
In this work, we propose and investigate a predator-prey model where the prey population is structured by sex and the predators (unstructured) depredate based on sex-bias. We provide conditions for the existence of equilibrium points and perform local stability analysis on them. We derive global stability conditions for the extinction state. We show the possible occurrence of Hopf and saddle-node bifurcations. Multiple Hopf bifurcations are observed as the sex-biased predation rate is varied. This variation also shows the opposite consequences in the densities of the sex-structured prey. Our results show that sex-biased predation can cause both stabilizing and destabilizing effects for certain parameter choices. It can also cause an imbalanced sex-ratio, which has ecological consequences. Furthermore when intraspecific competition among predators is minimized, it can lead to the extinction of prey. We discuss the ecological implications and application of our results to the biocontrol of invasive species susceptible to sex-biased predation.
Citation: Eric M. Takyi, Charles Ohanian, Margaret Cathcart, Nihal Kumar. Sex-biased predation and predator intraspecific competition effects in a prey mating system[J]. AIMS Mathematics, 2024, 9(1): 2435-2453. doi: 10.3934/math.2024120
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In this work, we propose and investigate a predator-prey model where the prey population is structured by sex and the predators (unstructured) depredate based on sex-bias. We provide conditions for the existence of equilibrium points and perform local stability analysis on them. We derive global stability conditions for the extinction state. We show the possible occurrence of Hopf and saddle-node bifurcations. Multiple Hopf bifurcations are observed as the sex-biased predation rate is varied. This variation also shows the opposite consequences in the densities of the sex-structured prey. Our results show that sex-biased predation can cause both stabilizing and destabilizing effects for certain parameter choices. It can also cause an imbalanced sex-ratio, which has ecological consequences. Furthermore when intraspecific competition among predators is minimized, it can lead to the extinction of prey. We discuss the ecological implications and application of our results to the biocontrol of invasive species susceptible to sex-biased predation.
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:I∈R→R3 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.
● T∧N=B, N∧B=T, and B∧T=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:I∈R→R3 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=Tqα∧u‖Tqα∧u‖,Bqα=Tqα∧Nqα, | (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]=[1000cosθsinθ0−sinθcosθ][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=⟨T′qα,Nqα⟩=−det[α′,α′′,u]‖α′∧u‖,()′=dds(),κ2=⟨T′qα,Bqα⟩=⟨α′,u⟩⟨α′,α′′⟩−⟨α′′,u⟩‖α′∧u‖,κ3=−⟨B′qα,Nqα⟩=⟨α′,u⟩det[α′,α′′,u]‖α′∧u‖2. | (2.6) |
Lemma 1. [19] Let s represent the arc length along the curve α:I∈R→R3 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:
[α′T′qα(s)N′qα(s)B′qα(s)]=[010000κ1(s)κ2(s)0−κ1(s)0κ3(s)0−κ2(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:I∈R→R3 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:I∈R→R3 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]=[0010ε0−ε00][TqαNqαBqα],ε=±1. | (2.9) |
Proof. Taking the s-derivative of (2.8) with respect to s, we have
dFαds=α′+φ′1Nqα+φ1N′qα+φ′2Bqα+φ2B′qα. | (2.10) |
Substituting from (2.7) into (2.10), we have
dFαds=dFαdsF⋅dsFds=(1−κ1φ1−κ2φ2)Tqα+(φ′1−κ3φ2)Nqα+(φ′2+κ3φ1)Bqα. | (2.11) |
Since the QFC represents the centers of the tangential oscillating spheres, the components of Tqα and Nqα vanish. Then,
1−κ1φ1−κ2φ2=0,φ′1−κ3φ2=0. | (2.12) |
Hence, we have
dFαdsF⋅dsFds=(φ′2+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,
TFq⋅dsFds=(φ′2+κ3φ1)Bqα. | (2.14) |
Taking the norm of the two sides of (2.14), then
dsFds=|φ′2+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=φ′2+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=TFq∧u‖TFq∧u‖=Bqα∧u‖Bqα∧u‖,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,
Bqα∧u=Tqα∧Nqα∧u=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=TFq∧NFq, 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:I⟶R3 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]=[0κF1κF2−κF10κF3−κF2−κF30][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|φ′2+κ3φ1|,κF2=⟨dTFqdsF,BFq⟩=εκ2|φ′2+κ3φ1|,κF3=−⟨dBFqdsF,NFq⟩=κ1|φ′2+κ3φ1|. | (2.23) |
Theorem 4. [17] Consider a unit speed curve α:I⟶R3 with its QFC Fα. Then, the quasi-focal curvatures φ1 and φ2 are given by
φ1=e−∫κ1κ3κ2ds(∫e∫κ1κ3κ2dsκ3κ2ds+C),φ2=1κ2−κ1κ2e−∫κ1κ3κ2ds(∫e∫κ1κ3κ2dsκ3κ2ds+C), | (2.24) |
where C is a constant of integration.
Definition 6. [21] Let γ(s):I⊂R→R3 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):I⊂R→R3 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):I⊂R→R3 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×R→R3 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):I⊂R→R3 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×R→R3 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):I⊂R→R3 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×R→R3 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 ψs∧ψv=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=ψs∧ψv‖ψs∧ψv‖. | (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=⟨nψ∧TFq,(TFq)′⟩,κn=⟨(TFq)′,nψ⟩,τg=⟨nψ∧∂n∂s,(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=L11L22−L212g11g22−g212, | (3.7) |
H=g11L22−2g12L12+g22L112(g11g22−g212), | (3.8) |
λ=det(F′α,η,η′)‖η′‖2,()′=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,j∂∂xi(√|det(II)|Lij∂∂xj(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,sv−12L22,ss12L11,sL12,s−12L11,vL12,v−12L22,sL11L1212L22,vL12L22|−|012L11,v12L22,s12L11,vL11L1212L22,sL12L22|), | (3.12) |
where (),v=∂∂v,(),vv=∂2∂v2,(),s=∂∂s,(),ss=∂2∂s2, and (),sv=∂2∂v∂s. 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:I∈R→R3 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′+(φ′1+vμ′2)Nqα+(φ1+vμ2)N′qα+(φ′2+vμ′1)Bqα+(φ2+vμ1)B′qα. | (4.4) |
Substituting from (2.7) into (4.4), then
ψ1,s=(1−κ1(φ1+vμ2)−κ2(φ2+vμ1))Tqα+(φ′1+vμ′2−κ3(φ2+vμ1))Nqα+(φ′2+vμ′1+κ3(φ1+vμ2))Bqα. | (4.5) |
Using relation (2.12), we obtain
ψ1,s=−v(κ1μ2+κ2μ1)Tqα+v(μ′2−κ3μ1)Nqα+(v(μ′1+κ3μ2)+φ′2+κ3φ1)Bqα. |
Choose
ξ1=−(κ1μ2+κ2μ1),ξ2=μ′2−κ3μ1,ξ3=μ′1+κ3μ2,μ1ξ3+μ2ξ2=0,ξ4=φ′2+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(((μ1ξ2−μ2ξ3)v−μ2ξ4)Tqα−vμ1ξ1Nqα+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=(λ1v−κ2ξ4)Tqα+(λ2v−κ3ξ4)Nqα+(λ3v+ξ′4)Bqα,ψ1,sv=ξ1Tqα+ξ2Nqα+ξ3Bqα,ψ1,vv=0, | (4.11) |
where
λ1=ξ′1−κ1ξ2−κ2ξ3,λ2=ξ′2+κ1ξ1−κ3ξ3,λ3=ξ′3+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(μ1ξ2−μ2ξ3)−λ2μ1ξ1+λ3μ2ξ1,A2=−λ1μ2ξ4−κ2ξ4(μ1ξ2−μ2ξ3)+μ1κ3ξ1ξ4+μ2ξ1ξ′4. | (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)=μ2Nqα−μ3Tqα. | (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=(1−κ1(φ1+vμ2)−κ2φ2−vμ′3)Tqα+(φ′1+vμ′2−κ3φ2−vκ1μ3)Nqα+(φ′2+κ3φ1+v(μ2κ3−μ3κ2))Bqα. |
Using relation (2.12), we obtain
ψ2,s=−v(μ′3+μ2κ1)Tqα+v(μ′2−μ3κ1)Nqα+(φ′2+κ3φ1+v(μ2κ3−μ3κ2))Bqα. |
Choose
˜ξ1=−(μ′3+μ2κ1),˜ξ2=μ′2−μ3κ1,μ2˜ξ2−μ3˜ξ1=0,˜ξ3=μ2κ3−μ3κ2,ξ4=φ′2+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)Tqα−μ3(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=(˜λ1v−κ2ξ4)Tqα+(˜λ2v−κ3ξ4)Nqα+(˜λ3v+ξ′4)Bqα,ψ2,sv=˜ξ1Tqα+˜ξ2Nqα+˜ξ3Bqα,ψ2,vv=0, | (4.24) |
where
˜λ1=˜ξ′1−κ1˜ξ2−κ2˜ξ3,˜λ2=˜ξ′2+κ1˜ξ1−κ3˜ξ3,˜λ3=˜ξ′3+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)ξ′4. | (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=(1−κ1φ1−κ2φ2−v(μ′3+μ1κ2))Tqα+(φ′1−κ3φ2−v(μ3κ1+μ1κ3))Nqα+(φ′2+κ3φ1+v(μ′1−μ3κ2))Bqα. |
Using relation (2.12), we obtain
ψ3,s=−v(μ′3+μ1κ2)Tqα−v(μ3κ1+μ1κ3)Nqα+(φ′2+κ3φ1+v(μ′1−μ3κ2))Bqα. |
Choose
ˆξ1=−(μ′3+μ1κ2),ˆξ2=−(μ3κ1+μ1κ3),ˆξ3=μ′1−μ3κ2,μ1ˆξ3−μ3ˆξ1=0,ξ4=φ′2+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=(ˆλ1v−κ2ξ4)Tqα+(ˆλ2v−κ3ξ4)Nqα+(ˆλ3v+ξ′4)Bqα,ψ3,sv=ˆξ1Tqα+ˆξ2Nqα+ˆξ3Bqα,ψ3,vv=0, | (4.37) |
where
ˆλ1=ˆξ′1−κ1ˆξ2−κ2ˆξ3,ˆλ2=ˆξ′2+κ1ˆξ1−κ3ˆξ3,ˆλ3=ˆξ′3+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)ˆξ2−ˆλ2(μ1ˆξ1+μ3ˆξ3),ˆA2=−(μ3ˆλ2+μ1κ2ˆξ2)ξ4+μ3ˆξ2ξ′4+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′+φ′1Nqα+φ1N′qα+φ′2Bqα+(φ2+v)B′qα. | (4.44) |
Substituting from (2.7) into (4.44), then
ψ4,s=(1−κ1φ1−κ2(φ2+v))Tqα+(φ′1−κ3(φ2+v))Nqα+(φ′2+κ3φ1)Bqα. | (4.45) |
Using relation (2.12), we obtain
ψ4,s=−vκ2Tqα−vκ3Nqα+(φ′2+κ3φ1)Bqα. |
Choose
ξ4=φ′2+κ3φ1,whereφ′2=(1−κ1φ1κ2)′. | (4.46) |
Then, we have
ψ4,s=−vκ2Tqα−vκ3Nqα+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+κ2Nqα√κ22+κ23. | (4.50) |
Lemma 23. The CSFF of the quasi-tangent developable surface are given as
L11=v√κ22+κ23(κ′2κ3−κ2κ′3−κ1(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(κ′2−κ1κ3))Tqα−(ξ4κ3+v(κ′3+κ1κ2))Nqα+(ξ′4−v(κ22+κ23))Bqα,ψ4,sv=−κ2Tqα−κ3Nqα,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=κ′2κ3−κ2κ′3−κ1(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=(1−κ1(φ1+v)−κ2φ2)Tqα+(φ′1−κ3φ2)Nqα+(φ′2+κ3φ1+vκ3)Bqα. | (4.57) |
Using relation (2.12), we obtain
ψ5,s=−vκ1Tqα+(vκ3+ξ4)Bqα,ξ4=φ′2+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)Tqα−vκ1Bqα√v2(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(κ′1+κ2κ3)−κ2ξ4)Tqα−(v(κ21+κ23)+κ3ξ4)Nqα+(v(κ′3−κ1κ2)+ξ′4)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=1√v2(κ21+κ23)+2vκ3ξ4+ξ24(v2(κ′1κ3−κ1κ′3+κ2(κ21+κ23))+v(ξ4(κ′1+2κ2κ3)−κ1ξ′4)+κ2ξ24),L12=κ1ξ4√v2(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(κ′1κ3−κ1κ′3+κ2(κ21+κ23))+v(ξ4(κ′1+2κ2κ3)−κ1ξ′4)+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=(1−κ1φ1−κ2φ2)Tqα+(φ′1−κ3φ2−vκ1)Nqα+(φ′2+κ3φ1−vκ2)Bqα. | (4.68) |
Using relation (2.12), we obtain
ψ6,s=−vκ1Nqα+(ξ4−vκ2)Bqα,ξ4=φ′2+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)v2−2vκ2ξ4+ξ24,g12=0,g22=1. | (4.71) |
Lemma 31. The normal vector nψ6 to the quasi-binormal ruled surface is given by
nψ6=−(ξ4−vκ2)Nqα+vκ1Bqα√v2(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(κ′1−κ2κ3)+κ3ξ4)Nqα+(ξ′4−v(κ′2+κ1κ3))Bqα,ψ6,sv=−κ1Nqα−κ2Bqα,ψ6,vv=0, | (4.73) |
Lemma 33. The CSFF of the quasi-binormal ruled surface are given as
L11=1√v2(κ21+κ22)+2vκ2ξ4+ξ24(v2(κ1κ′2−κ′1κ2+κ3(κ21+κ22))+v(ξ4(κ′1−2κ2κ3)−κ1ξ′4)+κ3ξ24),L12=κ1ξ4√v2(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(κ1κ′2−κ′1κ2+κ3(κ21+κ22))+v(ξ4(κ′1−2κ2κ3)−κ1ξ′4)+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(2∂∂s(∂∂vln√|K|)−∂∂v(L11L12∂∂vln√|K|)),KII=−12(L12)3(L12(2L12,sv−L11,vv)+L12,v(L11,v−2L12,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(coss−1),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
κ1=−23,κ2=0,κ3=√53. |
Using (2.12), then φ1=−32,φ2=0, and the QFC associated with the original curve is given by
Fα=(−16(4+5coss),−56sins,√53s), | (5.4) |
Using (2.21), the q-frame of the QFC Fα is given by
TFq=(−√53sins,√53coss,−23),NFq=(coss,sins,0),BFq=(23sins,−23coss,−√53). | (5.5) |
Using (2.23), the quasi-curvatures for the QFC are given as
κF1=−23,κF2=0,κF3=−43√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:
ψ1(s,v)=16(−√10vsins−(3√2v+5)coss−4,√10vcoss−(3√2v+5)sins,2(√5s−√2v)), |
for η(s)=1√2(TFq−NFq),μ1=−μ2=1√2. This surface is illustrated with Figure 1(a). The normal vector to the surface is
nψ1=(2√5(√2v+3)sins+6√2vcoss,6√2vsins−2√5(√2v+3)coss,−14√2v−15)3√56v2+60√2v+45. |
Lemma 35. The CFFF and CSFF for the osculating type of QRS are given, respectively, by
g11=79v2+53√2v+54,g12=−√52√2,g22=1,L11=2v(14v+15√2)9√56v2+60√2v+45,L12=√10√56v2+60√2v+45,L22=0. |
Lemma 36. The MC, GC, SMC, and SGC for the osculating type of QRS are given by
H=4(28v2+30√2v+45)(56v2+60√2v+45)3/2,K=−720(56v2+60√2v+45)2. |
HII=8(5488v4+11760√2v3+25844v2+14190√2v+5085)45(56v2+60√2v+45)3/2,KII=28v(784v3+1680√2v2+2430v+675√2)+810045(56v2+60√2v+45)3/2. |
Lemma 37. The geodesic curvature κg, normal curvature κn, and geodesic torsion τg of the QFC Fα on the surface ψ1 are given, respectively, according to Eq (3.4), as follows:
κg=−5(2√2v+3)3√56v2+60√2v+45,κn=−2√10v3√56v2+60√2v+45,τg=−2√5v(28v+15√2)9(56v2+60√2v+45). |
2. The normal type of quasi-ruled surfaces:
The normal type of quasi-ruled surface has the following parametrization:
ψ2(s,v)=16(√2v(3coss−2sins)−5coss−4,√2v(3sins+2coss)−5sins,√5(2s+√2v)), |
for η(s)=1√2(NFq−BFq),μ2=−μ3=1√2. This surface is illustrated with Figure 1(b).
The normal vector to the surface is
nψ2=√5(−(4v+6√2)sins+(6v−9√2)coss,(6v−9√2)sins+(4v+6√2)coss,6√5(5√2−13v3))6√26v2−30√2v+45. |
Lemma 38. The CFFF and CSFF for the normal type of QRS are given, respectively, by
g11=136(26v2−30√2v+45),g12=0,g22=1,L11=−118√65v2−75√2v+2252,L12=2√5√26v2−30√2v+45,L22=0. |
Lemma 39. The MC, GC, SMC, and SGC for the normal type of QRS are given by
H=−√102√26v2−30√2v+45,K=−720(26v2−30√2v+45)2, |
HII=−13(26v2−30√2v+45)+18036√10√26v2−30√2v+45,KII=−13√10720√26v2−30√2v+45, |
Lemma 40. The geodesic curvature κg, normal curvature κn, and geodesic torsion τg of the QFC Fα on the surface ψ2 are given, respectively, according to Eq (3.4) as follows:
κg=5(3√2−2v)6√26v2−30√2v+45,κn=5(3√2−2v)6√26v2−30√2v+45,τg=20√2v+78v2−90√2v+135−518. |
3. The rectifying type of quasi-ruled surfaces:
The rectifying type of quasi-ruled surface has the following parametrization:
ψ3(s,v)=16(−√2(√5+2)vsins−5coss−4,√2(√5+2)vcoss−5sins,2√5s+√2(√5−2)v), |
for η(s)=1√2(TFq−BFq),μ1=−μ3=1√2. This surface is illustrated with Figure 1(c).
The normal vector to the surface is
nψ3=(−2vsins−9√10coss,2vcoss−9√10sins,−2(4√5+9)v)3√8(4√5+9)v2+90. |
Lemma 41. The CFFF and CSFF for the rectifying type QRS are given, respectively, by
g11=118(4√5+9)v2+54,g12=−√52√2,g22=1,L11=−2(√5+2)v2−45√518√4(4√5+9)v2+45,L12=2√5+5√8(4√5+9)v2+90,L22=0. |
Lemma 42. The MC, GC, SMC, and SGC for the rectifying type of QRS are given by
H=360−8(√5+2)v22(4(4√5+9)v2+45)3/2,K=−180(4√5+9)(4(4√5+9)v2+45)2, |
HII=180−4(√5+2)v2(4(4√5+9)v2+45)3/2+−32(161√5+360)v4+360(143√5+320)v2−40500(√5+2)45√14v25−36√5+81(4(4√5+9)v2+45)2,KII=−4(4(17√5+38)v4+45(19√5+42)v2−2025)45(4(4√5+9)v2+45)3/2. |
Lemma 43. The geodesic curvature κg, normal curvature κn, and geodesic torsion τg of the QFC Fα on the surface ψ3 are given, respectively, according to Eq (3.4) as follows:
κg=−(2√5+5)√2v3√4(4√5+9)v2+45,κn=5√4(4√5+9)v2+45,τg=5√2(4√5+9)v3(4(4√5+9)v2+45). |
4. The quasi-tangent developable surface
The quasi-tangent developable surface has the following parametrization:
ψ4(s,v)=16(−2√5vsins−5coss−4,2√5vcoss−5sins,2(√5s−2v)), |
for η(s)=TFq. This surface is illustrated with Figure 2(a). The normal vector to the surface is
nψ4=13(2sins,−2coss,−√5). |
Lemma 44. The CFFF and CSFF for the quasi-tangent developable surface are given, respectively, by
g11=136(20v2+45),g12=−√52,g22=1,L11=2√59v,L12=0,L22=0. |
Lemma 45. The MC and GC for the quasi-tangent developable surface are given by
H=1√5v,K=0. |
Furthermore, the SMC and SGC are undefined.
Lemma 46. The geodesic curvature κg, normal curvature κn, and geodesic torsion τg of the QFC Fα on the surface ψ4 are given, respectively, according to Eq (3.4) as follows:
κg=−√53,κn=0,τg=0. |
Hence, the QFC Fα, which is the base curve of the quasi-tangent developable surface ψ4, 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:
ψ5(s,v)=16((6v−5)cos(s)−4,(6v−5)sin(s),2√5s), |
for η(s)=NFq. This surface is illustrated with Figure 2(b). The normal vector to the surface is
nψ5=√5√12v(3v−5)+45(−2sins,2coss,5−6v√5). |
Lemma 47. The CFFF and CSFF for the quasi-principal normal ruled surface are given, respectively, by
g11=112(12v2−20v+15),g12=0,g22=1,L11=0,L12=2√5√12v(3v−5)+45,L22=0. |
Lemma 48. The MC, GC, SMC, and SGC for the quasi-principal normal ruled surface are given by
H=0,K=−80(4v(3v−5)+15)2,HII=0,KII=0. |
Hence, the quasi-principal normal ruled surface is minimal, Ⅱ flat, and Ⅱ minimal.
Lemma 49. The geodesic curvature κg, normal curvature κn, and geodesic torsion τg of the QFC Fα on the surface ψ5 are given, respectively, according to Eq (3.4) as follows:
κg=15−10v3√12v(3v−5)+45,κn=0,τg=0. |
Hence, the QFC Fα, which is the base curve of the quasi-principal normal ruled surface ψ5, 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:
ψ6(s,v)=16(4vsins−5coss−4,−4vcoss−5sins,2√5(s−v)), |
for η(s)=BFq. This surface is illustrated with Figure 2(c). The normal vector to the surface is
nψ6=√53√16v2+45(9coss−4vsins,4vcoss+9sins,−8v√5). |
Lemma 50. The CFFF and CSFF for the quasi-binormal ruled surface are given, respectively, by
g11=136(16v2+45),g12=0,g22=1,L11=√518√16v2+45,L12=2√5√16v2+45,L22=0. |
Lemma 51. The MC, GC, SMC, and SGC for the quasi-binormal ruled surface are given by
H=√5√16v2+45,K=−720(16v2+45)2,HII=32v2+1359√5√16v2+45,KII=√16v2+459√5. |
Lemma 52. The geodesic curvature κg, normal curvature κn, and geodesic torsion τg of the QFC Fα on the surface ψ6 are given, respectively, according to Eq (3.4) as follows:
κg=0,κn=−5√16v2+45,τg=−40v48v2+135. |
Hence, the QFC Fα, which is the base curve of the quasi-binormal ruled surface ψ6, 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 |
CEFSF | Equations of Frenet-Serret frame |
FSF | Frenet-Serret frame |
GC | Gaussian curvature |
MC | Mean curvature |
QFC(s) | Quasi-focal curve(s) |
q-frame | Quasi-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.
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