BÄCKLUND TRANSFORMATIONS ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE
|
|
- Kory Bates
- 5 years ago
- Views:
Transcription
1 iauliai Math. Semin., 7 15), 2012, 4149 BÄCKLUND TRANSFORMATIONS ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE Murat Kemal KARACAN, Yilmaz TUNÇER Department of Mathematics, Usak University, Usak, Turkey; s: murat.karacan@usak.edu.tr, yilmaz.tuncer@usak.edu.tr Abstract. In this paper, we dene Bäcklund transformations of curves according to Bishop frame which preserving the natural curvatures under certain assumptions in Euclidean 3-space. Key wor and phrases: Bäcklund transformations, Bishop frame Mathematics Subject Classication: 53A Preliminaries In the 1890s, Bianchi, Lie, and nally Bäcklund looked at what are now called Bäcklund transformations of surfaces. In modern parlance, they begin with two surfaces in Euclidean space in a line congruence: there is a mapping between the surfaces M 1 and M 2 such that the line through any two corresponding points is tangent to both surfaces. Bäcklund proved that if a line congruence satised two additional conditions, that the line segment joining corresponding points has constant length, and that the normals at corresponding points form a constant angle, then the two surfaces are necessarily surfaces of constant negative curvature. He was also able to show that a Bäcklund transformation is integrable, in the sense that given a point on a surface of constant negative curvature and a tangent line segment at that point, a new surface of constant negative curvature can be found, containing the endpoint of the line segment,that is a Bäcklund transform of the original surface.
2 42 M. K. Karacan, Y. Tunçer The classical Bäcklund theorem studies the transformation of surfaces of constant negative curvature in R 3 by realizing them as the focal surfaces of a pseudo-spherical line congruence. The integrability theorem says that we can construct a new surface in R 3 with constant negative curvature from a given one. In 19, Tenenblat and Terng established a high dimension generalization of Bäcklund's theorem which is very interesting both for physical and mathematical reasons. After that, Chern and Terng customized Bäcklund theorem for ane surfaces 6. By the same year, this transformation was reduced to corresponding asymptotical lines by Terng 20 and following years Tenenblat expanded the Bäcklund transformation of two surfaces in R 3 1 to space forms 18. In 1990, Palmer constructed a Bäcklund transformation between spacelike and timelike surfaces of constant negative curvature in E At that decade, some researchers gave Bäcklund transformations on Weingarten surfaces 2, 5, 7 and 21. In 1998, Calini and Ivey 3 proposed a geometric realization of the Bäcklund transformation for the sine-gordon equation in the context of curves of constant torsion. Since the asymptotic lines on a pseudospherical surface have constant torsion, the Bäcklund transformation can be restricted to get a transformation that carries constant torsion curves to constant torsion curves. Later, the converse of the idea was proved and generalized for the n dimensional case by Nemeth 12. In 13, Nemeth studied a similar concept for constant torsion curves in the 3-dimensional constant curvature spaces. Shief and Rogers used an analogue of the classical Bäcklund transformation for the generation of soliton surfaces 16. In 8, Chou, Kouhua and Yongbo obtained the Bäcklund transformation on timelike surfaces with constant mean curvature in R 2,1. Zuo, Chen, Cheng studied Bäcklund theorems in three dimensional de Sitter space and anti-de Sitter space 22. Abdel-Baky presented the Minkowski versions of the Bäcklund theorem and its application by using the method of moving frames 1. Gurbuz studied Bäcklund transformations in R1 n 9. Using the same method Ozdemir and Coken have studied Bäcklund transformations of non-lightlike constant torsion curves in Minkowski 3-space 14. Also, Cengiz and Gurbuz have studied Bäcklund transformations of curves in the Galilean and Pseudo-Galilean spaces 4. In this paper, we show that a restriction of Bäcklund theorem on space curves satisfying the given three conditions preserves the rst and second curvaturesnatural curvatures) of the curves according to Bishop frame in Euclidean 3-space. This section is taken from 4.
3 Bäcklund transformations according to Bishop frame Introduction Let α : I R E 3 be an arbitrary curve in Euclidean 3-space E 3. Recall that the curve α is said to be of unit speed or parametrized by arc length function s) if α, α = 1, where, is the standard scalar inner) product of E 3 given by x, y = x 1 y 1 + x 2 y 2 + x 3 y 3 for each x = x 1, x 2, x 3 ), y = y 1, y 2, y 3 ) E 3. In particular, the norm of a vector x E 3 is given by x = x, x. Denote by {T s), Ns), Bs)} the moving Frenet frame along the unit speed curve α. Then the Frenet formulas are given by T N B = 0 κ 0 κ 0 τ 0 τ 0 Here T, N and B are called the tangent, the principal normal and the binormal vector el of the curves, respectively. κs) and τs) are called, curvature and torsion of the curve α, respectively 18. The ability to ride along a three-dimensional space curve and illustrate the properties of the curve, such as curvature and torsion, would be a great asset to mathematicians. The classic Serret-Frenet frame provides such ability, however, the Serret-Frenet frame does is not dened for all points along every curve. A new frame is needed for the kind of mathematical analysis that is typically done with computer graphics. The relatively parallel adapted frame or Bishop frame could provide the desired means to ride along any given space curve. The Bishop frame has many properties that make it ideal for mathematical research. Another area of interested about the Bishop frame is so-called normal development, or the graph of the twisting motion of Bishop frame. This information along with the initial position and orientation of the Bishop frame provide all of the information necessary to dene the curve. The Bishop frame may have applications in the area of Biology and Computer graphics. For example, it may be possible to compute information about the shape of sequences of DNA using a curve dened by the Bishop frame. The Bishop frame may also provide a new way to control virtual cameras in computer animations 11. The Bishop frame or parallel transport frame is an alternative approach to dening a moving frame that is well dened even when the curve has vanishing second derivative. We can parallel transport an orthonormal frame along a curve simply by parallel transporting each component of the frame. T N B.
4 44 M. K. Karacan, Y. Tunçer The parallel transport frame is based on the observation that, while T s) for a given curve model is unique, we may choose any convenient arbitrary basis Us), V s)) for the remainder of the frame, so long as it is in the normal plane perpendicular to T s) at each point. If the derivatives of Us), V s)) depend only on T s) and not each other, we can make Us) and V s) vary smoothly throughout the path regardless of the curvature. In addition, suppose the curve α is an arc length-parametrized C 2 curve. Suppose we have C 1 unit vector el U and V = T U along the curve α so that T, U = T, V = U, V = 0, i.e., T, U, V will be a smoothly varying right-handed orthonormal frame as we move along the curve to this point, the Frenet frame would work just ne if the curve were C 3 with κ 0). But now we want to impose the extra condition that U, V = 0.We say the unit rst normal vector eld U is parallel along the curve α. This means that the only change of U is in the direction of T. A Bishop frame can be dened even when a Frenet frame cannot e.g., when there are points with κ = 0) 10. Therefore, we have the alternative frame equations One can show that κs) = k1 2 + k2 2, T U V = 0 k 1 k 2 k k T U V. 2.1) ) θs) = arctan k2, k 1 0, τs) = dθs) k 1. So that k 1 and k 2 eectively correspond to a Cartesian coordinate system for the polar coordinates κ, θ with θ = τs). The orientation of the parallel transport frame includes the arbitrary choice of integration constant θ 0, which disappears from τ and hence from the Frenet frame) due to the dierentiation 1011, 17. Thus, the relation matrix may be expressed as T = T, Bishop curvatures are dened by N = U cos θs) V sin θs), B = U sin θs) + V cos θs). k 1 = κs) cos θs), k 2 = κs) sin θs).
5 Bäcklund transformations according to Bishop frame Bäcklund transformation according to Bishop frame Theorem 3.1. Suppose that ψ is a transformation between two curves α and β in Euclidean 3-space with β = ψα) such that in the corresponding points we have: a) the line segment βs)αs) at the intersection of the osculating planes of the curves has constant length r; b) the distance vector βs) αs) has the same angle γ π 2 tangent vectors of the curves; c) the binormals of the curves have the same constant angle ϕ 0. Then these curves are congruent with natural curvatures k β 1 = k α 1 = dγ, k β 2 = k α 2 = tan γ sin ϕ, r and the transformation of the curves is given by where C = k α 2 tan ϕ 2 β = α + 2C tan γ k2 α)2 + C 2 T α cos γ + U α sin γ), and γ is a solution of the dierential equation dγ = kβ 2 cos γ tan ϕ 2 kα 1 = k α 2 cos γ tan ϕ 2 kβ 1. with the Proof. Denote by T α, U α, V α ) and T β, U β, V β ) the Bishop frames of the curves α and β in the Euclidean 3-space E 3. Let V β be a unit second principal normal of β.if we denote by W1 α the unit vector of β α, then we can complete W1 α, V α and W1 α, V β to the positively oriented orthonormal frames W1 α, W 2 α, W 3 α) and W β 1, W β 2, W β 3 ), where W 3 α = V α, W β 3 = V β, and γ is the angle between W1 α and T α. The frames W1 α, W 2 α, W 3 α) and W β 1, W β 2, W β 3 ) can be obtained by rotating the framest α, U α, V α ) and T β, U β, V β ) around V α and V β with an angle γ, respectively. So, we can write W α 1 cos γ sin γ 0 T α W2 α = sin γ cos γ 0 U α W3 α V α and W α 1 W β 2 W β 3 = cos γ sin γ 0 sin γ cos γ T β U β V β.
6 46 M. K. Karacan, Y. Tunçer Similarly, for a rotation around W α 1 by the angle ϕ, From the above equations, we write W β 2 = W α 2 cos ϕ W α 3 sin ϕ, W β 3 = W α 2 sin ϕ + W α 3 cos ϕ. T β = cos 2 γ + sin 2 γ cos ϕ ) T α + cos γ sin γ) 1 cos ϕ) U α + sin γ sin ϕ) V α, 3.1) U β = cos γ sin γ) 1 cos ϕ) T α + sin 2 γ + cos 2 γ cos ϕ ) U α cos γ sin ϕ) V α, 3.2) V β = sin γ sin ϕ) T α + cos γ sin ϕ) U α + cos ϕ) V α. 3.3) Using 2.1) and 3.1), 3.2), 3.3), for T β, U β and V β, we have dt β du β dv β = k β 1 U β + k β 2 V β = k β 1 cos γ sin γ1 cos ϕ) kβ 2 sin γ sin ϕ T α + k β 1 sin 2 γ + cos 2 γ cos ϕ ) + k β 2 ) cos γ sin ϕ) + k β 2 cos ϕ kβ 1 cos γ sin ϕ V α, = k β 1 T β = k β 1 cos 2 γ + sin 2 γ cos ϕ ) T α k β 1 1 cos ϕ) cos γ sin γ) U α k β 1 sin γ sin ϕ) V α, = k β 2 T β = k β 2 cos 2 γ + sin 2 γ cos ϕ ) T α k β 2 1 cos ϕ) cos γ sin γ) U α k β 2 sin γ sin ϕ) V α, U α and taking derivative of T β, U β and V β in 3.1), 3.2), 3.3) with respect to s, we get dt β = 1 cos ϕ) + k1 α 2 dγ ) cos γ sin γ) k2 α sin γ sin ϕ) kα 1 cos 2 γ + sin 2 γ cos ϕ ) + cos 2 γ1 cos ϕ) dγ U α T α
7 Bäcklund transformations according to Bishop frame du β dv β = = + k2 α cos 2 γ + sin 2 γ cos ϕ ) + dγ cos 2 γ1 cos ϕ) dγ cos γ sin ϕ V α, kα 1 sin 2 γ + cos 2 γ cos ϕ ) + k2 α cos γ sin ϕ) T α + 1 cos ϕ) cos γ sin γ) dγ + kα 1 1 cos ϕ) cos γ sin γ) + k2 α 1 cos ϕ) cos γ sin γ) + sin γ sin ϕ) dγ cos γ sin ϕ) dγ sin γ sin ϕ) kα 1 ) k2 α cos ϕ) k α 1 + dγ T α V α, ) U α k α 2 sin γ sin ϕ) V α. Then, equating the two statements above, we obtain k β 2 = k2 α, dγ = k β 2 cos γ tan ϕ 2 kα 1. Similarly, using 2.1) and 3.1), 3.2), 3.3), we have k α 1 + k β 1 = 2 dγ, k α 1 = k β 1 = dγ. Now α is a unit speed curve. Dierentiating and substituting the distance vector β α) 2 = r 2, U α β α = r T α cos γ U α sin γ), 3.4) we nd that β is also a unit speed curve. Next, taking the derivative of 3.4), we obtain T β = r sin γ) k1 α + dγ ) T α + r sin γ) k1 α + dγ ) U α + rk2 α cos γ) V α. From this equation and the Bishop frames 3.1), 3.2) and 3.3), we get k β 2 = kα 2 = tan γ sin ϕ. r
8 48 M. K. Karacan, Y. Tunçer Then, rearranging this equality, we get r = tan γ sin ϕ k α 2 ) 2k2 α tan γ tan ϕ 2 = )). k2 α)2 1 + tan 2 ϕ 2 Finally, with the aid of 3.4), the Bäcklund transformation of the curves is β = α + 2C tan γ k2 α)2 + C 2 T α cos γ + U α sin γ). References 1 R. A. Abdel Baky, The Bäcklund's theorem in Minkowski 3-space R 3 1, Appl. Math. Comput., 160, ). 2 S. G. Buyske, Geometric aspects of Bäcklund transformations of weingarten submanifol, Pacic J. Math., 166, ). 3 A. Calini, T. Ivey, Bäcklund transformations and knots of constant torsion, J. Knot Theor. Ramif., 7, ). 4 S. Cengiz, N. Gurbuz, Bäcklund transformations of curves in the Galilean and pseudo-galilean spaces, v1.pdf 5 W. Chen, H. Li, Weingarten surfaces and Sine-Gordon equation, Sci. China Ser. A, 40, ). 6 S. S. Chern, C. L. Terng, An analogue of Bäcklund's theorem in ane geometry, Rocky Mt. J. Math., 10, ). 7 T. Chou, C. Xifang, Bäcklund transformation on surfaces with ak + bh = c, Chin. J. Contemp. Math., 18, ). 8 T. Chou, Z. Kouhua, T. Yongbo, Bäcklund transformation on surfaces with constant mean curvature in R 2 1, Acta Math. Sci., Ser. B, Engl. Ed., 233), ). 9 N. Gurbuz, Bäcklund transformations of constant torsion curves in R n 1, Hadronic J., 29, ). 10 A. J. Hanson, H. Ma, Parallel transport approach to curve framing, Indiana University, Techreports, TR425, January ).
9 Bäcklund transformations according to Bishop frame P. McCreary, Proseminar recap of Bishop frames, Department of Mathematics, Illinois University, June ). 12 S. Z. Nemeth, Bäcklund transformations of n-dimensional constant torsion curves, Publ. Math. Debrecen, 53, ). 13 S. Z. Nemeth, Bäcklund transformations of constant torsion curves in 3- dimensional constant curvature spaces, Ital. J. Pure Appl. Math., 7, ). 14 M. Ozdemir, A. C. Coken, Bäcklund transformation for non-lightlike curves in Minkowski 3-space, Chaos Solitons Fract., 42, ). 15 B. Palmer, Bäcklund transformations for surfaces in Minkowski space, J. Math. Phys, 31, ). 16 W. K. Schief, C. Rogers, Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces, P. Roy. Soc. Lond. A Mat., 455, ). 17 T. Shifrin, Dierential geometry: a rst course in curves and surfaces, 18 K. Tenenblat, Bäcklund theorems for submanifol of space forms and a generalized wave equation, Bull. Soc. Brasil. Mat., 16, ). 19 K. Tenenblat, C. L. Terng, Bäcklund's theorem for n-dimensional submanifol in R 2n 1, Ann. Math., 111, ). 20 C. L. Terng, A higher dimension generalization of the Sine-Gordon equation and its soluiton theory, Ann. Math., 111, ). 21 C. Xifang, T. Chou, Bäcklund transformations on Surfaces with k 1 m) k 2 m) = ±l 2 in R 2 1, J. Phys. A: Math. Gen, 30, ). 22 D. Zuo, Q. Chen, Y. Cheng, Bäcklund theorems in three-dimensional de Sitter space and anti-de Sitter space, J. Geom. Phys., 44, ). Received 23 November 2011
ON THE RULED SURFACES WHOSE FRAME IS THE BISHOP FRAME IN THE EUCLIDEAN 3 SPACE. 1. Introduction
International Electronic Journal of Geometry Volume 6 No.2 pp. 110 117 (2013) c IEJG ON THE RULED SURFACES WHOSE FRAME IS THE BISHOP FRAME IN THE EUCLIDEAN 3 SPACE ŞEYDA KILIÇOĞLU, H. HILMI HACISALIHOĞLU
More informationOn the dual Bishop Darboux rotation axis of the dual space curve
On the dual Bishop Darboux rotation axis of the dual space curve Murat Kemal Karacan, Bahaddin Bukcu and Nural Yuksel Abstract. In this paper, the Dual Bishop Darboux rotation axis for dual space curve
More informationParallel Transport Frame in 4 dimensional Euclidean Space E 4
Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611 CJMS. 3(1)(2014), 91-103 Parallel Transport Frame in 4 dimensional Euclidean
More informationNon-null weakened Mannheim curves in Minkowski 3-space
An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) Tomul LXIII, 2017, f. 2 Non-null weakened Mannheim curves in Minkowski 3-space Yilmaz Tunçer Murat Kemal Karacan Dae Won Yoon Received: 23.IX.2013 / Revised:
More informationSPLIT QUATERNIONS and CANAL SURFACES. in MINKOWSKI 3-SPACE
INTERNATIONAL JOURNAL OF GEOMETRY Vol. 5 (016, No., 51-61 SPLIT QUATERNIONS and CANAL SURFACES in MINKOWSKI 3-SPACE SELAHATTIN ASLAN and YUSUF YAYLI Abstract. A canal surface is the envelope of a one-parameter
More informationON OSCULATING, NORMAL AND RECTIFYING BI-NULL CURVES IN R 5 2
Novi Sad J. Math. Vol. 48, No. 1, 2018, 9-20 https://doi.org/10.30755/nsjom.05268 ON OSCULATING, NORMAL AND RECTIFYING BI-NULL CURVES IN R 5 2 Kazım İlarslan 1, Makoto Sakaki 2 and Ali Uçum 34 Abstract.
More informationHomothetic Bishop Motion Of Euclidean Submanifolds in Euclidean 3-Space
Palestine Journal of Mathematics Vol. 016, 13 19 Palestine Polytechnic University-PPU 016 Homothetic Bishop Motion Of Euclidean Submanifol in Euclidean 3-Space Yılmaz TUNÇER, Murat Kemal KARACAN and Dae
More informationSpherical Images and Characterizations of Time-like Curve According to New Version of the Bishop Frame in Minkowski 3-Space
Prespacetime Journal January 016 Volume 7 Issue 1 pp. 163 176 163 Article Spherical Images and Characterizations of Time-like Curve According to New Version of the Umit Z. Savcı 1 Celal Bayar University,
More informationSOME RELATIONS BETWEEN NORMAL AND RECTIFYING CURVES IN MINKOWSKI SPACE-TIME
International Electronic Journal of Geometry Volume 7 No. 1 pp. 26-35 (2014) c IEJG SOME RELATIONS BETWEEN NORMAL AND RECTIFYING CURVES IN MINKOWSKI SPACE-TIME KAZIM İLARSLAN AND EMILIJA NEŠOVIĆ Dedicated
More informationMannheim partner curves in 3-space
J. Geom. 88 (2008) 120 126 0047 2468/08/010120 7 Birkhäuser Verlag, Basel, 2008 DOI 10.1007/s00022-007-1949-0 Mannheim partner curves in 3-space Huili Liu and Fan Wang Abstract. In this paper, we study
More informationA METHOD OF THE DETERMINATION OF A GEODESIC CURVE ON RULED SURFACE WITH TIME-LIKE RULINGS
Novi Sad J. Math. Vol., No. 2, 200, 10-110 A METHOD OF THE DETERMINATION OF A GEODESIC CURVE ON RULED SURFACE WITH TIME-LIKE RULINGS Emin Kasap 1 Abstract. A non-linear differential equation is analyzed
More informationSmarandache Curves and Spherical Indicatrices in the Galilean. 3-Space
arxiv:50.05245v [math.dg 2 Jan 205, 5 pages. DOI:0.528/zenodo.835456 Smarandache Curves and Spherical Indicatrices in the Galilean 3-Space H.S.Abdel-Aziz and M.Khalifa Saad Dept. of Math., Faculty of Science,
More informationk type partially null and pseudo null slant helices in Minkowski 4-space
MATHEMATICAL COMMUNICATIONS 93 Math. Commun. 17(1), 93 13 k type partially null and pseudo null slant helices in Minkowski 4-space Ahmad Tawfik Ali 1, Rafael López and Melih Turgut 3, 1 Department of Mathematics,
More informationCERTAIN CLASSES OF RULED SURFACES IN 3-DIMENSIONAL ISOTROPIC SPACE
Palestine Journal of Mathematics Vol. 7(1)(2018), 87 91 Palestine Polytechnic University-PPU 2018 CERTAIN CLASSES OF RULED SURFACES IN 3-DIMENSIONAL ISOTROPIC SPACE Alper Osman Ogrenmis Communicated by
More informationThe Ruled Surfaces According to Type-2 Bishop Frame in E 3
International Mathematical Forum, Vol. 1, 017, no. 3, 133-143 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/imf.017.610131 The Ruled Surfaces According to Type- Bishop Frame in E 3 Esra Damar Department
More informationC-partner curves and their applications
C-partner curves and their applications O. Kaya and M. Önder Abstract. In this study, we define a new type of partner curves called C- partner curves and give some theorems characterizing C-partner curves.
More informationSLANT HELICES IN MINKOWSKI SPACE E 3 1
J. Korean Math. Soc. 48 (2011), No. 1, pp. 159 167 DOI 10.4134/JKMS.2011.48.1.159 SLANT HELICES IN MINKOWSKI SPACE E 3 1 Ahmad T. Ali and Rafael López Abstract. We consider a curve α = α(s) in Minkowski
More informationCHARACTERIZATION OF SLANT HELIX İN GALILEAN AND PSEUDO-GALILEAN SPACES
SAÜ Fen Edebiyat Dergisi (00-I) CHARACTERIZATION OF SLANT HELIX İN ALILEAN AND PSEUDO-ALILEAN SPACES Murat Kemal KARACAN * and Yılmaz TUNÇER ** *Usak University, Faculty of Sciences and Arts,Department
More informationFathi M. Hamdoon and A. K. Omran
Korean J. Math. 4 (016), No. 4, pp. 613 66 https://doi.org/10.11568/kjm.016.4.4.613 STUDYING ON A SKEW RULED SURFACE BY USING THE GEODESIC FRENET TRIHEDRON OF ITS GENERATOR Fathi M. Hamdoon and A. K. Omran
More informationINEXTENSIBLE FLOWS OF CURVES IN THE EQUIFORM GEOMETRY OF THE PSEUDO-GALILEAN SPACE G 1 3
TWMS J. App. Eng. Math. V.6, N.2, 2016, pp. 175-184 INEXTENSIBLE FLOWS OF CURVES IN THE EQUIFORM GEOMETRY OF THE PSEUDO-GALILEAN SPACE G 1 3 HANDAN ÖZTEKIN 1, HÜLYA GÜN BOZOK 2, Abstract. In this paper,
More informationA STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING FOR A NULL CURVE WITH CARTAN FRAME
Bull. Korean Math. Soc. 49 (), No. 3, pp. 635 645 http://dx.doi.org/.434/bkms..49.3.635 A STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING FOR A NULL CURVE WITH CARTAN FRAME N ihat Ayyildiz and Tunahan Turhan
More informationAvailable online at J. Math. Comput. Sci. 6 (2016), No. 5, ISSN:
Available online at http://scik.org J. Math. Comput. Sci. 6 (2016), No. 5, 706-711 ISSN: 1927-5307 DARBOUX ROTATION AXIS OF A NULL CURVE IN MINKOWSKI 3-SPACE SEMRA KAYA NURKAN, MURAT KEMAL KARACAN, YILMAZ
More informationTHOMAS A. IVEY. s 2 γ
HELICES, HASIMOTO SURFACES AND BÄCKLUND TRANSFORMATIONS THOMAS A. IVEY Abstract. Travelling wave solutions to the vortex filament flow generated byelastica produce surfaces in R 3 that carrymutuallyorthogonal
More informationPSEUDO-SPHERICAL EVOLUTES OF CURVES ON A TIMELIKE SURFACE IN THREE DIMENSIONAL LORENTZ-MINKOWSKI SPACE
PSEUDO-SPHERICAL EVOLUTES OF CURVES ON A TIMELIKE SURFACE IN THREE DIMENSIONAL LORENTZ-MINKOWSKI SPACE S. IZUMIYA, A. C. NABARRO AND A. J. SACRAMENTO Abstract. In this paper we introduce the notion of
More informationTHE BERTRAND OFFSETS OF RULED SURFACES IN R Preliminaries. X,Y = x 1 y 1 + x 2 y 2 x 3 y 3.
ACTA MATHEMATICA VIETNAMICA 39 Volume 31, Number 1, 2006, pp. 39-48 THE BERTRAND OFFSETS OF RULED SURFACES IN R 3 1 E. KASAP AND N. KURUOĞLU Abstract. The problem of finding a curve whose principal normals
More informationGeometry of Cylindrical Curves over Plane Curves
Applied Mathematical Sciences, Vol 9, 015, no 113, 5637-5649 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/101988/ams01556456 Geometry of Cylindrical Curves over Plane Curves Georgi Hristov Georgiev, Radostina
More information1-TYPE AND BIHARMONIC FRENET CURVES IN LORENTZIAN 3-SPACE *
Iranian Journal of Science & Technology, Transaction A, ol., No. A Printed in the Islamic Republic of Iran, 009 Shiraz University -TYPE AND BIHARMONIC FRENET CURES IN LORENTZIAN -SPACE * H. KOCAYIGIT **
More informationA Note on Inextensible Flows of Partially & Pseudo Null Curves in E 4 1
Prespacetime Journal April 216 Volume 7 Issue 5 pp. 818 827 818 Article A Note on Inextensible Flows of Partially & Pseudo Null Curves in E 4 1 Zühal Küçükarslan Yüzbaşı 1 & & Mehmet Bektaş Firat University,
More informationSurfaces of Arbitrary Constant Negative Gaussian Curvature and Related Sine-Gordon Equations
Mathematica Aeterna, Vol.1, 011, no. 01, 1-11 Surfaces of Arbitrary Constant Negative Gaussian Curvature and Related Sine-Gordon Equations Paul Bracken Department of Mathematics, University of Texas, Edinburg,
More informationSolutions for Math 348 Assignment #4 1
Solutions for Math 348 Assignment #4 1 (1) Do the following: (a) Show that the intersection of two spheres S 1 = {(x, y, z) : (x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 = r 2 1} S 2 = {(x, y, z) : (x x 2 ) 2
More informationConstant ratio timelike curves in pseudo-galilean 3-space G 1 3
CREAT MATH INFORM 7 018, No 1, 57-6 Online version at http://creative-mathematicsubmro/ Print Edition: ISSN 1584-86X Online Edition: ISSN 1843-441X Constant ratio timelike curves in pseudo-galilean 3-space
More informationInelastic Admissible Curves in the Pseudo Galilean Space G 3
Int. J. Open Problems Compt. Math., Vol. 4, No. 3, September 2011 ISSN 1998-6262; Copyright ICSRS Publication, 2011 www.i-csrs.org Inelastic Admissible Curves in the Pseudo Galilean Space G 3 1 Alper Osman
More informationExistence Theorems for Timelike Ruled Surfaces in Minkowski 3-Space
Existence Theorems for Timelike Ruled Surfaces in Minkowski -Space Mehmet Önder Celal Bayar University, Faculty of Science and Arts, Department of Mathematics, Muradiye Campus, 45047 Muradiye, Manisa,
More informationSurfaces Family with Common Smarandache Geodesic Curve According to Bishop Frame in Euclidean Space
MATHEMATICAL SCIENCES AND APPLICATIONS E-NOTES 4 (1 164-174 (016 c MSAEN Surfaces Family with Common Smarandache Geodesic Curve According to Bishop Frame in Euclidean Space Gülnur Şaffak Atalay* and Emin
More informationGeometric approximation of curves and singularities of secant maps Ghosh, Sunayana
University of Groningen Geometric approximation of curves and singularities of secant maps Ghosh, Sunayana IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish
More informationDifferential geometry of transversal intersection spacelike curve of two spacelike surfaces in Lorentz-Minkowski 3-Space L 3
Differential geometry of transversal intersection spacelike curve of two spacelike surfaces in Lorentz-Minkowski 3-Space L 3 Osmar Aléssio Universidade Estadual Paulista Júlio de Mesquita Filho - UNESP
More informationSmarandache curves according to Sabban frame of fixed pole curve belonging to the Bertrand curves pair
Smarandache curves according to Sabban frame of fixed pole curve belonging to the Bertrand curves pair Süleyman Şenyurt, Yasin Altun, and Ceyda Cevahir Citation: AIP Conference Proceedings 76, 00045 06;
More informationSurfaces with Parallel Normalized Mean Curvature Vector Field in 4-Spaces
Surfaces with Parallel Normalized Mean Curvature Vector Field in 4-Spaces Georgi Ganchev, Velichka Milousheva Institute of Mathematics and Informatics Bulgarian Academy of Sciences XX Geometrical Seminar
More informationThe equiform differential geometry of curves in 4-dimensional galilean space G 4
Stud. Univ. Babeş-Bolyai Math. 582013, No. 3, 393 400 The equiform differential geometry of curves in 4-dimensional galilean space G 4 M. Evren Aydin and Mahmut Ergüt Abstract. In this paper, we establish
More informationOn a family of surfaces with common asymptotic curve in the Galilean space G 3
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 2016), 518 523 Research Article On a family of surfaces with common asymptotic curve in the Galilean space G 3 Zühal Küçükarslan Yüzbaşı Fırat
More informationSmarandache Curves In Terms of Sabban Frame of Fixed Pole Curve. Key Words: Smarandache Curves, Sabban Frame, Geodesic Curvature, Fixed Pole Curve
Bol. Soc. Paran. Mat. s. v. 4 06: 5 6. c SPM ISSN-75-88 on line ISSN-00787 in press SPM: www.spm.uem.br/bspm doi:0.569/bspm.v4i.75 Smarandache Curves In Terms of Sabban Frame of Fixed Pole Curve Süleyman
More informationTransversal Surfaces of Timelike Ruled Surfaces in Minkowski 3-Space
Transversal Surfaces of Timelike Ruled Surfaces in Minkowski -Space Mehmet Önder Celal Bayar University, Faculty of Science and Arts, Department of Mathematics, Muradiye Campus, 45047, Muradiye, Manisa,
More informationOn Rectifying Dual Space Curves
On Rectifying Dual Space Curves Ahmet YÜCESAN, NihatAYYILDIZ, anda.ceylançöken Süleyman Demirel University Department of Mathematics 32260 Isparta Turkey yucesan@fef.sdu.edu.tr ayyildiz@fef.sdu.edu.tr
More informationON THE PARALLEL SURFACES IN GALILEAN SPACE
ON THE PARALLEL SURFACES IN GALILEAN SPACE Mustafa Dede 1, Cumali Ekici 2 and A. Ceylan Çöken 3 1 Kilis 7 Aral k University, Department of Mathematics, 79000, Kilis-TURKEY 2 Eskişehir Osmangazi University,
More informationTHE FUNDAMENTAL THEOREM OF SPACE CURVES
THE FUNDAMENTAL THEOREM OF SPACE CURVES JOSHUA CRUZ Abstract. In this paper, we show that curves in R 3 can be uniquely generated by their curvature and torsion. By finding conditions that guarantee the
More informationTimelike Rotational Surfaces of Elliptic, Hyperbolic and Parabolic Types in Minkowski Space E 4 with Pointwise 1-Type Gauss Map
Filomat 29:3 (205), 38 392 DOI 0.2298/FIL50338B Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Timelike Rotational Surfaces of
More informationLECTURE 6: PSEUDOSPHERICAL SURFACES AND BÄCKLUND S THEOREM. 1. Line congruences
LECTURE 6: PSEUDOSPHERICAL SURFACES AND BÄCKLUND S THEOREM 1. Line congruences Let G 1 (E 3 ) denote the Grassmanian of lines in E 3. A line congruence in E 3 is an immersed surface L : U G 1 (E 3 ), where
More informationClassifications of Special Curves in the Three-Dimensional Lie Group
International Journal of Mathematical Analysis Vol. 10, 2016, no. 11, 503-514 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2016.6230 Classifications of Special Curves in the Three-Dimensional
More informationCHARACTERIZATIONS OF SPACE CURVES WITH 1-TYPE DARBOUX INSTANTANEOUS ROTATION VECTOR
Commun. Korean Math. Soc. 31 016), No., pp. 379 388 http://dx.doi.org/10.4134/ckms.016.31..379 CHARACTERIZATIONS OF SPACE CURVES WITH 1-TYPE DARBOUX INSTANTANEOUS ROTATION VECTOR Kadri Arslan, Hüseyin
More informationTHE NATURAL LIFT CURVES AND GEODESIC CURVATURES OF THE SPHERICAL INDICATRICES OF THE TIMELIKE BERTRAND CURVE COUPLE
International Electronic Journal of Geometry Volume 6 No.2 pp. 88 99 (213) c IEJG THE NATURAL LIFT CURVES AND GEODESIC CURVATURES OF THE SPHERICAL INDICATRICES OF THE TIMELIKE BERTRAND CURVE COUPLE SÜLEYMAN
More informationArbitrary-Speed Curves
Arbitrary-Speed Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 12, 2017 The Frenet formulas are valid only for unit-speed curves; they tell the rate of change of the orthonormal vectors T, N, B with respect
More informationN C Smarandache Curve of Bertrand Curves Pair According to Frenet Frame
International J.Math. Combin. Vol.1(016), 1-7 N C Smarandache Curve of Bertrand Curves Pair According to Frenet Frame Süleyman Şenyurt, Abdussamet Çalışkan and Ünzile Çelik (Faculty of Arts and Sciences,
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More informationSPLIT QUATERNIONS AND SPACELIKE CONSTANT SLOPE SURFACES IN MINKOWSKI 3-SPACE
INTERNATIONAL JOURNAL OF GEOMETRY Vol. (13), No. 1, 3-33 SPLIT QUATERNIONS AND SPACELIKE CONSTANT SLOPE SURFACES IN MINKOWSKI 3-SPACE MURAT BABAARSLAN AND YUSUF YAYLI Abstract. A spacelike surface in the
More informationON HELICES AND BERTRAND CURVES IN EUCLIDEAN 3-SPACE. Murat Babaarslan 1 and Yusuf Yayli 2
ON HELICES AND BERTRAND CURVES IN EUCLIDEAN 3-SPACE Murat Babaarslan 1 and Yusuf Yayli 1 Department of Mathematics, Faculty of Arts and Sciences Bozok University, Yozgat, Turkey murat.babaarslan@bozok.edu.tr
More informationCharacterizations of the Spacelike Curves in the 3-Dimentional Lightlike Cone
Prespacetime Journal June 2018 Volume 9 Issue 5 pp. 444-450 444 Characterizations of the Spacelike Curves in the 3-Dimentional Lightlike Cone Mehmet Bektas & Mihriban Kulahci 1 Department of Mathematics,
More informationHOMEWORK 2 SOLUTIONS
HOMEWORK SOLUTIONS MA11: ADVANCED CALCULUS, HILARY 17 (1) Find parametric equations for the tangent line of the graph of r(t) = (t, t + 1, /t) when t = 1. Solution: A point on this line is r(1) = (1,,
More informationNull Bertrand curves in Minkowski 3-space and their characterizations
Note di Matematica 23, n. 1, 2004, 7 13. Null Bertrand curves in Minkowski 3-space and their characterizations Handan Balgetir Department of Mathematics, Firat University, 23119 Elazig, TURKEY hbalgetir@firat.edu.tr
More informationKilling Magnetic Curves in Three Dimensional Isotropic Space
Prespacetime Journal December l 2016 Volume 7 Issue 15 pp. 2015 2022 2015 Killing Magnetic Curves in Three Dimensional Isotropic Space Alper O. Öğrenmiş1 Department of Mathematics, Faculty of Science,
More informationSome Characterizations of Partially Null Curves in Semi-Euclidean Space
International Mathematical Forum, 3, 28, no. 32, 1569-1574 Some Characterizations of Partially Null Curves in Semi-Euclidean Space Melih Turgut Dokuz Eylul University, Buca Educational Faculty Department
More informationOn T-slant, N-slant and B-slant Helices in Pseudo-Galilean Space G 1 3
Filomat :1 (018), 45 5 https://doiorg/1098/fil180145o Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat On T-slant, N-slant and B-slant
More informationRelatively normal-slant helices lying on a surface and their characterizations
Hacettepe Journal of Mathematics and Statistics Volume 46 3 017, 397 408 Relatively normal-slant helices lying on a surface and their characterizations Nesibe MAC T and Mustafa DÜLDÜL Abstract In this
More informationDarboux vector and stress analysis of Winternitz frame
NTMSCI 6, No. 4, 176-181 (018) 176 New Trends in Mathematical Sciences http://dx.doi.org/10.085/ntmsci.019.39 Darboux vector and stress analysis of Winternitz frame Yilmaz Tuncer 1 and Huseyin Kocayigit
More informationDierential Geometry Curves and surfaces Local properties Geometric foundations (critical for visual modeling and computing) Quantitative analysis Algo
Dierential Geometry Curves and surfaces Local properties Geometric foundations (critical for visual modeling and computing) Quantitative analysis Algorithm development Shape control and interrogation Curves
More informationAbstract. In this paper we give the Euler theorem and Dupin indicatrix for surfaces at a
MATEMATIQKI VESNIK 65, 2 (2013), 242 249 June 2013 originalni nauqni rad research paper THE EULER THEOREM AND DUPIN INDICATRIX FOR SURFACES AT A CONSTANT DISTANCE FROM EDGE OF REGRESSION ON A SURFACE IN
More informationDifferential-Geometrical Conditions Between Geodesic Curves and Ruled Surfaces in the Lorentz Space
Differential-Geometrical Conditions Between Geodesic Curves and Ruled Surfaces in the Lorentz Space Nihat Ayyildiz, A. Ceylan Çöken, Ahmet Yücesan Abstract In this paper, a system of differential equations
More informationA new characterization of curves on dual unit sphere
NTMSCI 2, No. 1, 71-76 (2017) 71 Journal of Abstract and Computational Mathematics http://www.ntmsci.com/jacm A new characterization of curves on dual unit sphere Ilim Kisi, Sezgin Buyukkutuk, Gunay Ozturk
More informationSELECTED SAMPLE FINAL EXAM SOLUTIONS - MATH 5378, SPRING 2013
SELECTED SAMPLE FINAL EXAM SOLUTIONS - MATH 5378, SPRING 03 Problem (). This problem is perhaps too hard for an actual exam, but very instructional, and simpler problems using these ideas will be on the
More informationInextensible Flows of Curves in Minkowskian Space
Adv. Studies Theor. Phys., Vol. 2, 28, no. 16, 761-768 Inextensible Flows of Curves in Minkowskian Space Dariush Latifi Department of Mathematics, Faculty of Science University of Mohaghegh Ardabili P.O.
More informationThe General Solutions of Frenet s System in the Equiform Geometry of the Galilean, Pseudo-Galilean, Simple Isotropic and Double Isotropic Space 1
International Mathematical Forum, Vol. 6, 2011, no. 17, 837-856 The General Solutions of Frenet s System in the Equiform Geometry of the Galilean, Pseudo-Galilean, Simple Isotropic and Double Isotropic
More informationInterpolated Rigid-Body Motions and Robotics
Interpolated Rigid-Body Motions and Robotics J.M. Selig London South Bank University and Yuanqing Wu Shanghai Jiaotong University. IROS Beijing 2006 p.1/22 Introduction Interpolation of rigid motions important
More informationOn the Dual Darboux Rotation Axis of the Timelike Dual Space Curve
On the Dual Darboux Rotation Axis of the Timelike Dual Space Curve Ahmet Yücesan, A. Ceylan Çöken and Nihat Ayyildiz Abstract In this paper, the Dual Darboux rotation axis for timelike dual space curve
More informationContents. 1. Introduction
FUNDAMENTAL THEOREM OF THE LOCAL THEORY OF CURVES KAIXIN WANG Abstract. In this expository paper, we present the fundamental theorem of the local theory of curves along with a detailed proof. We first
More informationarxiv: v1 [math.dg] 1 Oct 2018
ON SOME CURVES WITH MODIFIED ORTHOGONAL FRAME IN EUCLIDEAN 3-SPACE arxiv:181000557v1 [mathdg] 1 Oct 018 MOHAMD SALEEM LONE, HASAN ES, MURAT KEMAL KARACAN, AND BAHADDIN BUKCU Abstract In this paper, we
More informationTHE DARBOUX TRIHEDRONS OF REGULAR CURVES ON A REGULAR SURFACE
International lectronic Journal of eometry Volume 7 No 2 pp 61-71 (2014) c IJ TH DARBOUX TRIHDRONS OF RULAR CURVS ON A RULAR SURFAC MRAH TUNÇ AND MİN OZYILMAZ (Communicated by Levent KULA) Abstract In
More informationNatural Lifts and Curvatures, Arc-Lengths of the Spherical Indicatries of the Evolute Curve in E 3
International Mathematical Forum, Vol. 9, 214, no. 18, 857-869 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/imf.214.448 Natural Lifts and Curvatures, Arc-Lengths of the Spherical Indicatries
More informationInextensible Flows of Curves in Lie Groups
CJMS. 113, 3-3 Caspian Journal of Mathematical Sciences CJMS University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-611 Inextensible Flows of Curves in Lie Groups Gökmen Yıldız a and
More informationInvestigation of non-lightlike tubular surfaces with Darboux frame in Minkowski 3-space
CMMA 1, No. 2, 58-65 (2016) 58 Communication in Mathematical Modeling and Applications http://ntmsci.com/cmma Investigation of non-lightlike tubular surfaces with Darboux frame in Minkowski 3-space Emad
More informationA local characterization for constant curvature metrics in 2-dimensional Lorentz manifolds
A local characterization for constant curvature metrics in -dimensional Lorentz manifolds Ivo Terek Couto Alexandre Lymberopoulos August 9, 8 arxiv:65.7573v [math.dg] 4 May 6 Abstract In this paper we
More informationAn Optimal Control Problem for Rigid Body Motions in Minkowski Space
Applied Mathematical Sciences, Vol. 5, 011, no. 5, 559-569 An Optimal Control Problem for Rigid Body Motions in Minkowski Space Nemat Abazari Department of Mathematics, Ardabil Branch Islamic Azad University,
More informationThe Natural Lift of the Fixed Centrode of a Non-null Curve in Minkowski 3-Space
Malaya J Mat 4(3(016 338 348 The Natural Lift of the Fixed entrode of a Non-null urve in Minkowski 3-Space Mustafa Çalışkan a and Evren Ergün b a Faculty of Sciences epartment of Mathematics Gazi University
More information8. THE FARY-MILNOR THEOREM
Math 501 - Differential Geometry Herman Gluck Tuesday April 17, 2012 8. THE FARY-MILNOR THEOREM The curvature of a smooth curve in 3-space is 0 by definition, and its integral w.r.t. arc length, (s) ds,
More informationDetermination of the Position Vectors of Curves from Intrinsic Equations in G 3
Applied Mathematics Determination of the Position Vectors of Curves from Intrinsic Equations in G 3 Handan ÖZTEKIN * and Serpil TATLIPINAR Department of Mathematics, Firat University, Elazig, Turkey (
More informationGeometric Modelling Summer 2016
Geometric Modelling Summer 2016 Exercises Benjamin Karer M.Sc. http://gfx.uni-kl.de/~gm Benjamin Karer M.Sc. Geometric Modelling Summer 2016 1 Dierential Geometry Benjamin Karer M.Sc. Geometric Modelling
More informationUnit Speed Curves. Recall that a curve Α is said to be a unit speed curve if
Unit Speed Curves Recall that a curve Α is said to be a unit speed curve if The reason that we like unit speed curves that the parameter t is equal to arc length; i.e. the value of t tells us how far along
More informationD Tangent Surfaces of Timelike Biharmonic D Helices according to Darboux Frame on Non-degenerate Timelike Surfaces in the Lorentzian Heisenberg GroupH
Bol. Soc. Paran. Mat. (3s.) v. 32 1 (2014): 35 42. c SPM ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v32i1.19035 D Tangent Surfaces of Timelike Biharmonic D
More informationTangent and Normal Vectors
Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in
More informationarxiv: v1 [math.dg] 12 Jun 2015
arxiv:1506.03938v1 [math.dg] 1 Jun 015 NOTES ON W-DIRECTION CURVES IN EUCLIDEAN 3-SPACE İlkay Arslan Güven 1,, Semra Kaya Nurkan and İpek Ağaoğlu Tor 3 1,3 Department of Mathematics, Faculty of Arts and
More informationMath 32A Discussion Session Week 5 Notes November 7 and 9, 2017
Math 32A Discussion Session Week 5 Notes November 7 and 9, 2017 This week we want to talk about curvature and osculating circles. You might notice that these notes contain a lot of the same theory or proofs
More informationOn the solution of differential equation system characterizing curve pair of constant Breadth by the Lucas collocation approximation
NTMSCI, No 1, 168-183 16) 168 New Trends in Mathematical Sciences http://dxdoiorg/185/ntmsci1611585 On the solution of differential equation system characterizing curve pair of constant Breadth by the
More informationHamdy N. Abd-Ellah and Abdelrahim Khalifa Omran
Korean J. Math. 5 (017), No. 4, pp. 513 535 https://doi.org/10.11568/kjm.017.5.4.513 STUDY ON BCN AND BAN RULED SURFACES IN E 3 Hamdy N. Abd-Ellah and Abdelrahim Khalifa Omran Abstract. As a continuation
More informationarxiv: v1 [math.dg] 22 Aug 2015
arxiv:1508.05439v1 [math.dg] 22 Aug 2015 ON CHARACTERISTIC CURVES OF DEVELOPABLE SURFACES IN EUCLIDEAN 3-SPACE FATIH DOĞAN Abstract. We investigate the relationship among characteristic curves on developable
More informationLectures on Quantum sine-gordon Models
Lectures on Quantum sine-gordon Models Juan Mateos Guilarte 1, 1 Departamento de Física Fundamental (Universidad de Salamanca) IUFFyM (Universidad de Salamanca) Universidade Federal de Matto Grosso Cuiabá,
More informationCurves from the inside
MATH 2401 - Harrell Curves from the inside Lecture 5 Copyright 2008 by Evans M. Harrell II. Who in the cast of characters might show up on the test? Curves r(t), velocity v(t). Tangent and normal lines.
More informationOn the Inclined Curves in Galilean 4-Space
Applie Mathematical Sciences Vol. 7 2013 no. 44 2193-2199 HIKARI Lt www.m-hikari.com On the Incline Curves in Galilean 4-Space Dae Won Yoon Department of Mathematics Eucation an RINS Gyeongsang National
More informationOn the Differential Geometric Elements of Mannheim Darboux Ruled Surface in E 3
Applied Mathematical Sciences, Vol. 10, 016, no. 6, 3087-3094 HIKARI Ltd, www.m-hiari.com https://doi.org/10.1988/ams.016.671 On the Differential Geometric Elements of Mannheim Darboux Ruled Surface in
More informationWeek 3: Differential Geometry of Curves
Week 3: Differential Geometry of Curves Introduction We now know how to differentiate and integrate along curves. This week we explore some of the geometrical properties of curves that can be addressed
More informationDIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE
International Electronic Journal of Geometry Volume 7 No. 1 pp. 44-107 (014) c IEJG DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE RAFAEL LÓPEZ Dedicated to memory of Proffessor
More informationThe Frenet Serret formulas
The Frenet Serret formulas Attila Máté Brooklyn College of the City University of New York January 19, 2017 Contents Contents 1 1 The Frenet Serret frame of a space curve 1 2 The Frenet Serret formulas
More informationCURVATURE VIA THE DE SITTER S SPACE-TIME
SARAJEVO JOURNAL OF MATHEMATICS Vol.7 (9 (20, 9 0 CURVATURE VIA THE DE SITTER S SPACE-TIME GRACIELA MARÍA DESIDERI Abstract. We define the central curvature and the total central curvature of a closed
More information