The present paper presents evolutions of spherical indicatrix of a space  curve according to the quasi-frame. Then, some geometric properties of these  surfaces constructed by evolutions have been obtained. At the end, illustrative examples of the spherical images of a space curve have been presented. 

Sariaydin, Muhammed T.

Yazla, Aziz

English

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NIŠ)Ser. Math. Inform. Vol. 35, No 3 (2020), 605-619https://doi.org/10.22190/FUMI2003605YON SURFACES CONSTRUCTED BY EVOLUTION ACCORDINGTO QUASI FRAMEAziz Yazla and Muhammed T. Sariaydin© by University of Nǐs, Serbia | Creative Commons Licence: CC BY-NC-NDAbstract. The present paper presents evolutions of spherical indicatrix of a spacecurve according to the quasi-frame. Then, some geometric properties of these surfacesconstructed by evolutions have been obtained. At the end, illustrative examples of thespherical images of a space curve have been presented.Keywords: space curve; spherical images; quasi-frame.1. IntroductionThe curves obtained with the help of a given space curve have been studied bymany researchers. Bertrand curve pairs, involute-evolute curve pairs and sphericalimages of a space curve can be given as examples of these curves, [7]. For example,Korpinar [4] investigated the surfaces constructed by the binormal spherical imageof a space curve. They derived the time evolution equations for the Frenet frameof binormal spherical image as a curve occurring on the sphere and gave somegeometric properties of these surfaces such as fundamental forms and curvatures.The spherical image of the curve moving with time occurs on a sphere. In [6], timeevolution equations of a space curve given with the quasi frame are obtained.In this paper, we have found relations between the motion of curves and themotion of their spherical image. We have obtained the Frenet elements of thespherical images of the curve given with the quasi frame. Then we have derived somegeometric properties of the surfaces constructed by the evolution of the sphericalimages of a space curve. At the end, we have given the illustrative examples of thespherical images of a space curve.Received February 24, 2019; accepted March 16, 20192010 Mathematics Subject Classification. Primary 47S10; Secondary 53A04, 53B30605brought to you by COREView metadata, citation and similar papers at core.ac.uk606 A. Yazla and M. T. Sariaydin2. PreliminariesIn this section, we present the Frenet frame and the quasi frame along a spacecurve which are given by Soliman in [6]. Also, we give some geometric propertiesfor these frames.Let r = r(s) be a space curve parameterized with arc-length in R3. The Frenetframe of r consists of the vectors T,N,B which are given byT = r′(s),N =r′′(s)‖r′′(s)‖,B = T×N,where T is the tangent vector, N is the normal vector and B is the binormal vectorof the curve r.The curvature κ and the torsion τ are given byκ = ‖r′′(s)‖ ,τ =det(r′, r′′, r′′′)‖r′′(s)‖2.The quasi frame of a space curve r = r(s) which is parameterized with arc-lengthconsists of the vectors Tq,Nq,Bq. They are given byTq = T,Nq =T×−→k∥∥∥T×−→k ∥∥∥ ,Bq = T×Nq,where−→k is the projection vector which can be chosen as−→k = (1, 0, 0) or−→k =(0, 1, 0) or−→k = (0, 0, 1). In this paper, we choose the projection vector−→k = (0, 0, 1).Nq and Bq are called the quasi normal vector and the quasi binormal vector, re-spectively.Let θ be the angle between the normal N and the quasi normal Nq. The quasiformulas are given by, [1],∂∂s TqNqBq = 0 k1 k2−k1 0 k3−k2 −k3 0 TqNqBq ,where ki are called the quasi curvatures (1 ≤ i ≤ 3) which are given byk1 = κ cosθ =〈T′q,Nq〉,k2 = −κ sinθ =〈T′q,Bq〉,k3 = θ′ + τ = −〈Nq,B′q〉.On Surfaces Constructed by Evolution According to Quasi Frame 6073. The Spherical Images of a Space CurveIn this section, we give the representation of the Frenet frame, curvature andtorsion for spherical images of the curve in terms of the quasi frame and curvaturesof the curve.Given a space curve r parameterized with arc-length in R3. Let T be the unittangent vector of r. When we take−−→PQ = T; while the moving point P is drawingthe curve r, the moving point Q draws a curve on the unit sphere. This curve iscalled the spherical image of the tangent to the curve r. The spherical image ofthe normal and the binormal to the curve are defined similarly. Now we give theseconcepts according to the quasi frame of the curve.Definition 3.1. Let r = r(s) be a space curve parameterized with arc-length inR3. The following space curves lie on a unit spherer1(s) = Tq(s),r2(s) = Nq(s),r3(s) = Bq(s)and they are called the spherical image of the tangent, the quasi normal and thequasi binormal to the curve, respectively.3.1. Spherical Image of TqLet{Tq,Nq,Bq}be the quasi frame of the curve r = r(s) parameterized witharc-length and {T,N,B} be the Frenet frame of the curve r1(s) = Tq(s). The quasicurvatures of the curve r are denoted by k1, k2, k3 and the curvature and the torsionof the curve r1 are denoted by κ and τ, respectively.Theorem 3.1. The Frenet elements of r1 can be given in terms of the quasi ele-ments of r as follows: TNB = 0k1√k21+k22k2√k21+k22A1√U1B1√U1C1√U1K1√V1L1√V1M1√V1 TqNqBqκ = (1 +K1(k21 + k22)3 )12 ,τ =W1V1,608 A. Yazla and M. T. SariaydinwhereA1 = −(k21 + k22)2,B1 = k′1k22 − k1k2k′2 − k21k2k3 − k32k3,C1 = k31k3 + k21k′2 + k1k22k3 − k1k′1k2,K1 = k21k3 + k22k3 + k1k′2 − k′1k2,L1 = −k2(k21 + k22),M1 = k1(k21 + k22),U1 =(k21 + k22)4+(k21 + k22) (k21k3 + k22k3 + k1k′2 − k′1k2)2,V1 =(k21 + k22)3+(k21k3 + k22k3 + k1k′2 − k′1k2)2,W1 = 3(k1 (k′1)2k2 + k′1k′2k22 − k21k′1k′2 − k1k2 (k′2)2)+(k21 + k22) (k1k′′2 + k21k′3 + k22k′3 − k′′1k2 − k1k′1k3 − k2k′2k3).Proof. By simple calculations one can easily get the first, the second and the thirdderivatives of r1 as follows:r′1(s) = k1Nq + k2Bq,r′′1 (s) = −(k21 + k22)Tq + (k′1 − k2k3)Nq + (k1k3 + k′2)Bq,r′′′1 (s) = −3(k1k′1 + k2k′2)Tq + (k′′1 − 2k′2k3 − k2k′3 − k1k23 − k1k22 − k31)Nq+(k′′2 + 2k′1k3 + k1k′3 − k2k23 − k2k21 − k32)Bq.Then, it is easy to compute the following:‖r′1‖ =√k21 + k22,r′1 × r′′1 = K1Tq + L1Nq +M1Bq,‖r′1 × r′′1‖ =√V1,det(r′1, r′′1 , r′′′1 ) = 〈r′1 × r′′1 , r′′′1 〉 = W1.Using the Frenet formulas, one can easily obtain the Frenet elements of r1 in termsof the quasi elements of r as indicated in the theorem.3.2. Spherical Image of NqLet{Tq,Nq,Bq}be the quasi frame of the curve r = r(s) parameterized witharc-length and {T,N,B} be the Frenet frame of the curve r2(s) = Nq(s). The quasicurvatures of the curve r are denoted by k1, k2, k3 and the curvature and the torsionof the curve r2 are denoted by κ and τ, respectively.Theorem 3.2. The Frenet elements of r2 can be given in terms of the quasi ele-On Surfaces Constructed by Evolution According to Quasi Frame 609ments of r as follows: TNB =−k1√k21+k230 k3√k21+k23A2√U2B2√U2C2√U2K2√V2L2√V2M2√V2 TqNqBqκ = (1 +K2(k21 + k23)3 )12 ,τ =W2V2,whereA2 = k1k3k′3 − k21k2k3 − k′1k23 − k2k33,B2 = −k21(k21 + k23),C2 = k21k′3 − k1k′1k3 − k31k2 − k1k2k23−,K2 = k3(k21 + k23),L2 = k1k′3 − k′1k3 − k21k2 − k2k23,M2 = k1(k21 + k23),U2 =(k21 + k23)4+(k21 + k23) (k1k′3 − k′1k3 − k21k2 − k2k23)2,V2 =(k21 + k23)3+(k1k′3 − k′1k3 − k21k2 − k2k23)2,W2 = 3(k1 (k′1)2k3 + k′1k′3k23 − k21k′3k′3 − k1k3 (k′3)2)+(k21 + k23) (k1k′′3 − k′′1k3 − k′2k23 − k21k′2 + k2k′3k3 + k1k′1k2).Proof. The calculations can be made similar to the proof of the first theorem.3.3. Spherical Image of BqLet{Tq,Nq,Bq}be the quasi frame of the curve r = r(s) parameterized witharc-length and {T,N,B} be the Frenet frame of the curve r3(s) = Bq(s). The quasicurvatures of the curve r are denoted by k1, k2, k3 and the curvature and the torsionof the curve r3 are denoted by κ and τ, respectively.Theorem 3.3. The Frenet elements of r3 can be given in terms of the quasi ele-ments of r as follows: TNB =−k2√k22+k230 −k3√k22+k23A3√U3B3√U3C3√U3K3√V3L3√V3M3√V3 TqNqBqκ = (1 +K3(k22 + k23)3 )12 ,τ =W3V3,610 A. Yazla and M. T. SariaydinwhereA3 = k2k3k′3 + k1k22k3 + k1k33 − k′2k23,B3 = k2k′2k3 − k1k2k23 − k1k32 − k′3k22,C3 = −(k22 + k23)2,K3 = k3(k22 + k23),L3 = −k2(k22 + k23),M3 = k21k2 + k1k23 + k2k′3 − k′2k3,U3 =(k22 + k23)4+(k22 + k23) (k2k′3 − k′2k3 + k1k22 + k1k23)2,V3 =(k22 + k23)3+(k2k′3 − k′2k3 + k1k22 + k1k23)2,W3 = 3(k2 (k′2)2k3 + k′2k′3k23 − k22k′2k′3 − k2k3 (k′3)2)+(k22 + k23) (k′1k22 + k′1k23 + k′′3k2 − k3k′′2 − k1k2k′2 − k1k3k′3).Proof. The calculations can be made similar to the proof of the first theorem.An evolving curve can be thought as a collection of curves parameterized bytime. This means that each curve in the collection has a space parameter s anda time parameter t, [3]. The following definitions can be given according to quasiframe in R3 considering references [6] and [7].(3.1)∂∂s TqNqBq = 0 k1 k2−k1 0 k3−k2 −k3 0 TqNqBq ,(3.2)∂∂t TqNqBq = 0 λ µ−λ 0 ν−µ −ν 0 TqNqBq .Applying the compatibility condition∂∂s∂∂t TqNqBq = ∂∂t∂∂s TqNqBq ,in the light of the equations (3.1) and (3.2) one can easily get0(∂k1∂t − νk2 + µk3 −∂λ∂s) (∂k2∂t + νk1 − λk3 +∂µ∂s)−(∂k1∂t − νk2 + µk3 −∂λ∂s)0(∂k3∂t − µk1 + λk2 −∂ν∂s)−(∂k2∂t + νk1 − λk3 +∂µ∂s)−(∂k3∂t − µk1 + λk2 −∂ν∂s)0 = 03×3.On Surfaces Constructed by Evolution According to Quasi Frame 611Thus, the compatibility condition becomes∂k1∂t= νk2 − µk3 +∂λ∂s,∂k2∂t= λk1 − νk3 −∂µ∂s,∂k3∂t= µk1 − λk2 +∂ν∂s.4. Surfaces Constructed by the Evolution of the Spherical Images of aSpace CurveIn this section, we study the surfaces constructed by the evolution of the spher-ical image of the tangent, spherical image of the quasi normal and spherical imageof the quasi binormal to the curve.4.1. Surfaces Constructed Using the Spherical Image of the TangentThe equation of surfaces constructed by the spherical image of the tangent isgiven byΨ = Tq(s, t).Theorem 4.1. Under the assumption µk1−λk2 > 0, the Gaussian curvature K1,the mean curvature H1 and the principal curvatures k11 and k21 of Ψ are given byK1 = 1, H1 = −1, k11 = −1, k21 = −1.Proof. The tangent space to the surface is spanned byΨs = k1Nq + k2Bq,(4.1)Ψt = λNq + µBq,where the lower indices show partial differentiation. Then the unit normal to Ψ isgiven byNΨ =Ψs ×Ψt‖Ψs ×Ψt‖= Tq.Using the equations (3.1), (3.2) and (4.1), the second order derivatives are calculatedand given byΨss = −(k21 + k22)Tq + ((k1)s − k2k3) Nq + ((k2)s + k1k3) Bq,Ψtt = −(λ2 + µ2)Tq + (λt − µν) Nq + (µt + λν)Bq,Ψst = −(λk1 + µk2)Tq + (λs − µk3) Nq + (λk3 − µs)Bq.The components of the first fundamental form gij , (1 ≤ i, j ≤ 2) are obtained asfollows:g11 = 〈Ψs,Ψs〉 = k21 + k22,g12 = 〈Ψs,Ψt〉 = λk1 + µk2,g22 = 〈Ψt,Ψt〉 = λ2 + µ2.612 A. Yazla and M. T. SariaydinThe components of the second fundamental form lij , (1 ≤ i, j ≤ 2) are obtained asfollows:l11 = 〈Ψss,NΨ〉 = −(k21 + k22),l12 = 〈Ψst,NΨ〉 = − (λk1 + µk2) ,l22 = 〈Ψtt,NΨ〉 = −(λ2 + µ2).Thus, we get the following equalities:K1 =l11l22 − l212g11g22 − g212= 1,H1 =l11g22 − 2l12g12 + l22g112 (g11g22 − g212)= −1,k11 = H1 +√H21 −K1 = −1,k21 = H1 −√H21 −K1 = −1.4.2. Surfaces Constructed Using the Spherical Image of the QuasiNormalThe equation of surfaces constructed by the spherical image of the quasi normalis given byφ = Nq(s, t).Theorem 4.2. Under the assumption νk1− λk3 > 0, the Gaussian curvature K2,the mean curvature H2 and the principal curvatures k12 and k22 of φ are given byK2 = 1, H2 = −1, k12 = −1, k22 = −1.Proof. The tangent space to the surface is spanned byφs = −k1Tq + k3Bq,(4.2)φt = −λTq + νBq,where the lower indices show partial differentiation. Then the unit normal to φ isgiven byNφ =φs × φt‖φs × φt‖= Nq.Using the equations (3.1), (3.2) and (4.2), the second order derivatives are calculatedand given byφss = − ((k1)s + k2k3) Tq −(k21 + k23)Nq + ((k3)s − k1k2) Bq,φtt = − (λt + µν) Tq −(λ2 + ν2)Nq + (νt − λµ)Bq,φst = − (λs + νk2) Tq − (λk1 + νk3) Nq + (νs − λk2)Bq.On Surfaces Constructed by Evolution According to Quasi Frame 613The components of the first fundamental form gij , (1 ≤ i, j ≤ 2) are obtained asfollows:g11 = 〈φs, φs〉 = k21 + k23,g12 = 〈φs, φt〉 = λk1 + νk3,g22 = 〈φt, φt〉 = λ2 + ν2.The components of the second fundamental form lij , (1 ≤ i, j ≤ 2) are obtained asfollows:l11 = 〈φss,Nφ〉 = −(k21 + k23),l12 = 〈φst,Nφ〉 = − (λk1 + νk3) ,l22 = 〈φtt,Nφ〉 = −(λ2 + ν2).Thus, we get the following equalities:K2 =l11l22 − l212g11g22 − g212= 1,H2 =l11g22 − 2l12g12 + l22g112 (g11g22 − g212)= −1,k12 = H2 +√H22 −K2 = −1,k22 = H2 −√H22 −K2 = −1.4.3. Surfaces Constructed Using the Spherical Image of the QuasiBinormalThe equation of surfaces constructed by the spherical image of the quasi binormalis given byϕ = Bq(s, t).Theorem 4.3. Under the assumption νk2−µk3 > 0, the Gaussian curvature K3,the mean curvature H3 and the principal curvatures k13 and k23 of ϕ are given byK3 = 1, H3 = −1, k13 = −1, k23 = −1.Proof. The tangent space to the surface is spanned byϕs = −k2Tq − k3Nq,(4.3)ϕt = −µTq − νNq,614 A. Yazla and M. T. Sariaydinwhere the lower indices show partial differentiation. Then the unit normal to ϕ isgiven byNϕ =ϕs × ϕt‖ϕs × ϕt‖= Bq.Using the equations (3.1), (3.2) and (4.3), the second order derivatives are calculatedand given byϕss = (k1k3 − (k2)s) Tq − ((k3)s + k1k2) Nq −(k22 + k23)Bq,ϕtt = (λν − µt) Tq − (νt + λµ)Nq −(µ2 + ν2)Bq,ϕst = (µs + νk1) Tq − (νs − µk1)Nq − (µk2 + νk3)Bq.The components of the first fundamental form gij , (1 ≤ i, j ≤ 2) are obtained asfollows:g11 = 〈ϕs, ϕs〉 = k22 + k23,g12 = 〈ϕs, ϕt〉 = µk2 + νk3,g22 = 〈ϕt, ϕt〉 = µ2 + ν2.The components of the second fundamental form lij , (1 ≤ i, j ≤ 2) are obtained asfollows:l11 = 〈ϕss,Nϕ〉 = −(k22 + k23),l12 = 〈ϕst,Nϕ〉 = − (µk2 + νk3) ,l22 = 〈ϕtt,Nϕ〉 = −(µ2 + ν2).Thus, we get the following equalities:K3 =l11l22 − l212g11g22 − g212= 1,H3 =l11g22 − 2l12g12 + l22g112 (g11g22 − g212)= −1,k13 = H3 +√H23 −K3 = −1,k23 = H3 −√H23 −K3 = −1.5. ExamplesWe give two illustrative examples to the spherical images of a regular space curveaccording to quasi frame.On Surfaces Constructed by Evolution According to Quasi Frame 615Example 5.1. Let us consider the space curve α which is defined byα : R −→ R3,α(t) = ((2 + cos t+ sin t) sin t cos(sin(10t)),(2 + cos t+ sin t) sin t sin(sin(10t)),(2 + cos t+ sin t) cos t).Calculating the first derivative of α, one can easily see that∥∥α′(t)∥∥ 6= 0for all t ∈ R. So we can say that α is a regular space curve. In the light of the quasiformulas, one can easily obtain the quasi frame{Tq,Nq,Bq}of α. The graphics of thecurve α and its spherical images are given below.Fig. 5.1: The curve α(a) (b)Fig. 5.2: The spherical image of tangent of curve α616 A. Yazla and M. T. Sariaydin(a) (b)Fig. 5.3: The spherical image of the quasi-normal of curve α(a) (b)Fig. 5.4: The spherical image of the quasi-binormal of curve αExample 5.2. Let us consider the space curve β which is defined in [5] byβ : R −→ R3,β(t) = (−185sin(− t4) +245sin(9t4),−185cos(− t4) +245cos(9t4),35cos t).Calculating the first derivative of β, one can easily see that∥∥β′(t)∥∥ 6= 0for all t ∈ R. So we can say that β is a regular space curve. In the light of the quasiformulas, one can easily obtain the quasi frame{Tq,Nq,Bq}of β. The graphics of thecurve β and its spherical images are given below.On Surfaces Constructed by Evolution According to Quasi Frame 617Fig. 5.5: The curve β(a) (b)Fig. 5.6: The spherical image of tangent of curve β618 A. Yazla and M. T. Sariaydin(a) (b)Fig. 5.7: The spherical image of the quasi-normal of curve β(a) (b)Fig. 5.8: The spherical image of the quasi-binormal of curve βREFERENCES1. M. Dede, C. Ekici and A. Gorgulu: Directional q-frame along a space curve. IntJour Adv Res Comp Scie Soft Eng IJARCSSE. 5(12) (2015), 775-780.On Surfaces Constructed by Evolution According to Quasi Frame 6192. M. Dede, C. Ekici and H. Tozak: Directional tubular surfaces. International Journalof Algebra. 9(12) (2015), 527-535.3. R. A. Hussien and G. M. Samah: Generated Surfaces via Inextensible Flows ofCurves. Journal of Applied Mathematics. (2016).4. T. Korpinar and E. Turhan: Time Evolution Equations for Surfaces Generatedvia Binormal Spherical Image in Terms of Inextensible Flows. Journal of DynamicalSystems and Geometric Theories. 12(2) (2014), 145-157.5. L. Kula, N. Ekmekci, Y. Yayli and K. Ilarslan: Characterizations of Slant Helicesin Euclidean 3-Space. Turk. J. Math.. 34(2) (2010), 261–273.6. M. A. Soliman: Evolutions of the Ruled Surfaces via the Evolution of Their DirectrixUsing Quasi Frame along a Space Curve. Journal of Applied Mathematics and Physics.(2018).7. M. A. Soliman, N. H. Abdel-All, R. A. Hussien and T. Y. Shaker: SurfacesGenerated via the Evolution of Spherical Image of a Space Curve. Korean Journal ofMathematics. 26(3) (2018), 425-437.Aziz YAZLASelcuk University, Faculty of SceinceDepartment of MathematicsP. O. Box 42130Konya/Turkeyaziz.yazla@selcuk.edu.trMuhammed T. SARIAYDINSelcuk University, Faculty of SceinceDepartment of MathematicsP.O. Box 42130Konya/Turkeytalatsariaydin@gmail.com

ON SURFACES CONSTRUCTED BY EVOLUTION ACCORDING TO QUASI FRAME

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ON SURFACES CONSTRUCTED BY EVOLUTION ACCORDING TO QUASI FRAME

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