Differential geometry of transversal intersection spacelike curve of two spacelike surfaces in Lorentz-Minkowski 3-Space L 3
|
|
- Gerald Carr
- 5 years ago
- Views:
Transcription
1 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 Campus de Ilha Solteira Ilha Solteira, SP, Brasil osmar@mat.feis.unesp.br Irwen Valle Guadalupe Universidade Vale do Rio Verde de Três Corações- UNINCOR Av. Castelo Branco 82-Chácara das Rosas Cep , Três Corações, MG, Brasil irwenguadalupe@terra.com.br Recently Kiehn [3] showed the surprising result that Falaco solitons (which looks like wormholes structures in a swimming pool) can be represented as maximal surfaces in a 3- dimensional Lorentz-Minkowski space L 3. This suggests that study of spacelike surfaces in L 3 their intersections is a worth enterprise. Here we study some issues concerning the differential geometry of the surface-surface intersection curves in L 3 taking advantage of some recent advances in geometrical algorithms, representations algebraic computer methods which have been used in problems in different areas such as computer graphics visualization; computer vision image-based modelling; computer-aided design manufacturing; motion planning, kinematics, robotics, animation. These studies which use concepts from the theory of complex variables, quaternions, Clifford algebras, Lie groups; line spherical geometry, dual representations, Laguerre geometry the cyclographic map,, of course, in our case we need also to know the main properties of the geometry of curves surfaces in L 3 ) have furnished new insights, some exact solutions to problems that were formerly subject to approximations, unifying frameworks for seemingly disparate ideas, other unexpected benefits. The surface surface intersection (SSI), is a fundamental problem in computational geometry geometric modelling of complex shapes. For general parametric surface intersections, the most commonly used methods include subdivision marching. Marching-based algorithms begin by finding a starting point on a intersection curve, proceed to march along the curve. Most marching methods make use of the local differential geometry or Taylor series expansions around each point of the intersection curve in order to give a direction a control over each step in the procedure. Two types of surfaces, parametric implicit, are commonly used in geometric modelling systems. Those kinds of surfaces lead to three types of surface-surface intersection problems: parametric-parametric, implicit-implicit parametric-implicit. In general, what it is wanted in such problems is to determine the intersection curve between two given surfaces. To compute the intersection curve with precision efficiency, approaches of superior order are necessary, that is, it is necessary to obtain the geometric properties of the intersection curves. While the differential geometry of a parametric curve can be found in many textbooks such as, e.g., in (do Carmo [1], Struik [7]; Wilmore [8]), there is only a scarce literature on the differential geometry of intersection curves. Willmore describes how to obtain the unit tangent vector t, the unit principal normal vector n, the unit binormal vector b, as well as the curvature k the torsion τ of a intersection curve of two implicit surfaces. However, Ye
2 Maekawa [10] provides t, n, b, k, τ, algorithms for the evaluation of higher-order derivatives for transversal as well as tangential intersections for all three types of intersection problems. In the present paper we study the differential geometry of a transversal intersection spacelike curve resulting from the intersection of two parametric spacelike surfaces in Lorentz- Minkowski 3-space L 3, since we did not find any literature the literature dealing with this problem, which according to previous observation may be an important one also in Physics. The motivation comes from the fact that in Walrave s thesis ([9]), he studied the moving Frenet frames of curves in Minkowski space. The main difference between the study of curves surfaces in Euclidean space E 3 curves surfaces in Lorentz-Minkowski space L 3, is that the frame is not unique. As the local properties of curves are directly related with the Frenet frame Frenet-Serret equations, then we have indeed a good reason to analyze this issue. The reaming of the paper is organized as follows: In Section 1 introduces some notation definitions, reviews some aspects of the differential geometry in Lorentz-Minkowski 3-space L 3. In Section 2 we compute the curvature k (Proposition 2) the torsion τ (Proposition 3) of the transversal intersection spacelike curve resulting from the intersection of two parametric spacelike surfaces in L 3. In Section 3 we give some examples which we think illustrate the contents of Proposition 2, Proposition 3. 1 Review of differential geometry in Lorentz-Minkowski 3-space L 3. The Lorentz-Minkowsk 3-space L 3 is the real vector space R 3 provided with the Lorentz metric [6] given by v w = x 1 y 1 + x 2 y 2 + x 3 y 3 (1) where v = x 1 e 1 + x 2 e 2 + x 3 e 3 w = y 1 e 1 +y 2 e 2 +y 3 e 3 are any two vectors of L 3 {e 1, e 2, e 3 } is an oriented basis. Recall that a vector v 0 in L 3 can be a spacelike, a timelike or a null (lightlike), if respectively holds v v > 0, v v < 0 or v v = 0. In particular, the vector v = 0 is a spacelike. If v = (x 1, x 2, x 3 ) is in L 3 we define the norm of v by v = (v v) 1 2 = 2 x 1 x 1 + x 2 x 2 + x 3 x 3 (2) Two vectors u v in L 3 are said to be orthogonal if u v = 0. A vector u in L 3 which satisfies u u = ±1 is called a unit vector. A basis {v 1, v 2, v 3 } on L 3 is called an orthogonal basis if the vectors v i, i = 1, 2, 3 are mutually orthogonal unit vectors; specifically v i v j = 1 if i = j = 1 1 if i = j = 2, 3 0 if i j (3) Now, if {v 1, v 2, v 3 } is a orthogonal basis on L 3 such that v 1 v 1 < 0 v i v i > 0, i = 2, 3,then for each v in L 3, we have ([4]) v = v v 1 v 1 v 1 v 1 + v v 2 v 2 v 2 v 2 + v v 3 v 3 v 3 v 3 consequently, if {e 1, e 2, e 3 } is a orthonormal basis on L 3,we obtain v = (v e 1 )e 1 + (v e 2 )e 2 + (v e 3 )e 3 (4) Lemma 1 [5] If v is a timelike vector in L 3 u is orthogonal to v then u must be a spacelike vector. Let F be the set of all timelike vectors in L 3. For u in F, C(u) = {v F u v < 0} is the timecone of L 3 containing u. The opposite ttimecone is C( u) = C(u) = {v F u v > 0}. Since u is spacelike (Lemma1), F is the disjoint union of these two timecones. Lemma 2 [6]. Timelike vectors v w in L 3 are in the same timecone if only if v w < 0. Many features of inner product space have novel analogues in L 3. For e-xample, in an inner product space the Schwarz inequality permits the definition of the angle θ between v w as the unique number such that v w = v w cos θ. An analogous Lorentz- Minkowski result is a follows.
3 Proposition 1 [6]. Let v w be timelike vectors in L 3. Then v.w v w, with equality if only if v w are collinear. If v w are in the same timecone of L 3, there is a unique θ 0, called the hyperbolic angle between v w, such that v w = v w cosh(θ) If v w are not in the same timecone of L 3, there is a unique θ 0, called the hyperbolic angle between v w, such that v w = v w cosh(θ) We also recall that the vector product [2, 4] of u v (in that order) is the unique vector u v L 3 defined by u v = e 1 e 2 e 3 u 1 u 2 u 3 v 1 v 2 v 3 (5) where {e 1, e 2, e 3 } is the canonical basis of L 3 u = (u 1, u 2, u 3 ) v = (v 1, v 2, v 3 ). We can easily check that the triple scalar product of the three vectors u, v, w is given by w (u v) = det w 1 w 2 w 3 u 1 u 2 u 3 v 1 v 2 v 3 (6) where w = (w 1, w 2, w 3 ). Recall that the vector product is not associative, that moreover we have the following properties ( u y v y (u v) (x y) = det u x v x ) (7) where u, v, x, y are arbitrary vectors in L 3 (u v) w = (v w)u (u w)v; u, v, w L 3 (8) Remark 1. If u v are spacelike vectors of L 3 then u v is a timelike vector of L 3. We also recall that an arbitrary curve c = c(s) can locally be a spacelike, timelike or null (lightlike), if all of its velocity vectors c (s) are respectively spacelike, timelike or null. [6] A non-null curve c(s) is said to be parameterized by pseudo arclength parameter s, if hold c (s).c (s) = ±1. In this case, the curve c is said to be of unit speed. Let be a spacelike curve parametrized by arc length s. Therefore c is a spacelike unit vector, ie, c = 1, this implies that c c = c 2 = 1. Then c c = 0 (9) Depending of the vector c we consider the following three cases(see [9]): case 1. c c > 0. The number k(s) = c = c c is called the curvature de c at s. At points where k(s) 0, a unit vector n(s) in the direction c (s) is well defined by the equation c (s) = k(s)n(s) (10) From Eq. 9, we can see that c (s) is normal to c (s). Thus, n(s) is normal to c (s) it is an spacelike vector is called the normal vector at s. We shall denote by t(s) = c (s) the spacelike unit tangent vector of c at s. Thus from Eq. 10 we have The binormal vector t (s) = k(s)n(s) (11) b(s) = t(s) n(s) (12) is the unique timelike unit vector perpendicular to the spacelike(osculating) plane {t(s), n(s)} at every point c(s) of c such that {t, n, b} has the same orientation as L 3. Since b(s) is a unit vector, the length b (s) measures the rate of change of the neighboring osculating planes with the osculating plane at s. To compute b (s) we observe that, on the one h, b (s) is normal to b(s) that, on the other h, b (s) = t(s) n (s)+t (s) n(s) = t(s) n (s) (13) that is, b (s) is normal to t(s). It follows that b (s) is parallel to n(s) we may write b (s) = τ(s)n(s) (14) For some function (Warning: Many authors write - τ(s) instead of our τ(s) ). The number
4 τ (s) defined by Eq.( 14) is called the torsion of c at s. Let us summarize our position. To each value of the parameter s, we have associated three orthogonal unit vectors t(s), n(s), b(s). The trihedron thus formed is referred to as the Frenet trihedron at s. The derivatives t (s) = kn, b (s) = τn of the vectors t(s) b(s), when expressed in the basis {t, n, b}, yield geometrical entities (curvature k torsion τ ) which give us information about the behavior of c in a neighborhood of s. The search for other local geometrical entities would lead us to compute n (s). Using Eq(4) we have n = (n t)t + (n n)n (n b)b = kt + τb (15) we obtain again the curvature the torsion. For later use we shall call the equations the Frenet formulas. case 2. c c < 0. t = kn n = kt + τb b = τn (16) The normal vector n(s) is the unit timelike vector. The binormal vector b(s) is the unique spacelike unit vector perpendicular to the plane {t(s), n(s)} at every point c(s) of c such that {t, n, b} has the same orientation as L 3. The Frenet formulas are case 3. c c = 0. t = kn n = kt + τb b = τn (17) To rule out straight lines points of inflexion on c, we shall suppose that c 0. The normal vector n(s) is then the vector c (s). The binormal vector b(s) is the unique null vector perpendicular to t(s) at every point c(s) of c, such that n b = 1. The Frenet formulas are t = kn n = τn b = kt τb (18) where the curvature k can only take two values; 0 when c is a straight line, or 1 in all other cases. If c is a straight line, then c (s) = 0 = t which means that k = 0. If c is not a straight line, then exist an interval I on which c 0, n(s) is defined as n(s) = t (s), thus k = 1.{t, n, b} is a pseudo-orthonormal basis in L 3, which means that n = a 1 t+a 2 n+a 3 b, b = b 1 t+b 2 n+b 3 b. From n n = n t = b b = 0 we get respectively that a 1 = a 3 = b 2 = 0. Considering that n b = 1 t b = 0 we get by differentiation that n b + n b = 0 b t + b t = 0 which means that a 2 = b 3 b 1 = k = 1 Concluding, we see that in this case there is only one curvature a 2 = τ Now let us evaluate the third derivative c (s). By differentiating equation c = t = kn in the three cases,we obtain c (s) = k n + kn (19) where we can replace n by the second equation of the Frenet formulas. We have case1 : c (s) = k 2 t + k n + kτb (20) case2 : c (s) = k 2 t + k n + kτb (21) case3 : c (s) = τn (22) The torsion can be obtained from Eq.(20), Eq.(21) Eq.(22) as
5 case1 : τ = b c k case2 : τ = b c k (23) (24) case3 : τ = b c (25) Recall that an arbitrary plane in L 3 is spacelike if the induced metric is Riemannian. We also recall that an arbitrary regular surface X = X(u, v) is called a spacelike surface if the tangent plane at any point is spacelike this case the vectors X u X v are spacelike vectors of the tangent plane, then we have that (X u X v ) X u = (X u X v ) X v = 0, therefore X u X v is a timelike vector. The surface normal vector is perpendicular to the tangent plane hence at any point the unit normal vector is given by N = Xu Xv Xu Xv therefore N is a timelike unit vector of L 3. (26) 2 Curvature torsion of transversal intersection spacelike curve of two spacelike surfaces in L 3. In this section we compute the curvature k torsion τ of transversal intersection spacelike curvature of two spacelike surfaces in L Transversal intersection spacelike curve. Let X A = X A (u, v) X B = X B (u, v) be the two spacelike parametric surfaces. Let c = c(s) be the transversal intersection spacelike curve of both surfaces X A X B. This means that the spacelike tangent vector of the transversal intersection spacelike curve c lies on the tangent planes of both surfaces. Therefore it can be obtained as the cross product of the unit surface normal vectors of the surfaces at p = c(s) T = NA N B N A N B (27) where N A N B are the timelike unit normal vectors to spacelike surfaces X A X B, respectively. 2.2 Curvature of transversal intersection spacelike curve. The curvature vector c of the transversal intersection spacelike curve at p, being perpendicular to T, must lie in the normal plane spanned by N A N B. Thus we can express it as c = αn A + βn B (28) where α β are the coefficients that we need to determine. We know that normal curvature at p in direction T is the projection of the curvature vector c = kn onto the timelike unit surface normal vector N at p, therefore by projecting Eq.(28) onto the timelike normals of both surfaces we have k A n = α βcosh(θ) k B n = αcosh(θ) β (29) if N A N B are in the same timecone of L 3 k A n = α + βcosh(θ) k B n = αcosh(θ) β (30) if N A N B are not in the same timecone of L 3. Where θ is the angle between the timelike unit normals vectors N A N B is evaluated by cosh(θ) = N A N B, if N A N B are in the same timecone of L 3 cosh(θ) = N A N B, if N A N B are not in the same timecone of L 3. We have the following Proposition 2 Suppose that c = c(s) is a transversal intersection spacelike curve of two spacelike surfaces X A X B c spacelike or timelike vector. Then the curvature k of the curve c is given by k 2 = (ka n ) 2 + (k B n ) 2 ± 2k A n k B n cosh(θ) (31)
6 Proof. Solving the coefficients α β from linear systems Eq.(29) Eq.(30), we have α = ka n ± kn B cosh(θ) sinh 2, β = kb n ± kn A cosh(θ) (θ) (32) Substituting Eq.(32) in Eq.(28) we have c = ka n ± kn B cosh(θ) sinh 2 N A + kb n ± kn A cosh(θ) (θ) sinh 2 N B (θ) (33) Now using the same ideas that Ye Maekawa [10] we can evaluate the two normal curvatures kn A kn B at p therefore we obtain the curvature vector from Eq.(33). Consequently, the curvature of the intersection spacelike curve c at p can be calculated using Eq.(10) Eq.(33) as follows. k 2 = ( ) kn A 2 ( ) + k B 2 n ± 2k A n kn B cosh(θ) λ A n = γ + δ cosh(θ), λ B n = γ cosh(θ) δ. (38) if N A N B are not in the same timecone of L 3. We have the following Proposition 3 Suppose that c = c(s) is a transversal spacelike curve of two spacelike surfaces X A X B, if c is spacelike, timelike or null vector. Then the torsion of the curve c is given by τ = case 1: 1 k [(λ A ) n ± λb n cosh(θ) b N A + (λ B ) n ± λa n cosh(θ) b N B] (39) case 2: 2.3 Torsion of transversal intersection spacelike curve. Since the timelike unit normal vectors N A N B lie on the normal plane, the term k n + kτb in Eq.(20) Eq.(21) the term τn in Eq.(22) can be replaced by γn A + δn B. Thus 1 k case 3: 1 [(λ A ) n ± λb n cosh(θ) b N A + (λ B ) n ± λa n cosh(θ) b N B] (40) [(λ A ) n ± λb n cosh(θ) b N A + (λ B ) n ± λa n cosh(θ) b N B] (41) case1 : c (s) = k 2 t + γn A + δn B. (34) case2 : c (s) = k 2 t + γn A + δn B. (35) case3 : c (s) = γn A + δn B. (36) Now, if we project c (s) onto the timelike unit surface normal N at p denote by λ n, we obtain λ A n = γ δ cosh(θ), λ B n = γ cosh(θ) δ. (37) if N A N B are in the same timecone of L 3 where the binormal vector b is evaluated in the three cases the curvature k is evaluated by Eq.(31). Proof. Solving the coefficients from linear system Eqs.(37) (38), we have γ = λa n ± λ B n cosh θ sinh 2, δ = λb n ± λ A n cosh θ θ (42) substituting in Eqs.(34), (35), (36), we have case1: c = κ 2 t+ λa n ± λ B n cosh θ N A + λa n ± λ B n cosh θ (43) N B
7 case2: c = κ 2 t+ λa n ± λ B n cosh θ case3: c = λa n ± λ B n cosh θ N A + λa n ± λ B n cosh θ sinh 2 N B θ (44) N A + λa n ± λ B n cosh θ sinh 2 N B θ (45) 3 Example Figure 1: Intersection X A (u, v) X B (r, w) For illustrative proposition[3], we present the example. Example 1 The parametric surface X A is a Catenoid given by X A (u, v) = (u, sinh(u)sin(v), sinh(u)cos(v)) parametric surface X B is Helicoid given by X B (r, w) = ( w, cosh(r)cos(w), cosh(r)sin(w)). Point of the intersection curve is P ( , , ) where u = v = 1 are domain points of the surface X A r = w = are domain points of the surface X B. N A = { , , }; N B = { , , }; cosh θ = ; t = { , , }; k A n = 0; k B n = ; c = { , , }; k = ; λ A n = ; λ B n = ; c = { , , }; n = { , , }; b = { , , }; τ = [2] S. Izumiga, Timelike Hipersurfaces in the Sitter Space Lagendrian Singularities, Hokkaido University preprints series in Mathematics, Sapporo, Japan, [3] R. M. Kiehn, Falaco Solitons- Black-Holes in a Swiming Pool, [4] C. M. C. Lopes, Superfícies de Tipo Espaço com Vetor Curvatura Média Nulo em L 3 e L 4, Master thesis,, IME Universidade de São Paulo, [5] Naber, G. L. (1988), Spacetime Singularities An Introduction, London Mathematical Society, Students texts 11. [6] B. O Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, [7] J. D. Struik, Lectures on Classical Differential Geometry, Addison Wesley, Cambridge, MA, [8] T. J. Willmore, An Introduction to Differential Geometry. Clarendon Press, Oxford, [9] J. Walrave, Curves surfaces in Minkowski space. Doctoral Thesis, K.U. Leuven. Fac. Science, Leuven, References [1] M. P. Carmo, Differential Geometry of Curves Surfaces, Prentice-Hall, Englewood Cliffs, NJ, [10] X. Ye T. Maekawa, Differential geometry of Intersection Curves of Two Surfaces, Computer Aided Geometric Desgin, 16, (1999),
Investigation 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 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 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 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 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 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 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 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 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 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 informationOn Natural Lift of a Curve
Pure Mathematical Sciences, Vol. 1, 2012, no. 2, 81-85 On Natural Lift of a Curve Evren ERGÜN Ondokuz Mayıs University, Faculty of Arts and Sciences Department of Mathematics, Samsun, Turkey eergun@omu.edu.tr
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 informationarxiv:gr-qc/ v1 31 Dec 2005
On the differential geometry of curves in Minkowski space arxiv:gr-qc/0601002v1 31 Dec 2005 J. B. Formiga and C. Romero Departamento de Física, Universidade Federal da Paraíba, C.Postal 5008, 58051-970
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 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 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 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 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 informationOn the curvatures of spacelike circular surfaces
Kuwait J. Sci. 43 (3) pp. 50-58, 2016 Rashad A. Abdel-Baky 1,2, Yasin Ünlütürk 3,* 1 Present address: Dept. of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, P.O. Box 126300, Jeddah
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 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 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 informationBÄCKLUND TRANSFORMATIONS ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE
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, 64200 Usak,
More informationFOCAL SET OF CURVES IN THE MINKOWSKI SPACE
FOCAL SET OF CURVES IN THE MINKOWSKI SPACE A. C. NABARRO AND A. J. SACRAMENTO Abstract. We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the
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 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 informationON THE CURVATURE THEORY OF NON-NULL CYLINDRICAL SURFACES IN MINKOWSKI 3-SPACE
TWMS J. App. Eng. Math. V.6, N.1, 2016, pp. 22-29. ON THE CURVATURE THEORY OF NON-NULL CYLINDRICAL SURFACES IN MINKOWSKI 3-SPACE BURAK SAHINER 1, MUSTAFA KAZAZ 1, HASAN HUSEYIN UGURLU 3, Abstract. This
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 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 informationON 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 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 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 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 informationHoro-tight immersions of S 1
CADERNOS DE MATEMÁTICA 06, 129 134 May (2005) ARTIGO NÚMERO SMA#226 Horo-tight immersions of S 1 Marcelo Buosi * Faculdades Federais Integradas de Diamantina, Rua da Glória 187, 39100-000, Diamantina,
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 Geometric Applications of Timelike Quaternions
Some Geometric Applications of Timelike Quaternions M. Özdemir, A.A. Ergin Department of Mathematics, Akdeniz University, 07058-Antalya, Turkey mozdemir@akdeniz.edu.tr, aaergin@akdeniz.edu.tr Abstract
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 informationSpecial Curves and Ruled Surfaces
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 203-212. Special Curves and Ruled Surfaces Dedicated to Professor Koichi Ogiue on his sixtieth birthday
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 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 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 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 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 informationCubic Helices in Minkowski Space
Cubic Helices in Minkowski Space Jiří Kosinka and Bert Jüttler Johannes Kepler University, Institute of Applied Geometry, Altenberger Str. 69, A 4040 Linz, Austria Abstract We discuss space like and light
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 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 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 informationTHE NATURAL LIFT CURVE OF THE SPHERICAL INDICATRIX OF A TIMELIKE CURVE IN MINKOWSKI 4-SPACE
Journal of Science Arts Year 5, o (, pp 5-, 5 ORIGIAL PAPER HE AURAL LIF CURVE OF HE SPHERICAL IDICARIX OF A IMELIKE CURVE I MIKOWSKI -SPACE EVRE ERGÜ Manuscript received: 65; Accepted paper: 55; Published
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 informationLinear Ordinary Differential Equations
MTH.B402; Sect. 1 20180703) 2 Linear Ordinary Differential Equations Preliminaries: Matrix Norms. Denote by M n R) the set of n n matrix with real components, which can be identified the vector space R
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 informationG 1 Hermite Interpolation by Minkowski Pythagorean Hodograph Cubics
G 1 Hermite Interpolation by Minkowski Pythagorean Hodograph Cubics Jiří Kosinka and Bert Jüttler Johannes Kepler University, Institute of Applied Geometry, Altenberger Str. 69, A 4040 Linz, Austria Abstract
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 informationON THE DETERMINATION OF A DEVELOPABLE TIMELIKE RULED SURFACE. Mustafa KAZAZ, Ali ÖZDEMİR, Tuba GÜROĞLU
SDÜ FEN EDEBİYAT FAKÜLTESİ FEN DERGİSİ (E-DERGİ). 008, (), 7-79 ON THE DETERMINATION OF A DEVELOPABLE TIMELIKE RULED SURFACE Mustafa KAZAZ, Ali ÖZDEMİR, Tuba GÜROĞLU Department of Mathematics, Faculty
More informationTRANSVERSAL SURFACES OF TIMELIKE RULED SURFACES IN MINKOWSKI 3-SPACE IR
IJRRAS () November 0 wwwarpapresscom/volumes/volissue/ijrras 08pf TRANSVERSAL SURFACES OF TIMELIKE RULED SURFACES IN MINKOWSKI -SPACE Mehmet Öner Celal Bayar University, Faculty of Science an Arts, Department
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 informationMinkowski spacetime. Pham A. Quang. Abstract: In this talk we review the Minkowski spacetime which is the spacetime of Special Relativity.
Minkowski spacetime Pham A. Quang Abstract: In this talk we review the Minkowski spacetime which is the spacetime of Special Relativity. Contents 1 Introduction 1 2 Minkowski spacetime 2 3 Lorentz transformations
More informatione 2 = e 1 = e 3 = v 1 (v 2 v 3 ) = det(v 1, v 2, v 3 ).
3. Frames In 3D space, a sequence of 3 linearly independent vectors v 1, v 2, v 3 is called a frame, since it gives a coordinate system (a frame of reference). Any vector v can be written as a linear combination
More informationCharacterizations of a helix in the pseudo - Galilean space G
International Journal of the Phsical ciences Vol 59), pp 48-44, 8 August, 00 Available online at http://wwwacademicjournalsorg/ijp IN 99-950 00 Academic Journals Full Length Research Paper Characterizations
More informationEikonal slant helices and eikonal Darboux helices in 3-dimensional pseudo-riemannian manifolds
Eikonal slant helices and eikonal Darboux helices in -dimensional pseudo-riemannian maniolds Mehmet Önder a, Evren Zıplar b a Celal Bayar University, Faculty o Arts and Sciences, Department o Mathematics,
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 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 informationHouston Journal of Mathematics c 2010 University of Houston Volume 36, No. 3, 2010
Houston Journal of Mathematics c 2010 University of Houston Volume 36, No. 3, 2010 SINGULARITIES OF GENERIC LIGHTCONE GAUSS MAPS AND LIGHTCONE PEDAL SURFACES OF SPACELIKE CURVES IN MINKOWSKI 4-SPACE LINGLING
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 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 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 informationOn the Fundamental Forms of the B-scroll with Null Directrix and Cartan Frame in Minkowskian 3-Space
Applied Mathematical Sciences, Vol. 9, 015, no. 80, 3957-3965 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.015.5330 On the Fundamental Forms of the B-scroll with Null Directrix and Cartan
More informationOn constant isotropic submanifold by generalized null cubic
On constant isotropic submanifold by generalized null cubic Leyla Onat Abstract. In this paper we shall be concerned with curves in an Lorentzian submanifold M 1, and give a characterization of each constant
More informationTIMELIKE BIHARMONIC CURVES ACCORDING TO FLAT METRIC IN LORENTZIAN HEISENBERG GROUP HEIS 3. Talat Korpinar, Essin Turhan, Iqbal H.
Acta Universitatis Apulensis ISSN: 1582-5329 No. 29/2012 pp. 227-234 TIMELIKE BIHARMONIC CURVES ACCORDING TO FLAT METRIC IN LORENTZIAN HEISENBERG GROUP HEIS 3 Talat Korpinar, Essin Turhan, Iqbal H. Jebril
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 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 informationM -geodesic in [5]. The anologue of the theorem of Sivridağ and Çalışkan was given in Minkowski 3-space by Ergün
Scholars Journal of Phsics Mathematics Statistics Sch. J. Phs. Math. Stat. 5; ():- Scholars Academic Scientific Publishers (SAS Publishers) (An International Publisher for Academic Scientific Resources)
More informationON THE SCALAR AND DUAL FORMULATIONS OF THE CURVATURE THEORY OF LINE TRAJECTORIES IN THE LORENTZIAN SPACE. 1. Introduction
J. Korean Math. Soc. 43 (2006), No. 6, pp. 1339 1355 ON THE SCALAR AND DUAL FORMULATIONS OF THE CURVATURE THEORY OF LINE TRAJECTORIES IN THE LORENTZIAN SPACE N ihat Ayyıldız and Ahmet Yücesan Abstract.
More informationOn the Invariants of Mannheim Offsets of Timelike Ruled Surfaces with Timelike Rulings
Gen Math Notes, Vol, No, June 04, pp 0- ISSN 9-784; Copyright ICSRS Publication, 04 wwwi-csrsorg Available free online at http://wwwgemanin On the Invariants of Mannheim Offsets of Timelike Ruled Surfaces
More informationMath 302 Outcome Statements Winter 2013
Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a
More informationResearch Article Normal and Osculating Planes of Δ-Regular Curves
Abstract and Applied Analysis Volume 2010, Article ID 923916, 8 pages doi:10.1155/2010/923916 Research Article Normal and Osculating Planes of Δ-Regular Curves Sibel Paşalı Atmaca Matematik Bölümü, Fen-Edebiyat
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 informationSpacelike Salkowski and anti-salkowski Curves With a Spacelike Principal Normal in Minkowski 3-Space
Int. J. Open Problem Compt. Math., Vol., No. 3, September 009 ISSN 998-66; Copyright c ICSRS Publication, 009 www.i-cr.org Spacelike Salkowki and anti-salkowki Curve With a Spacelike Principal Normal in
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 informationSpacetime and 4 vectors
Spacetime and 4 vectors 1 Minkowski space = 4 dimensional spacetime Euclidean 4 space Each point in Minkowski space is an event. In SR, Minkowski space is an absolute structure (like space in Newtonian
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 informationUNIVERSITY OF DUBLIN
UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429
More informationDual Smarandache Curves of a Timelike Curve lying on Unit dual Lorentzian Sphere
MATHEMATICAL SCIENCES AND APPLICATIONS E-NOTES 4 () -3 (06) c MSAEN Dual Smarandache Curves of a Timelike Curve lying on Unit dual Lorentzian Sphere Tanju Kahraman* and Hasan Hüseyin Uğurlu (Communicated
More informationThere is a function, the arc length function s(t) defined by s(t) = It follows that r(t) = p ( s(t) )
MATH 20550 Acceleration, Curvature and Related Topics Fall 2016 The goal of these notes is to show how to compute curvature and torsion from a more or less arbitrary parametrization of a curve. We will
More informationQing-Ming Cheng and Young Jin Suh
J. Korean Math. Soc. 43 (2006), No. 1, pp. 147 157 MAXIMAL SPACE-LIKE HYPERSURFACES IN H 4 1 ( 1) WITH ZERO GAUSS-KRONECKER CURVATURE Qing-Ming Cheng and Young Jin Suh Abstract. In this paper, we study
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 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 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 informationHow big is the family of stationary null scrolls?
How big is the family of stationary null scrolls? Manuel Barros 1 and Angel Ferrández 2 1 Departamento de Geometría y Topología, Facultad de Ciencias Universidad de Granada, 1807 Granada, Spain. E-mail
More informationIndex. Bertrand mate, 89 bijection, 48 bitangent, 69 Bolyai, 339 Bonnet s Formula, 283 bounded, 48
Index acceleration, 14, 76, 355 centripetal, 27 tangential, 27 algebraic geometry, vii analytic, 44 angle at a corner, 21 on a regular surface, 170 angle excess, 337 angle of parallelism, 344 angular velocity,
More informationLORENTZIAN MATRIX MULTIPLICATION AND THE MOTIONS ON LORENTZIAN PLANE. Kırıkkale University, Turkey
GLASNIK MATEMATIČKI Vol. 41(61)(2006), 329 334 LORENTZIAN MATRIX MULTIPLICATION AND THE MOTIONS ON LORENTZIAN PLANE Halı t Gündoğan and Osman Keçı lı oğlu Kırıkkale University, Turkey Abstract. In this
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 informationThe spacetime of special relativity
1 The spacetime of special relativity We begin our discussion of the relativistic theory of gravity by reviewing some basic notions underlying the Newtonian and special-relativistic viewpoints of space
More informationM435: INTRODUCTION TO DIFFERENTIAL GEOMETRY
M435: INTRODUCTION TO DIFFERENTIAL GEOMETRY MARK POWELL Contents 1. Introduction 1 2. Geometry of Curves 2 2.1. Tangent vectors and arc length 3 2.2. Curvature of plane curves 5 2.3. Curvature of curves
More informationTHE CHARACTERIZATIONS OF GENERAL HELICES IN THE 3-DIMEMSIONAL PSEUDO-GALILEAN SPACE
SOOCHOW JOURNAL OF MATHEMATICS Volume 31, No. 3, pp. 441-447, July 2005 THE CHARACTERIZATIONS OF GENERAL HELICES IN THE 3-DIMEMSIONAL PSEUDO-GALILEAN SPACE BY MEHMET BEKTAŞ Abstract. T. Ikawa obtained
More informationINTRODUCTION TO GEOMETRY
INTRODUCTION TO GEOMETRY ERIKA DUNN-WEISS Abstract. This paper is an introduction to Riemannian and semi-riemannian manifolds of constant sectional curvature. We will introduce the concepts of moving frames,
More informationAngle contraction between geodesics
Angle contraction between geodesics arxiv:0902.0315v1 [math.ds] 2 Feb 2009 Nikolai A. Krylov and Edwin L. Rogers Abstract We consider here a generalization of a well known discrete dynamical system produced
More informationTRIGONOMETRY IN EXTENDED HYPERBOLIC SPACE AND EXTENDED DE SITTER SPACE
Bull. Korean Math. Soc. 46 (009), No. 6, pp. 1099 1133 DOI 10.4134/BKMS.009.46.6.1099 TRIGONOMETRY IN EXTENDED HYPERBOLIC SPACE AND EXTENDED DE SITTER SPACE Yunhi Cho Abstract. We study the hyperbolic
More informationClass Meeting # 12: Kirchhoff s Formula and Minkowskian Geometry
MATH 8.52 COURSE NOTES - CLASS MEETING # 2 8.52 Introduction to PDEs, Spring 207 Professor: Jared Speck Class Meeting # 2: Kirchhoff s Formula and Minkowskian Geometry. Kirchhoff s Formula We are now ready
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 information