In the article, the parametric expressions of the dual ruled surfaces are expressed in terms of the vectorial moments of the Frenet vectors. The integral invariants of these surfaces are calculated. It is seen that the dual parts of these invariants can be stated by the real terms. Finally, we present examples of the ruled surfaces with bases such as helix and Viviani’s curves

Çalışkan, Abdussamet

Şenyurt, Süleyman

English

Digital Commons @ PVAMU (Prairie View A&M Univ)

Applications and Applied Mathematics: An International Journal (AAM) Volume 17 Issue 2 Article 18 12-2022 (R1518) The Dual Spherical Curves and Surfaces in Terms of Vectorial Moments Süleyman Şenyurt Ordu University Abdussamet Çalışkan Ordu University Follow this and additional works at: https://digitalcommons.pvamu.edu/aam  Part of the Geometry and Topology Commons Recommended Citation Şenyurt, Süleyman and Çalışkan, Abdussamet (2022). (R1518) The Dual Spherical Curves and Surfaces in Terms of Vectorial Moments, Applications and Applied Mathematics: An International Journal (AAM), Vol. 17, Iss. 2, Article 18. Available at: https://digitalcommons.pvamu.edu/aam/vol17/iss2/18 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact hvkoshy@pvamu.edu. Available athttp://pvamu.edu/aamAppl. Appl. Math.ISSN: 1932-9466Applications and AppliedMathematics:An International Journal(AAM)Vol. 17, Issue 2 (December 2022), pp. 591 – 600The Dual Spherical Curves and SurfacesIn Terms of Vectorial Moments1Süleyman Şenyurt and 2Abdussamet Çalışkan1Faculty of Arts and SciencesDepartment of Mathematics Ordu University, Ordu, Türkiye2Fatsa Vocational SchoolAccounting and Tax Applications, Ordu University, Ordu, Türkiye1senyurtsuleyman52@gmail.com; 2abdussamet65@gmail.comReceived: August 7, 2021; Accepted: October 16, 2021AbstractIn the article, the parametric expressions of the dual ruled surfaces are expressed in terms of thevectorial moments of the Frenet vectors. The integral invariants of these surfaces are calculated. Itis seen that the dual parts of these invariants can be stated by the real terms. Finally, we presentexamples of the ruled surfaces with bases such as helix and Viviani’s curves.Keywords: Closed ruled surface; Gauss curvature; Drall; Dual spherical curves; Dual angle ofpitch; Vectorial momentMSC 2010 No.: 53A04, 53A051. IntroductionA surface that includes at least one 1-parameter family of straight lines is the ruled surface inIR3 (Do Carmo (1976)). By investigating one parameter spherical motion in dual indicatricecurves, Yaylı and Saraçoğlu (2011) obtain some special ruled surfaces. By considering Frenet andBlaschke frames, unit dual spherical curves are studied with ruled surfaces (Yaylı and Saraçoğlu5911?enyurt and Çal??kan: Dual Spherical Curves and SurfacesPublished by Digital Commons @PVAMU, 2022592 S. Şenyurt and A. Çalışkan(2012)). Tunçer (2017) introduces the w-dual curves, constructed by the Frenet vectors of a reg-ular curve, and gives some properties about these curves. Yanlin and Donghe (2016) define thenotion of evolutes of dual spherical curves and find some results. In Oral and Kazaz (2017), theauthors examine slant ruled surfaces as dual. Using the vectorial moment, the ruled surface drawnby the dual pole curve is examined in Şenyurt and Çalışkan (2020). İncesu (2021) finds interestingresults for ruled surfaces corresponding to Bézier curves on the dual sphere and explains theseresults with examples. Aslan et al. (2021) examines the ruled surface with quaternions in both realand dual spaces, and the new expressions are obtained on the transforms of the geometric shapes.There exist studies by many authors on dual invariants of the ruled surface (Hacısalihoğlu (1972);Gürsoy (1990b); Güven et al. (2011); Rashad (2002); Rashad (2003)).In this study, the ruled surface is examined in terms of dual spaces, and the dual parts of thecorresponding ruled surface are expressed in real terms. The new results are found about the dualinvariants of the ruled surface. In the examples, ruled surfaces drawn by special curves such ashelix and Viviani’s curves are given, and the invariants of these surfaces are examined.2. Main ResultsLet f(ϖ), g(ϖ) and h(ϖ) be at least C3− functions. α(ϖ) can be written in the form ofα(ϖ) = f(ϖ)T (ϖ) + g(ϖ)N(ϖ) + h(ϖ)B(ϖ), (1)as a linear combination of the Frenet vectors. By differentiating both side of (1), we obtainf ′(ϖ)− g(ϖ)κ(ϖ) = 1, h′(ϖ) + g(ϖ)τ(ϖ) = 0, g′(ϖ) + f(ϖ)κ(ϖ)− h(ϖ)τ(ϖ) = 0. (2)The geometric location of T̂ = T + εT ∗, N̂ = N + εN∗ and B̂ = B + εB∗ vectors draws closedcurves on the dual sphere. These curves are called dual spherical curves. The dual expressions ofthe closed ruled surfaces areψT̂ (ϖ, v) = βT (ϖ) + vT (ϖ), βT (ϖ) = T (ϖ) ∧ T∗(ϖ),ψN̂(ϖ, v) = βN(ϖ) + vN(ϖ), βN(ϖ) = N(ϖ) ∧N∗(ϖ), (3)ψB̂(ϖ, v) = βB(ϖ) + vB(ϖ), βB(ϖ) = B(ϖ) ∧B∗(ϖ).Vectorial moments of T,N,B are given, respectively, byT ∗ = α ∧ T = hN − gB, N∗ = α ∧N = −hT + fB, B∗ = α ∧B = gT − fN. (4)Using (4) in (3), we obtain ruled surfaces in terms of linear combination of Frenet vectors as2Applications and Applied Mathematics: An International Journal (AAM), Vol. 17 [2022], Iss. 2, Art. 18https://digitalcommons.pvamu.edu/aam/vol17/iss2/18AAM: Intern. J., Vol. 17, Issue 2 (December 2022) 593follows:ψT̂ (ϖ, v) = T (ϖ) ∧ T∗(ϖ) + vT (ϖ) = vT (ϖ) + g(ϖ)N(ϖ) + h(ϖ)B(ϖ),ψN̂(ϖ, v) = N(ϖ) ∧N∗(ϖ) + vN(ϖ) = f(ϖ)T (ϖ) + vN(ϖ) + h(ϖ)B(ϖ),ψB̂(ϖ, v) = B(ϖ) ∧B∗(ϖ) + vB(ϖ) = f(ϖ)T (ϖ) + g(ϖ)N(ϖ) + vB(ϖ).Theorem 2.1.Let ψT̂ (ϖ, v), ψN̂(ϖ, v) and ψT̂ (ϖ, v) be the ruled surfaces corresponding to the (T̂ ), (N̂) and (B̂)curves. Dralls of the closed ruled surfaces are given byPT̂ = 0, PN̂ =τκ2 + τ 2, PB̂ =1τ·Proof:We know that the drall is calculated byPT̂ =det((T ∧ T ∗)′, T, T ′)∥T ′∥2· (5)Using (2), it can be written that(T ∧ T ∗)′ = (gN + hB)′ = −gκT − fκN.If above value is substituted into (5), we can writePT̂ =det((T ∧ T ∗)′, T, T ′)∥T ′∥2=1κ2∣∣∣∣∣∣−gκ −fκ 01 0 00 κ 0∣∣∣∣∣∣= 0.Similarly, with the values of(N ∧N∗)′ = (fT + hB)′ = (1 + gκ)T − (−fκ+ hτ)N + (−gτ)B,(B ∧B∗)′ = (fT + gN)′ = T + hτN + gτB,3?enyurt and Çal??kan: Dual Spherical Curves and SurfacesPublished by Digital Commons @PVAMU, 2022594 S. Şenyurt and A. Çalışkandralls of the surfaces corresponding to the (N̂) curve and (B̂) are, respectively,PN̂ =det((N ∧N∗)′, N,N ′)∥N ′∥2=1κ2 + τ 2∣∣∣∣∣∣1 + gκ fκ− hτ ′ −gτ0 1 0−κ 0 τ∣∣∣∣∣∣=τκ2 + τ 2,PB̂ =det((B ∧B∗)′, B,B′)∥B′∥2=1τ 2∣∣∣∣∣∣1 hτ gτ0 0 10 −τ 0∣∣∣∣∣∣=1τ· ■Theorem 2.2.Let ψT̂ (ϖ, v), ψN̂(ϖ, v) and ψT̂ (ϖ, v) be the ruled surfaces corresponding to the (T̂ ), (N̂) and (B̂)curves. The dual angles of pitch (d.a.p.) of closed ruled surfaces areΛT̂ = λT − ε(gλB +∮f ′),ΛN̂ = −ε(hλT − fλB +∮g′),ΛB̂ = λB − ε(−gλT +∮h′).Proof:The d.a.p. of the surface (T̂ ) isΛT̂ = −⟨D, T̂ ⟩ = −⟨d+ εd∗, T + ε(hN − gB)⟩= −∮τ − ε(−g∮κ+∮f ′)= λT − ε(gλB +∮f ′).4Applications and Applied Mathematics: An International Journal (AAM), Vol. 17 [2022], Iss. 2, Art. 18https://digitalcommons.pvamu.edu/aam/vol17/iss2/18AAM: Intern. J., Vol. 17, Issue 2 (December 2022) 595Similarly, d.a.p. of the surfaces corresponding to the (N̂) and (B̂) areΛN̂ = −⟨D, N̂⟩ = −⟨d+ εd∗, N + ε(−hT + fB)⟩= −ε(hλT − fλB +∮g′),ΛB̂ = −⟨D, B̂⟩ = −⟨d+ εd∗, B + ε(gT − fN)⟩= λB − ε(−gλT +∮h′). ■Example 2.1.Let α(ϖ) = 1√2(− cosϖ,− sinϖ,ϖ) be a circular helix curve. Then, we obtainT (ϖ) =1√2(sinϖ,− cosϖ, 1), N(ϖ) = (cosϖ, sinϖ, 0),B(ϖ) =1√2(− sinϖ, cosϖ, 1),κ(ϖ) =1√2, τ(ϖ) =1√2.Considering the dual expression of a ruled surface, we getψT̂ (ϖ, v) = −√24(ϖ sinϖ + 2 cosϖ − 2v sinϖ,−ϖ cosϖ + 2 sinϖ + 2v cosϖ,−ϖ − 2v),ψN̂(ϖ, v) = (v cosϖ, v sinϖ,1√2ϖ),ψB̂(ϖ, v) =√24(w sinϖ − 2 cosϖ − 2v sinϖ,−w cosϖ − 2 sinϖ + 2v cosϖ,ϖ + 2v).These ruled surfaces are shown in Figure 1. Let us find the functions f(ϖ), g(ϖ), h(ϖ). From theequation (1), we can write− cosϖ = f(ϖ) sinϖ + g(ϖ)√2 cosϖ − h(ϖ) sinϖ,− sinϖ = −f(ϖ) cosϖ + g(ϖ)√2 sinϖ + h(ϖ) cosϖ,ϖ = f(ϖ) + g(ϖ).The solutions of f(ϖ), g(ϖ), h(ϖ) are given byf(ϖ) =ϖ2, g(ϖ) = − 1√2, h(ϖ) =ϖ2·5?enyurt and Çal??kan: Dual Spherical Curves and SurfacesPublished by Digital Commons @PVAMU, 2022596 S. Şenyurt and A. ÇalışkanBy taking into account consideration of the above equation, d.a.p. are given byΛT̂ = λT + ε(1√2λB −12∮dϖ)= λT + ε(1√2λB −12LT ),ΛN̂ = εϖ2(−λT + λB),ΛB̂ = λB − ε(1√2λT +12∮dϖ)= λB − ε(1√2λT +12LT ).(i) (ii) (iii)Figure 1. The figures (i), (ii) and (iii) show closed ruled surfaces corresponding to the (T̂ ), (N̂) and (B̂) dual sphericalcurves, respectively. Base curves of these surfaces are helix curve.Example 2.2.Let α(ϖ) = (45cosϖ, 1− sinϖ,−35cosϖ) be a curve. Then, it is easy to show thatT (ϖ) =(−45sinϖ,− cosϖ, 35sinϖ), N(ϖ) =(−45cosϖ, sinϖ,35cosϖ),B(ϖ) =(−35, 0,−45), κ(ϖ) = 1, τ(ϖ) = 0.Considering the dual expression of a ruled surface, we obtain closed ruled surfaces asψT̂ (ϖ, v) =(− 45((sinϖ − 1) cosϖ + v sinϖ), (sinϖ − 1) sinϖ − v cosϖ,35((sinϖ − 1) cosϖ + v sinϖ)),ψN̂(ϖ, v) =(45cosϖ sinϖ − v45cosϖ, cos 2ϖ + v sinϖ,−35cosϖ sinϖ + v35cosϖ),ψB̂(ϖ, v) =(45cos2ϖ − v35, 1− sinϖ,−35cos2ϖ − v45).6Applications and Applied Mathematics: An International Journal (AAM), Vol. 17 [2022], Iss. 2, Art. 18https://digitalcommons.pvamu.edu/aam/vol17/iss2/18AAM: Intern. J., Vol. 17, Issue 2 (December 2022) 597These ruled surfaces are shown in Figure 2. Let us find the functions f(ϖ), g(ϖ), h(ϖ). FromEquation (1), we can write4 cosϖ = −4f(ϖ) sinϖ − 4g(ϖ) cosϖ − 3h(ϖ),1− sinϖ = −f(ϖ) cosϖ + g(ϖ) sinϖ,−3 cosϖ = 3f(ϖ) sinϖ + 3g(ϖ) cosϖ − 4h(ϖ).The solutions of f(ϖ), g(ϖ), h(ϖ) are given byf(ϖ) = − cosϖ, g(ϖ) = sinϖ − 1, h(ϖ) = 0.By taking into account the above equation, d.a.p. of closed ruled surfaces areΛT̂ = λT − ε((sinϖ − 1)λB +∮cosϖ), ΛN̂ = −ε(cosϖλB +∮cosϖ),ΛB̂ = λB − ε((1− sinϖ)λT + c),where c is an arbitrary constant known as the integration constant.(i) (ii) (iii)Figure 2. The figures (i), (ii) and (iii) show closed ruled surfaces corresponding to the (T̂ ), (N̂) and (B̂) dual sphericalcurves, respectively.Example 2.3.The Viviani’s curve is formed by the intersection of a cylinder and a sphere. It is parametrized byα(ι) =(a(1 + cos ι), a sin ι, 2a sinι2).Here, 2a is the radius of the sphere (Gray et al. (2017)). The expression for the Frenet invariants ofViviani’s curve are given by7?enyurt and Çal??kan: Dual Spherical Curves and SurfacesPublished by Digital Commons @PVAMU, 2022598 S. Şenyurt and A. ÇalışkanT (ι) =(−√2 sin ι√3 + cos ι,√2 cos ι√3 + cos t,√2 cos ι2√3 + cos ι),N(ι) =(−3− 12 cos ι− cos 2ι√88 cos ι+ 162 + 6 cos 2ι,−12 sin ι− sin 2ι√88 cos ι+ 162 + 6 cos 2ι,2√2 sin ι2√81 + 44 cos t+ 3 cos 2ι),B(ι) =(3 sin ι2+ sin 3ι2√26 + 6 cos ι,−2√2 cos3 ι2√13 + 3 cos ι,2√2√13 + 3 cos ι),κ(ι) =√3 cos ι+ 13a(cos ι+ 3)32, τ(ι) =6 cos ι23a cos ι+ 13a·Considering the dual expression of a ruled surface, we can write closed ruled surfaces asψT̂ (ι, v) =(2a cos2ι2− v√2 sin ι√3 + cos ι, a sin ι+ v√2 cos ι√3 + cos ι, 2a sinι2+ v√2 cos ι2√3 + cos ι),ψN̂(ι, v) =(−2a(2 cos8 ι2+ 9 cos6 ι2+ 4 cos4 ι2− 17 cos2 ι2+ 2)3 cos4 ι2+ 8 cos2 ι2+ 5− v 3 + 12 cos ι+ cos 2ι√88 cos ι+ 162 + 6 cos 2ι,−2a sin ι(cos6 ι2+ 5 cos4 ι2+ 5 cos2 ι2− 5)3 cos4 ι2+ 8 cos2 ι2+ 5− v 12 sin ι+ sin 2ι√88 cos ι+ 162 + 6 cos 2ι,32a(cos4 ι2+ 3 cos2 ι2+ 1)sin ι222 cos ι+ 39 + 3 cos2 ι+ v2√2 sin ι2√81 + 44 cos ι+ 3 cos 2ι),ψB̂(ι, v) =(4a(cos6 ι2+ 3 cos4 ι2+ cos2 ι2− 1)3 cos2 ι2+ 5+ v3 sin ι2+ sin 3ι2√26 + 6 cos ι,a(4 sin 3ι2cos3 ι2(cos ι+ 1) + 3 sin 2ι cos2 ι2+ 13 sin ι cos2 ι226 + 6 cos ι+6 sin 3ι2sin ι sin ι2− cos2 3ι2sin ι+ 26 sin ι26 + 6 cos ι− v2√2 cos3 ι2√13 + 3 cos ι,2a(cos2 ι2+ 1)sin ι23 cos2 ι2+ 5+ v2√2√13 + 3 cos ι).These closed ruled surfaces are shown in Figure 3.8Applications and Applied Mathematics: An International Journal (AAM), Vol. 17 [2022], Iss. 2, Art. 18https://digitalcommons.pvamu.edu/aam/vol17/iss2/18AAM: Intern. J., Vol. 17, Issue 2 (December 2022) 599(i) (ii) (iii)Figure 3. The figures (i), (ii) and (iii) show ruled surfaces corresponding to the (T̂ ), (N̂) and (B̂) dual spherical curves,respectively. Base curves of these surfaces are Vivian’s curve.3. ConclusionIn this study, the ruled surfaces corresponding to dual curves are written as a linear combinationof Frenet vectors. According to the conditions, the dralls and the dual angles of the pitch of thesurfaces are calculated. It is seen that the dual parts of the dual angles of pitch can be written inreal terms. The fact that dual elements can be expressed in real terms will be a guide in the studiesabout dual space.REFERENCESAbbena, E., Salamon, S. and Gray A. (2005). Modern Differential Geometry of Curves and Sur-faces with Mathematica, CRC Press, United States.Aslan, S., Bekar, M. and Yaylı, Y. (2021). Ruled surfaces constructed by quaternions, Journal ofGeometry and Physics, Vol. 161, No. 104048, pp. 1–9.Blaschke, W. (1945). 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(R1518) The Dual Spherical Curves and Surfaces in Terms of Vectorial Moments

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(R1518) The Dual Spherical Curves and Surfaces in Terms of Vectorial Moments

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