Euler - Savary s Formula on Minkowski Geometry

Transcription

1 Euler - Saary s Formula on Minkowski Geometry T. Ikawa Dedicated to the Memory of Grigorios TSAGAS ( ) President of Balkan Society of Geometers ( ) Abstract We consider a base cure a rolling cure and a roulette on Minkowski plane and gie the relation between the curatures of these three cures. This formula is a generalization of the Euler - Saary s formula of Euclidean plane. Mathematics Subject Classifications: 53A35 53B30 Key words: base cure curature Euler - Saary s formula rolling cure roulette. Introduction On the Euclidean plane E we consider two cures c B and c R. Let P be a point relatie to c R. When c R rolles without splitting along c B the locus of the point P makes a cure say c L. On this set of cures c B c R c L are called the base cure rolling cure and roulette respectiely. For example if c B is a straight line c R is a quadratic cure and P is a focus of c R then c L is the Delaunay cure that are used to study surfaces of reolution with the constant mean curature. Since this rolling situation makes up three cures it is natural to ask questions: what is the relation between the curatures of these cures when gien two cures can we find the third one? Many geometers studied these questions and generalized the situation [3]. Today the relation of the curatures is called as the Euler - Saary s formula. Howeer the rolling situation on the Minkowski geometry is not studied yet. Only the Delaunay cure is considered to study surfaces of reolution with the constant mean curature []. The purpose of this paper is to gie answers to the aboementioned general questions on the Minkowski geometry. After the preliminaries of section in section 3 we consider the associated cure that is the key concept to study the roulette for the roulette is one of associated cures of the base cure. Section 4 is deoted to gie the Euler - Saary s formula on the Minkowski plane. In the final section we determine the third cure from other two. Balkan Journal of Geometry and Its Applications Vol.8 No. 003 pp c Balkan Society of Geometers Geometry Balkan Press 003.

2 3 T. Ikawa Preliminaries Let L be the Minkowski plane with metric g = (+ ). A ector X of L is said to be spacelike if g(x X) > 0 or X = 0 timelike if g(x X) < 0 and null if g(x X) = 0 and X 0. A cure c is a smooth mapping c : I L from an open interal I into L. Let t be a parameter of c. By c(t) = (x(t) y(t)) we denote the orthogonal coordinate representation of c(t). The ector field dc ( dx dt = dt dy ) =: X is called the tangent dt ector field of the cure c(t). If the tangent ector field X of c(t) is a spacelike timelike or null then the cure c(t) is called spacelike timelike or null respectiely. In the rest of this paper we mostly consider non-null cures. When the tangent ector field X is non-null we can hae the arc length parameter s and hae the Frenet formula (.) dx ds = ky dy ds = kx where k is the curature of c(s) (cf. []). The ector field Y is called the normal ector field of the cure c(s). Remark that we hae the same representation of the Frenet formula regardless of whether the cure is spacelike or timelike. If φ(s) is the slope angle of the cure then we hae dφ ds = k. 3 Associated cure In this section we gie general formulas of the associated cure. Let c(s) be a non-null cure with the arc length parameter s and {X Y } the Frenet frame of c(s). If we put (3.) c A = c(s) + u (s)x + u (s)y then c A (s) generally makes a cure. This cure is called the associated cure of c(s). Remark that {u (s) u (s)} is a relatie coordinate of c A (s) with respect to {c(s) X Y }. If we put dc A ds = δu ds X + δu ds Y then since dc A ds = dc ds + du ds X +u dx ds + du ( ds Y +u dy ds = + du ) ( ds + ku X + ku + du ) ds Y by irtue of (.) we hae (3.) δu ds = du ds + ku + δu ds = du ds + ku. Let s A be the arc length parameter of c A. Then from

3 Euler - Saary s Formula on Minkowski Geometry 33 dc A ds = dc A ds = X + Y := du ds + ku + := du ds + ku the Frenet frame {Z W } of c A has following equations; (3.3) dz = k A W dw = k A Z where k A is the curature of c A. Let θ (resp. ω) be the slope angle of c (resp. c A ). Then k A = dω = dω ( ds = k + dφ ) (3.4) ds where φ = ω θ. If c A is space-like then we can put cosh φ = sinh φ =. Since ( ) dφ ds = d cosh ds (3.4) reduces to k A = ( k + ) where dash represents the deriatie with respect to s. If c A is time-like since sinh φ = we hae ( k A = k + ) 4 Euler - Saary s formula In this section we consider the roulette and gie the Euler - Saary s formula. Let c B (resp. c R ) be the base (resp. rolling) cure and k B (resp. k R ) the curature of c B (resp. c R ). Let P be a point relatie to c R. By c L we denote the roulette of the locus of P. We can consider that c L is an associated cure of c B then the relatie coordinate {x y} of c L with respect to c B satisfies

4 34 T. Ikawa (4.) = dx + k B y + δy = dy + k B x by irtue of (3.). Since c R rolles without splitting along c B at each point of contact we can consider {x y} is a relatie coordinate of c L with respect to c R for a suitable parameter s R. In this case the associated cure is reduced to a point P. Hence it follows that (4.) (4.3) so (4.4) = dx + k R y + = 0 = dx + k R y = 0. Substituting these equations into (4.) we hae = (k B k R )y δy = x y. δy = (k B k R )x Proposition 4. Let c R rolles without splitting along c B from the starting time t = 0. Then at each time t = t 0 of this motion the normal at the point c L (t 0 ) passes through the point of contact c B (t 0 ) = c R (t 0 ). Suppose that c L is spacelike. Then from (4.3) (4.5) ( 0 < ) ( ) δy = (k B k R ) (y x ). Hence we can put Differentiating these equations we hae x = sinh φ y = cosh φ. dx = dr sinh φ + r cosh φ dφ = k R r cosh φ dy = dr cosh φ + r sinh φ dφ = k R r sinh φ by irtue of (4.). From these equations it follows that r dφ = cosh φ k r r. Therefore substituting this equation into (3.4) we hae rk L = ± cosh φ r k B k R.

5 Euler - Saary s Formula on Minkowski Geometry 35 If c L is timelike by similar calculation we hae rk L = ± + sinh φ r k B k R. We can easily see that the case c L is null makes a contradiction. Theorem 4. On the Minkowski plane L suppose that a cure c R rolles without splitting along a cure c B. Let c L be a locus of a point P that is relatie to c R. Let Q be a point on c L and R a point of contact of c B and c R corresponds to Q relatie to the rolling relation. By (r φ) we denote a polar coordinate of Q with respect to the origin R and the base line c B R. Then curatures k B k R and k L of c B c R and c L respectiely satisfies rk L = ± rk L = ± + cosh φ r k B k R sinh φ r k B k R 5 Determining the cure (when c L (when c L is space like) is time like). Since the roulette is a locus of a point it is determined by the base cure and the rolling cure. In theis section we consider the conerse problem. First suppose that a base cure c B and a roulette c L is gien. Let (x(s B ) y(s B )) be the orthogonal coordinates of the base cure c B with the arc length parameter s B. For a point Q of c B draw the normal to the roulette c L. Let R be the foot of this normal with the orthogonal coordinate (f(s B ) g(s B )). Then the length of QR is (5.) QR = (f(s B ) x(s B )) (g(s B ) y(s B )). If we consider (5.) on the rolling cure c R this equation represents the length of the point P relatie to c R and a point of c R. Hence the orthogonal coordinate (u(s B ) (s B )) of c R is gien by the equations u(s B ) (s B ) = (f(s B ) x(s B )) (g(s B ) y(s B )) ( du ) ( ) d = ± the sign of ± depends on spacelike or timelike of c R. Next suppose that a rolling cure c R and a roulette c L is gien. Let (x(s L ) y(s L )) be the orthogonal coordinate of c L with arclength parameter s L. Suppose that the polar coordinate r(s R ) of c R is gien by the arc length parameter s R of c R. ( dy Since the normal of c L is dx ) a point (u ) of the base cure c B is gien ds L ds L by

6 36 T. Ikawa (5.) Then from we hae du = dx ds L ± dr ds L d = dy ds L ± dr ds L u = x(s L ) ± r(s R ) dy ds L = y(s L ) ± r(s R ) dx ds L dy ± r d ( dy ds L s L ds L ( dx dx ± r d ds L s L ds L du = dx ( ± rk L ) ds L ± dr dy ds L ds L d = dy ( ± rk L ) ds L ± dr dx ds L ds L where k L is the curature of c L. Since s R is also the arc length of c B it follows that ( du ) ( ) d = ) dsl ) dsl ( ) dsl ( ± rk L ) dr = ± where the sign of ± depends on spacelike or timelike of c B. From this differential equation we can sole s L = s L (s R ). Substituting this equation into (5.) we can hae the orthogonal coordinate of c B. The solability of these differential equations is easily checked. For example we hae solutions like that : c B is x-axis c R is quadratic cure and c L is Delaunay cure (cf. []). References [] J. Hano and K. Nomizu Surfaces of reolution with constant mean curature in Lorentz-Minkowski space Tohoku Math. J. 36 (984) [] T. Ikawa On cures and submanifolds in an indefinite-riemannian manifold Tsukuba J. Math. 9 (985) [3] T. Kubota Ueber die Bewegung einer Kure die auf einer anderen Kure rollt Science Rep. of the Tôhoku Imperial Uni. 9 (90) [4] B. O Neill Semi-Riemannian Geometry with Applications to Relatiity Academic Press 983. Nihon Uniersity Dep. of Math. School of Medicine Itabashi Tokyo Japan 73 tikawa@med.nihon-u.ac.jp