}{36 \sqrt{10} \sqrt{26 v ^2-30 \sqrt{2} v +45}}, \;\; \boldsymbol{K}_{II} = -\frac{13 \sqrt{10}}{720} \sqrt{26 v ^2-30 \sqrt{2} v +45}, \end{equation*}

    Lemma 40. The geodesic curvature \kappa_g , normal curvature \kappa_n , and geodesic torsion \tau_g of the QFC F_\alpha on the surface \psi_2 are given, respectively, according to Eq (3.4) as follows:

    \begin{equation*} \kappa_g = \frac{5 \left(3 \sqrt{2}-2 v\right)}{6 \sqrt{26 v ^2-30 \sqrt{2} v +45}}, \; \kappa_n = \frac{5 \left(3 \sqrt{2}-2 v\right)}{6 \sqrt{26 v ^2-30 \sqrt{2} v +45}}, \; \tau_g = \frac{20 \sqrt{2} v +}{78 v ^2-90 \sqrt{2} v +135}-\frac{5}{18}. \end{equation*}

    3. The rectifying type of quasi-ruled surfaces:

    The rectifying type of quasi-ruled surface has the following parametrization:

    \begin{equation*} \psi_{3}( s , v ) = \frac{1}{6}\left(-\sqrt{2} \left(\sqrt{5}+2\right) v \sin s -5 \cos s -4, \sqrt{2} \left(\sqrt{5}+2\right) v \cos s -5 \sin s , 2 \sqrt{5} s +\sqrt{2} \left(\sqrt{5}-2\right) v \right), \end{equation*}

    for \eta( s ) = \frac{1}{\sqrt{2}}(\boldsymbol{T}_{q}^{F} -\boldsymbol{B}_{q}^{F}), \; \; \mu_1 = -\mu_3 = \frac{1}{\sqrt{2}} . This surface is illustrated with Figure 1(c).

    The normal vector to the surface is

    n_{\psi_{3}} = \frac{\left(-2 v \sin s -9 \sqrt{10} \cos s , 2 v \cos s -9 \sqrt{10} \sin s , -2 \left(4 \sqrt{5}+9\right) v \right)}{3 \sqrt{8 \left(4 \sqrt{5}+9\right) v ^2+90}}.

    Lemma 41. The CFFF and CSFF for the rectifying type QRS are given, respectively, by

    \begin{equation*} \begin{split} \boldsymbol{g}_{11} & = \frac{1}{18} \left(4 \sqrt{5}+9\right) v ^2+\frac{5}{4}, \quad \boldsymbol{g}_{12} = -\frac{\sqrt{5}}{2\sqrt{2}}, \quad \boldsymbol{g}_{22} = 1, \\ \boldsymbol{L}_{11} & = \frac{-2 \left(\sqrt{5}+2\right) v ^2-45 \sqrt{5}}{18 \sqrt{4 \left(4 \sqrt{5}+9\right) v ^2+45}}, \quad \boldsymbol{L}_{12} = \frac{2 \sqrt{5}+5}{\sqrt{8 \left(4 \sqrt{5}+9\right) v ^2+90}}, \quad \boldsymbol{L}_{22} = 0. \end{split} \end{equation*}

    Lemma 42. The MC, GC, SMC, and SGC for the rectifying type of QRS are given by

    \begin{equation*} \boldsymbol{H} = \frac{360-8 \left(\sqrt{5}+2\right) v ^2}{2 \left(4 \left(4 \sqrt{5}+9\right) v ^2+45\right)^{3/2}}, \quad \boldsymbol{K} = -\frac{180 \left(4 \sqrt{5}+9\right)}{\left(4 \left(4 \sqrt{5}+9\right) v ^2+45\right)^2}, \end{equation*}
    \begin{equation*} \begin{split} \boldsymbol{H}_{II}& = \frac{180-4 \left(\sqrt{5}+2\right) v ^2}{\left(4 \left(4 \sqrt{5}+9\right) v ^2+45\right)^{3/2}}+\frac{-32 \left(161 \sqrt{5}+360\right) v ^4+360 \left(143 \sqrt{5}+320\right) v ^2-40500 \left(\sqrt{5}+2\right)}{45 \sqrt{\frac{1}{\frac{4 v ^2}{5}-36 \sqrt{5}+81}} \left(4 \left(4 \sqrt{5}+9\right) v ^2+45\right)^2}, \\ \boldsymbol{K}_{II} & = -\frac{4 \left(4 \left(17 \sqrt{5}+38\right) v ^4+45 \left(19 \sqrt{5}+42\right) v ^2-2025\right)}{45 \left(4 \left(4 \sqrt{5}+9\right) v ^2+45\right)^{3/2}}. \end{split} \end{equation*}

    Lemma 43. The geodesic curvature \kappa_g , normal curvature \kappa_n , and geodesic torsion \tau_g of the QFC F_\alpha on the surface \psi_3 are given, respectively, according to Eq (3.4) as follows:

    \begin{equation*} \kappa_g = \frac{-\left(2 \sqrt{5}+5\right) \sqrt{2} v}{3 \sqrt{4 \left(4 \sqrt{5}+9\right) v ^2+45}}, \; \kappa_n = \frac{5}{\sqrt{4 \left(4 \sqrt{5}+9\right) v ^2+45}}, \; \tau_g = \frac{5 \sqrt{2} \left(4 \sqrt{5}+9\right) v }{3 \left(4 \left(4 \sqrt{5}+9\right) v ^2+45\right)}. \end{equation*}

    4. The quasi-tangent developable surface

    The quasi-tangent developable surface has the following parametrization:

    \begin{equation*} \psi_{4}( s , v ) = \frac{1}{6}\left(-2 \sqrt{5} v \sin s -5 \cos s -4, 2 \sqrt{5} v \cos s -5 \sin s , 2 (\sqrt{5} s -2 v )\right), \end{equation*}

    for \eta( s ) = \boldsymbol{T}_{q}^{F} . This surface is illustrated with Figure 2(a). The normal vector to the surface is

    n_{\psi_{4}} = \frac{1}{3}(2 \sin s , -2 \cos s , -\sqrt{5} ).
    Figure 2.  Visualization of the quasi-tangent, quasi-principal normal, and quasi-binormal ruled surfaces. The green curve represents the original curve, and the blue curve represents the quasi-focal curve for s \in [0, 6\pi] , and v \in [0, 5] .

    Lemma 44. The CFFF and CSFF for the quasi-tangent developable surface are given, respectively, by

    \begin{equation*} \begin{split} \boldsymbol{g}_{11} & = \frac{1}{36}(20\; v ^2+45), \quad \boldsymbol{g}_{12} = -\frac{\sqrt{5}}{2}, \quad \boldsymbol{g}_{22} = 1, \\ \boldsymbol{L}_{11} & = \frac{2\sqrt{5} }{9} \; v , \quad \boldsymbol{L}_{12} = 0, \quad \boldsymbol{L}_{22} = 0. \end{split} \end{equation*}

    Lemma 45. The MC and GC for the quasi-tangent developable surface are given by

    \begin{equation*} \boldsymbol{H} = \frac{1}{\sqrt{5} \; v }, \quad \boldsymbol{K} = 0. \end{equation*}

    Furthermore, the SMC and SGC are undefined.

    Lemma 46. The geodesic curvature \kappa_g , normal curvature \kappa_n , and geodesic torsion \tau_g of the QFC F_\alpha on the surface \psi_4 are given, respectively, according to Eq (3.4) as follows:

    \begin{equation*} \kappa_g = -\frac{\sqrt{5}}{3 } , \qquad \kappa_n = 0 , \qquad \tau_g = 0. \end{equation*}

    Hence, the QFC F_\alpha , which is the base curve of the quasi-tangent developable surface \psi_{4} , is both an asymptotic line and a principal line at any point (s, v) on the surface.

    5. The quasi-principal normal ruled surface:

    The quasi-principal normal ruled surface has the following parametrization:

    \begin{equation*} \psi_{5}( s , v ) = \frac{1}{6}\left((6 v -5) \cos (s)-4, (6 v -5) \sin (s), 2 \sqrt{5} s\right), \end{equation*}

    for \eta( s ) = \boldsymbol{N}_{q}^{F} . This surface is illustrated with Figure 2(b). The normal vector to the surface is

    n_{\psi_{5}} = \frac{\sqrt{5}}{\sqrt{12 v (3 v -5)+45}}\left(-2 \sin s , 2 \cos s , \frac{5-6 v }{\sqrt{5}}\right).

    Lemma 47. The CFFF and CSFF for the quasi-principal normal ruled surface are given, respectively, by

    \begin{equation*} \begin{split} \boldsymbol{g}_{11} & = \frac{1}{12} (12 v ^2-20 v +15), \quad \boldsymbol{g}_{12} = 0, \quad \boldsymbol{g}_{22} = 1, \\ \boldsymbol{L}_{11} & = 0, \quad \boldsymbol{L}_{12} = \frac{2 \sqrt{5}}{\sqrt{12 v (3 v -5)+45}}, \quad \boldsymbol{L}_{22} = 0. \end{split} \end{equation*}

    Lemma 48. The MC, GC, SMC, and SGC for the quasi-principal normal ruled surface are given by

    \begin{equation*} \boldsymbol{H} = 0, \quad \boldsymbol{K} = -\frac{80}{(4 v (3 v -5)+15)^2}, \;\boldsymbol{H}_{II} = 0, \;\;\boldsymbol{K}_{II} = 0. \end{equation*}

    Hence, the quasi-principal normal ruled surface is minimal, Ⅱ flat, and Ⅱ minimal.

    Lemma 49. The geodesic curvature \kappa_g , normal curvature \kappa_n , and geodesic torsion \tau_g of the QFC F_\alpha on the surface \psi_5 are given, respectively, according to Eq (3.4) as follows:

    \begin{equation*} \kappa_g = \frac{15-10 v}{3 \sqrt{12 v (3 v -5)+45}}, \; \kappa_n = 0, \; \tau_g = 0. \end{equation*}

    Hence, the QFC F_\alpha , which is the base curve of the quasi-principal normal ruled surface \psi_{5} , is both an asymptotic line and a principal line at any point (s, v) on the surface.

    6. The quasi-binormal ruled surface:

    The quasi-binormal ruled surface has the following parametrization:

    \begin{equation*} \psi_{6}( s , v ) = \frac{1}{6}\left(4 v \sin s -5 \cos s -4, -4 v \cos s -5 \sin s , 2 \sqrt{5} ( s - v )\right), \end{equation*}

    for \eta( s ) = \boldsymbol{B}_{q}^{F} . This surface is illustrated with Figure 2(c). The normal vector to the surface is

    n_{\psi_{6}} = \frac{\sqrt{5}}{3 \sqrt{16 v ^2+45}}\left(9 \cos s -4 v \sin s , 4 v \cos s +9 \sin s , -\frac{8 v }{\sqrt{5}}\right).

    Lemma 50. The CFFF and CSFF for the quasi-binormal ruled surface are given, respectively, by

    \begin{equation*} \begin{split} \boldsymbol{g}_{11} & = \frac {1} {36} (16 v ^2 + 45), \quad \boldsymbol{g}_{12} = 0, \quad \boldsymbol{g}_{22} = 1, \\ \boldsymbol{L}_{11} & = \frac{\sqrt{5}}{18} \sqrt{16 v ^2+45}, \quad \boldsymbol{L}_{12} = \frac{2 \sqrt{5}}{\sqrt{16 v ^2+45}}, \quad \boldsymbol{L}_{22} = 0. \end{split} \end{equation*}

    Lemma 51. The MC, GC, SMC, and SGC for the quasi-binormal ruled surface are given by

    \begin{equation*} \boldsymbol{H} = \frac{\sqrt{5}}{\sqrt{16 v ^2+45}}, \;\boldsymbol{K} = -\frac{720}{\left(16 v ^2+45\right)^2}, \; \boldsymbol{H}_{II} = \frac{32 v ^2+135}{9 \sqrt{5} \sqrt{16 v ^2+45}}, \; \boldsymbol{K}_{II} = \frac{\sqrt{16 v ^2+45}}{9 \sqrt{5}}. \end{equation*}

    Lemma 52. The geodesic curvature \kappa_g , normal curvature \kappa_n , and geodesic torsion \tau_g of the QFC F_\alpha on the surface \psi_6 are given, respectively, according to Eq (3.4) as follows:

    \begin{equation*} \kappa_g = 0 , \qquad \kappa_n = -\frac{5}{\sqrt{16 v ^2+45}}, \qquad \tau_g = -\frac{40 v }{48 v ^2+135}. \end{equation*}

    Hence, the QFC F_\alpha , which is the base curve of the quasi-binormal ruled surface \psi_{6} , is a geodesic curve at any point (s, v) on the surface.

    Effective visual aids are crucial for understanding complex geometric constructs. This section provides detailed visualizations of the constructed QRS to enhance comprehension. These visualizations are created using Mathematica 13, a powerful tool for generating high-quality graphics in differential geometry.

    This research presents a comprehensive study of quasi-ruled surfaces based on quasi-focal curves in 3-dimensional Euclidean space. The definitions of q-frame, quasi-focal curves, and quasi-ruled surfaces and their detailed analysis provide a new perspective on the construction of these surfaces in differential geometry.

    In this work, we have introduced and defined several novel types of QRS based on the QFC as the base curve and utilized the q-frame of the QFC to describ the rulings. These novel types of QRS include:

    ● Osculating type of quasi-ruled surface: This type of QRS has the ruling lies in the osculating plane of the base curve QFC.

    ● Normal type of quasi-ruled surface: This type of QRS has a ruling that lies in the normal plane of the base curve QFC.

    ● Rectifying type of quasi-ruled surface: This type of QRS has a ruling that lies in the rectifying plane of the base curve QFC.

    ● Quasi-tangent developable surfaces: This type of QRS has a ruling that parallels the quasi-tangent vector of the QFC.

    ● Quasi-principal normal ruled surfaces: This type of QRS has a ruling that parallels the quasi-principal normal vector of the QFC.

    ● Quasi-binormal ruled surfaces: This type of QRS has a ruling that parallels the quasi-binormal vector of the QFC.

    Some geometric properties are specified and analyzed for these types of QRS, including curvatures MC, GC, SMC, and SGC. These geometric properties contribute to the theoretical understanding of these surfaces. These novel types of quasi-ruled surfaces provide a rich framework for studying the geometric properties of surfaces constructed from the quasi-focal curves. Each type of QRS has unique characteristics based on the orientation of the ruling and the base curve. This classification allows for a deeper understanding of the intrinsic and extrinsic properties of these surfaces, which can be further explored in various applications, such as differential geometry, computer-aided design, and geometric modeling.

    The abbreviations used in this manuscript are illustrated by

    CFFF Coefficients of the first fundamental form
    CSFF Coefficients of the second fundamental form
    CEFSFEquations of Frenet-Serret frame
    FSF Frenet-Serret frame
    GC Gaussian curvature
    MC Mean curvature
    QFC(s)Quasi-focal curve(s)
    q-frameQuasi-frame
    QRS Quasi-ruled surface(s)
    SMC Second mean curvature
    SGC Second Gaussian curvature

    Samah Gaber: Investigation, writing the original draft, writing the review, editing, software; Asmahan Essa Alajyan: Investigation, writing the original draft, writing the review, editing; Adel H. Sorour: Investigation, writing the original draft, writing the review, editing. All authors have read and agreed to the published version of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in creating this article.

    The authors sincerely thank the reviewers for their insightful comments and constructive suggestions. Their valuable feedback has been instrumental in refining the content and improving the quality of this paper.

    The authors declare that they have no conflict of interest.



    [1] Y. Tuncer, N. Ekmekci, A study on ruled surfaces in Euclidean 3-space, J. Dyn. Syst. Geom. The., 8 (2010), 49–57. https://doi.org/10.1080/1726037X.2010.10698577 doi: 10.1080/1726037X.2010.10698577
    [2] I. A. Guven, S. Kaya, H. H. Hacisalihoglu, On closed ruled surfaces concerned with dual Frenet and Bishop frames, J. Dyn. Syst. Geom. The., 9 (2011), 67–74. https://doi.org/10.1080/1726037X.2011.10698593 doi: 10.1080/1726037X.2011.10698593
    [3] Y. Yu, H. Liu, S. D. Jung, Structure and characterization of ruled surfaces in Euclidean 3-space, Appl. Math. Comput., 233 (2014), 252–259. https://doi.org/10.1016/j.amc.2014.02.006 doi: 10.1016/j.amc.2014.02.006
    [4] Y. Tuncer, Ruled surfaces with the Bishop frame in Euclidean 3-space, Gen. Math. Notes, 26 (2015), 74–83. https://doi.org/10.1007/s40010-018-0546-y doi: 10.1007/s40010-018-0546-y
    [5] M. Masal, A. Z. Azak, The ruled surfaces according to type-2 Bishop frame in the Euclidean 3-space \mathbb{E}^3, Math. Sci. Appl. E-Notes, 3 (2018), 74–83. https://doi.org/10.36753/mathenot.421334 doi: 10.36753/mathenot.421334
    [6] G. Y. Senturk, S. Yuce, Characteristic properties of the ruled surface with Darboux frame in \mathbb{E}^3, Kuwait J. Sci. Eng., 42 (2015), 14–33. Available from: https://www.researchgate.net/publication/278686054_Characteristic_properties_of_the_ruled_surface_with_Darboux_frame_in_E3.
    [7] Y. Unluturk, M. Cimdiker, C. Ekici, Characteristic properties of the parallel ruled surfaces with Darboux frame in Euclidean 3-space, CMMA, 1 (2016), 26–43. Available from: https://dergipark.org.tr/en/pub/cmma/issue/27697/292077.
    [8] H. Liu, Y. Liu, S. D. Jung, Ruled invariants and associated ruled surfaces of a space curve, Appl. Math. Comput., 348 (2019), 479–486. https://doi.org/10.1016/j.amc.2018.12.011 doi: 10.1016/j.amc.2018.12.011
    [9] U. G. Kaymanli, C. Ekici, M. Dede, Directional evolution of the ruled surfaces via the evolution of their directrix using q-frame along a timelike space curve, Avrupa Bilim ve Teknoloji Derg., 20 (2019), 392–396. https://doi.org/10.31590/ejosat.681674 doi: 10.31590/ejosat.681674
    [10] S. Ouarab, A. O. Chahdi, Some characteristic properties of ruled surfaces with Frenet frame of an arbitrary non-cylindrical ruled surface in Euclidean 3-space, Int. J. Appl. Phys. Math., 10 (2020), 16–24. https://doi.org/10.17706/ijapm.2020.10.1.16-24 doi: 10.17706/ijapm.2020.10.1.16-24
    [11] C. Ekici, G. U. Kaymanli, S. Okur, A new characterization of ruled surfaces according to q-frame vectors in Euclidean 3-space, Int. J. Math. Combin., 3 (2017), 20–31. Available from: https://fs.unm.edu/IJMC/A_New_Characterization_of_Ruled_Surfaces_According_to_q-Frame_Vectors_in_Euclidean_3-Space.pdf.
    [12] S. Senyurt, D. Canli, E. Can, Smarandache-based ruled surfaces with the Darboux vector according to Frenet frame in \mathbb{E}^3, J. New Theory, 39 (2022), 8–18. Available from: https://www.researchgate.net/publication/362039701_Smarandache-Based_Ruled_Surfaces_with_the_Darboux_Vector_According_to_Frenet_Frame_in_E3#fullTextFileContent.
    [13] H. K. Samanci, The quasi-frame of the rational and polynomial Bezier curve by algorithm method in Euclidean space, Eng. Computations, 40 (2023), 1698–1722. https://doi.org/10.1108/EC-04-2022-0193 doi: 10.1108/EC-04-2022-0193
    [14] G. U. Kaymanli, C. Ekici, Evolutions of the ruled surfaces along a spacelike space curve, Punjab Univ. J. Math., 54 (2022), 221–232. https://doi.org/10.52280/pujm.2022.540401 doi: 10.52280/pujm.2022.540401
    [15] B. Pal, S. Kumar, Ruled like surfaces in three-dimensional Euclidean space, Ann. Math. Inform., 59 (2022), 83–101. https://doi.org/10.33039/ami.2022.12.011 doi: 10.33039/ami.2022.12.011
    [16] S. Gaber, A. H. Sorour, A. A. A. Salam, Construction of ruled surfaces from the W-curves and their characterizations in \mathbb{R}^3, Symmetry, 16 (2023), 509. https://doi.org/10.3390/sym16050509 doi: 10.3390/sym16050509
    [17] T. Korpinar, On quasi focal curves with quasi frame in space, Bol. Soc. Parana. Mat., 41 (2023), 1–3. https://doi.org/10.5269/bspm.50873 doi: 10.5269/bspm.50873
    [18] O. N. Barrett, Elementary differential geometry, 2 Eds., Academic Press, 2006. Available from: http://staff.ustc.edu.cn/spliu/2018DG/ONeill.pdf.
    [19] M. Dede, C. Ekici, A. Gorgulu, Directional q-frame along a space curve, IJARCSSE, 5 (2015), 775–780. https://doi.org/10.31590/ejosat.681674 doi: 10.31590/ejosat.681674
    [20] M. Dede, C. Ekici, H. Tozak, Directional tubular surfaces, Int. J. Algebr., 9 (2015), 527–535. https://doi.org/10.12988/ija.2015.51274 doi: 10.12988/ija.2015.51274
    [21] M. D. Carmo, Differential geometry of curves and surfaces, NJ, Englewood Cliffs: Prentice Hall, 1976. Available from: http://www2.ing.unipi.it/griff/files/dC.pdf.
    [22] O. B. Kalkan, S. Senyurt, Osculating type ruled surfaces with type-2 Bishop frame in \mathbb{E}^3, Symmetry, 16 (2024), 498. https://doi.org/10.3390/sym16040498 doi: 10.3390/sym16040498
    [23] O. Kaya, M. Onder, Generalized normal ruled surface of a curve in the Euclidean 3-space, Acta U. Sapientiae-Math., 13 (2021), 217–238. https://doi.org/10.2478/ausm-2021-0013 doi: 10.2478/ausm-2021-0013
    [24] Z. Isbilir, B. D. Yazici, M. Tosun, Generalized rectifying ruled surfaces of special singular curves, An. Sti. U. Ovid. Co-Mat., 31 (2023), 177–206. https://doi.org/10.2478/auom-2023-0038 doi: 10.2478/auom-2023-0038
    [25] S. Verpoort, The geometry of the second fundamental form: Curvature properties and variational aspects, Ph. D. Thesis, Belgium: Katholieke Universiteit Leuven, 2008. Available from: https://lirias.kuleuven.be/bitstream/1979/1779/2/hierrrissiedan!.pdf.
    [26] D. W. Yoon, On the second Gaussian curvature of ruled surfaces in Euclidean 3-space, Tamkang J. Math., 37 (2006), 221–226. https://doi.org/10.5556/j.tkjm.37.2006.167 doi: 10.5556/j.tkjm.37.2006.167
    [27] D. E. Blair, T. Koufogiorgos, Ruled surfaces with vanishing second Gaussian curvature, Monatsh. Math., 113 (1992), 177–181. https://doi.org/10.1007/BF01641765 doi: 10.1007/BF01641765
  • Reader Comments
  • © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(532) PDF downloads(53) Cited by(0)

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog