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 37410-000, 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
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.
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
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
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 3. 2.1 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)
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
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 ( 3.71239, 17.2195, 11.0565) where u = 3.71238898038469 v = 1 are domain points of the surface X A r = 3.71119496890542 w = 3.71238898038469 are domain points of the surface X B. N A = { 1.00119, 0.0411205, 0.0264032}; N B = { 1.0012, 0.0264347, 0.0411697}; cosh θ = 1.00239; t = { 0.0345545, 0.213668, 0.977517}; k A n = 0; k B n = 0.00239373; c = {0.00119543, 0.0338267, 0.00735167}; k = 0.0345957; λ A n = 0.0000826152; λ B n = 0.000165823; c = { 0.0000828123, 0.00092071, 0.00142271}; n = {0.0345545, 0.977771, 0.212503}; b = { 1.00119, 0.0411205, 0.0264032}; τ = 0.00238802. [2] S. Izumiga, Timelike Hipersurfaces in the Sitter Space Lagendrian Singularities, Hokkaido University preprints series in Mathematics, Sapporo, Japan, 2005. [3] R. M. Kiehn, Falaco Solitons- Black-Holes in a Swiming Pool, http://www22.pair.com/csdc/pdf/zeromean.pdf [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, 2002. [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, 1983. [7] J. D. Struik, Lectures on Classical Differential Geometry, Addison Wesley, Cambridge, MA, 1950. [8] T. J. Willmore, An Introduction to Differential Geometry. Clarendon Press, Oxford, 1959. [9] J. Walrave, Curves surfaces in Minkowski space. Doctoral Thesis, K.U. Leuven. Fac. Science, Leuven, 1985. References [1] M. P. Carmo, Differential Geometry of Curves Surfaces, Prentice-Hall, Englewood Cliffs, NJ, 1976. [10] X. Ye T. Maekawa, Differential geometry of Intersection Curves of Two Surfaces, Computer Aided Geometric Desgin, 16, (1999), 767-788.