Representation Formulas of Curves in a Two- and Three-Dimensional Lightlike Cone

Transcription

1 Reult. Math. 59 (011), c 011 Springer Bael AG /11/ publihed online April, 011 DOI / y Reult in Mathematic Repreentation Formula of Curve in a Two- and Three-Dimenional Lightlike Cone Huili Liu and Qingxian Meng Dedicated to Profeor Heinrich Wefelcheid on the occaion of hi 70th birthday Abtract. In thi paper we give the repreentation formula for pacelike curve in two-dimenional lightlike cone Q and three-dimenional lightlike cone Q 3. Uing thee formula we dicu the propertie and tructure of cone curve in Q and Q 3. Some example are alo given. Mathematic Subject Claification (010). 53A04, 53B30, 53C1. Keyword. Lightlike cone, cone curve, repreentation formula, tructure function, cone curvature, cone torion, Riccati equation. 1. Introduction In the tudy of curve theory in claical differential geometry of Euclidean pace or Minkowki pace, it i well known that the Frenet moving frame i uually a powerful method. However, when we deal with ome pecial curve, the Frenet moving frame might be le convenient or tranparent, rep. For intance, uing the tandard Frenet moving frame, the curve on the unit phere in Euclidean 3-pace are determined by the Frenet formula, the curvature κ(), torion τ() and the condition τ() κ() + d d ( 1 d τ() d ( 1 κ() )) =0. Supported by NSFC (No ), Chern Intitute of Mathematic and NEU.

2 438 H. Liu and Q. Meng Reult. Math. But if we chooe a pherical Frenet frame, thee condition can be replaced by the pherical Frenet formula and pherical curvature function [4]. Therefore, the choice of the moving frame field and the reearch method ometime are very important and efficient for the tudy of pecial curve. In [1], the author tudied cone curve and gave the notation of the cone curvature function etc., and alo ome example of cone curve in Minkowki pace. Some application of the theory of cone curve in Minkowki 3-pace are given in [ 4]. In thi paper, we tudy cone curve in a two-dimenional lightlike cone Q and a three-dimenional lightlike cone Q 3 with a new method. We will preent the repreentation formula of the curve in Q and Q 3 and define the tructure function of the cone curve. Our main reult are: Theorem A (Repreentation of cone curve in Q ). Let x() :I Q E 3 1 be a pacelike curve in Q with arc length parameter. Thenx() can be written a x() =(x 1,x,x 3 )= f (f f 1,,f + f 1 )= 1 f f 1 (f 1, f,f +1) for ome non contant function f() which i called tructure function of x(). The tructure function f() and the cone curvature function κ() of x() atify κ() = 1 [(log f ) ] [(log f ) ], where f = df d. Theorem B (Repreentation of cone curve in Q 3 ). Let x() :I Q 3 E 4 1 be a pacelike curve in Q 3 with arc length parameter. Thenx() can be written a x() =(x 1,x,x 3,x 4 )=ρ(f,g, 1 f g, 1+f + g ) for ome function f() and g(), called tructure function of x(). Here ρ() atifie 4ρ ()[f ()+g ()] = 1. The tructure function f(), g(), the cone curvature function κ() and the cone torion function τ() of x() atify κ() = 1 [(log ρ) ] + (log ρ) 1 θ, τ() =±(θ (log ρ) + θ ), where θ = ( 1+ f g ) 1 ( ) f g.

3 Vol. 59 (011) Repreentation of Cone Curve 439 Together with the relation between the cone curvature function and the tructure function, uually they are characterized by certain differential equation, we give ome example of cone curve and alo the olution of ome differential equation.. Repreentation Formula of Curve in Q In the following we ue the notation and concept from [1]. Let x : I Q E 3 1 be a pacelike curve in a two-dimenional lightlike cone Q of the Minkowki 3-pace E 3 1 with arc length parameter. We write x =(x 1,x,x 3 ) and have x 1 + x x 3 =0. Then from x 1 x 3 = x we get x 1 + x 3 x = x x 1 x 3, or x 1 + x 3 x = x x 1 x 3. Without lo of generality, for a curve x : I Q E 3 1 with x = x() = (x 1,x,x 3 ), we may aume that x 1 + x 3 = x = f(), (.1) x x 1 x 3 and x =ρ(). (.) From (.1) and (.) weget x 1 + x 3 =ρf, x 1 x 3 = ρf 1, (.3) x =ρ, therefore, x 1 = ρ(f f 1 ), x =ρ, (.4) x 3 = ρ(f + f 1 ). That i, the curve x : I Q E 3 1 can be written a x = x() =(x 1,x,x 3 )=ρ(f f 1,,f + f 1 ). (.5) From (.5) wehave x = ρ (f f 1,,f + f 1 )+ρf (1 + f, 0, 1 f ). From now on we write ẋ = x = dx d, ẍ = x = d x d, etc. Then, if i the arc length parameter of the curve x(), we have 4ρ f f =1. (.6)

4 440 H. Liu and Q. Meng Reult. Math. Therefore (by an appropriate tranformation, if neceary) we may aume that and get the following concluion. ρ() = f() f () (.7) Theorem.1. Let x : I Q E 3 1 be a pacelike curve in Q with arc length parameter. Thenx = x() =(x 1,x,x 3 ) can be written a x() = f (f f 1,,f + f 1 )= 1 f f 1 (f 1, f,f + 1) (.8) for ome non contant function f() and f = df d. Definition.1. The function f() in Theorem.1 i called tructure function of the cone curve x : I Q E 3 1 with arc length parameter. From (.8) we have Then ẋ =x = f f (f 1, f,f +1)+(f,1,f), ẍ =x =(f 3 f f f )(f 1, f,f +1) f 1 f (f,1,f)+f (1, 0, 1). ẍ, ẍ = f We put and get and f f ( f 3 f f ) f = 3f f +f 1 f. (.9) y() = ẍ() 1 ẍ(), ẍ() x(), (.10) y() = ẍ() 1 ẍ(), ẍ() x() = 1 4 f 3 f(f 1, f,f +1)+f 1 f (f,1,f) f (1, 0, 1) = 1 f f x + f 1 f (f,1,f) f (1, 0, 1), (.11) y, y = x, x = y, ẋ =0, x, y =1. (.1) Definition.. Let x = x() :I Q E 3 1 be a pacelike curve in Q with arc length parameter. Then y(), defined by (.10), i alo a curve in Q and i called aociated curve (or dual curve) of the curve x().

5 Vol. 59 (011) Repreentation of Cone Curve 441 Uing α() = ẋ() we know that {x(),α(),y()} form an aymptotic orthonormal frame along the curve x(), and the cone Frenet formula of x() are given by [1] ẋ() =α(), α() =κ()x() y(), (.13) ẏ() = κ()α(). From (.9), (.10) and (.13) we know that the cone curvature function κ() of x() atifie κ() = 1 ẍ(), ẍ() = 3 f = 1 f f f 1 f + f f f f 1 f = 1 [(log f ) ] [(log f ) ]. (.14) Thi i an Riccati equation of ξ() = (log f ) and the linear term of it i zero. Remark 1. For the Riccati equation ξ () = dξ() = 1 d ξ () κ(), (.15) we put ( ) 1 η() = exp ξ()d, (.16) then Eq. (.15) can be written a η () 1 κ()η() =0. (.17) Thi i a linear equation of econd order for η(). If κ() < 0, Eq. (.17) ha the olution ( ) 1 κ co θ η() =a 1 exp in θ d, (.18) where θ atifie θ = κ + κ in θ co θ (.19) κ and a 1 i contant. If κ() > 0, Eq. (.17) ha the olution ( ) 1 κ coh θ η() =a 1 exp inh θ d, (.0)

6 44 H. Liu and Q. Meng Reult. Math. where θ atifie κ θ = + κ inh θ coh θ. (.1) κ We will give a proof in the Appendix. The following concluion give ome pecial olution of Eq. (.14) and then the pecial curve in Q E 3 1. Theorem.. Let x : I Q E 3 1 be a pacelike curve in Q with arc length parameter and tructure function f(). If the curve x i planar, the tructure function f() atifie [(log f ) ] (log f ) =κ() =c = contant, (.) and x i quadric. Exactly, 1. when a = c<0,x i an ellipe; f() = a a tan ;. when c =0,xi a parabola; f() = a ; 3. when a = c>0, xi a hyperbola; f() = a a tanh. Proof. (a) The Riccati equation (.) can be directly olved by eparation of variable, a κ() = contant. Thi give the aertion. (b) Alternatively, uing Theorem.3 in [1], the aertion of Theorem. follow alo from (.14). Remark. By the parameter tranformation + 0, where 0 i contant, we omit an integration contant here and later. From (.14) and the fundamental theorem of cone curve we know that the curve, determined by f() and f()+f 0, are congruent, where f 0 i a contant. So we omit alo another integration contant here and later. Corollary.3. Let x = x() :I Q E 3 1 be a pacelike curve with arc length parameter. 1. If x() i an ellipe, it can be written a ( a x() = in a 1 a co, 1 a in a, a in a + 1 ) a co. (.3). If x() i a parabola, it can be written a ( a x() = a,, a ) +. (.4) a 3. If x() i a hyperbola, it can be written a ( a x() = a inh 1 a coh, 1 a inh a, a a inh + 1 ) a coh. (.5) Theorem.4. Let x : I Q E 3 1 be a pacelike curve in Q with arc length parameter and tructure function f(). If the curve x i a non planar helix, the tructure function f() atifie

7 Vol. 59 (011) Repreentation of Cone Curve 443 [(log f ) ] (log f ) =κ() =a( + b) (.6) and can be written a (by an appropriate parameter tranformation) Cae 1. f() = c or f() = c,forc 0, ±1 and a = c 1; Cae. f() = c log log or f() = c,forc 0anda = 1. Proof. We ue three method to prove thi theorem. (a) When κ() = a( + b),eq.(.17) i an Euler equation of econd order. Then olving (.17) and together with (.16), ξ() = (log f ) and a parameter tranformation, we can get Theorem.4. (b) From (.14), Theorem.1 and.4 of [1], Example 1 and of [], by the lengthy, but elementary calculation, we can get the concluion of thi theorem. (c) Let κ() =a( + b). When a<0, from (.19) wehave θ 1 ( ) = a in θ. (.7) ( + b) When a>0, from (.1) wehave θ 1 ( ) = a inh θ. (.8) ( + b) Solving thee equation and uing the relation between θ(), η(), ξ() and f(), we can get the concluion of Theorem.4. Remark 3. For Example 1 and of [], the contant c in the denominator hould be c in the formula (3), (4), (7) and (8). That i, the curve of Cae 1 of Theorem.4 i x() = c ((c c ),, ( c + c )); (.9) and the curve of Cae of Theorem.4 i x() = log ( c log log c,, c log + log ). (.30) c 3. Repreentation Formula of Curve in Q 3 Let x : I Q 3 E 4 1 be a pacelike curve in the three-dimenional lightlike cone Q 3 of the Minkowki 4-pace E 4 1 with arc length parameter. We put x =(x 1,x,x 3,x 4 ) and have x 1 + x + x 3 x 4 =0. Then from x 1 (ix ) = (x 3 x 4)weget x 1 + ix x 3 + x 4 = x 3 x 4 x 1 ix, or x 1 + ix x 3 x 4 = x 3 + x 4 x 1 ix. (3.1)

8 444 H. Liu and Q. Meng Reult. Math. Without lo of generality, for a curve x : I Q 3 E 4 1 with x = x() = (x 1,x,x 3,x 4 ), we may aume that x 1 + ix = x 3 x 4 = f()+ig(), (3.) x 3 + x 4 x 1 ix and x 3 + x 4 =ρ(). (3.3) Then x 1 + ix = x 3 + x 4 1 = x 3 x 4 x 1 ix f() ig(). (3.4) From (3.), (3.3) and (3.4) weget x 1 + ix =ρ(f + ig), x 1 ix =ρ(f ig), (3.5) x 3 + x 4 =ρ, x 3 + x 4 =ρ(f + g ). Then x 1 =ρf, x =ρg, x 3 = ρ(1 f g (3.6) ), x 4 = ρ(1 + f + g ). That i, the curve x : I Q 3 E 4 1 can be written a x = x() =(x 1,x,x 3,x 4 )=ρ(f,g, 1 f g, 1+f + g ). (3.7) From (3.7) wehave x = ρ (f,g, 1 f g, 1+f + g )+ρ(f,g, ff gg,ff + gg ). (3.8) Then i the arc length parameter of the curve x() mean that 4ρ (f + g)=1. (3.9) Theorem 3.1. Let x : I Q 3 E 4 1 be a pacelike curve in Q 3 with arc length parameter. Thenx can be written a x = x() =(x 1,x,x 3,x 4 )=ρ(f,g, 1 f g, 1+f + g ) (3.10) for ome function f() and g(). Hereρ() atifie 4ρ ()[f ()+g()] = 1. Definition 3.1. The function f()andg() in Theorem 3.1 are called the tructure function of the cone curve x : I Q 3 E 4 1 with arc length parameter. From (3.9), by a tranformation if neceary, we may aume that 1 ρ() = (3.11) f ()+g()

9 Vol. 59 (011) Repreentation of Cone Curve 445 or { f () =ρ() 1 in θ(), g () =ρ() 1 co θ(). (3.1) Putting y() = ẍ() 1 ẍ(), ẍ() x(), (3.13) we have y, y = x, x = y, ẋ =0, x, y =1. (3.14) By (3.1) we have alo { f = ρ ρ in θ + ρ 1 θ co θ, g = ρ ρ co θ ρ 1 θ in θ, (3.15) and { f =(ρ 3 ρ ρ ρ ρ 1 θ)inθ +(ρ 1 θ ρ ρ θ )coθ, g =(ρ 3 ρ ρ ρ ρ 1 θ)coθ (ρ 1 θ ρ ρ θ )inθ. (3.16) From (3.8) we get ẍ = ρ (f,g, 1 f g, 1+f + g ) +4ρ (f,g, ff gg,ff + gg ) +ρ(f,g, f ff g gg,f + ff + g + gg ), (3.17)... x = ρ (f,g, 1 f g, 1+f + g ) +6ρ (f,g, ff gg,ff + gg ) +6ρ (f,g, f ff g gg,f + ff + g + gg ) +ρ(f,g, 3f f ff 3g g gg, 3f f + ff +3g g + gg ). (3.18) Together with (3.1), (3.15) and (3.16) we obtain ẍ, ẍ =16ρ ( f + g ) +4ρ ( f + g) ( 8ρρ f + g) +16ρρ (f f + g g ) = ρ ρ + θ ρ 1 ρ, (3.19) here tan θ = f () g (), (3.0)

10 446 H. Liu and Q. Meng Reult. Math. and... x,... x =36ρ (f + g)+36ρ (f + g) +4ρ (f + g) 4ρ ρ (f + g) 4ρρ (f f + g g )+7ρ ρ (f f + g g ) +4ρρ (f f + g g )+4ρρ (f f + g g ) =4ρ ρ + ρ 4 ρ 4 +3ρ ρ θ 4ρ 3 ρ ρ 4ρ 1 ρ θ + θ 4 +ρ 1 ρ θ θ + θ. (3.1) Put α() = ẋ() and chooe β() uch that Then from (3.13) we have det(x(),α(),β(),y()) = 1. α() =ẍ() = 1 ẍ(), ẍ() x() y() =κ()x() y(). (3.) Therefore, the Frenet formula of the curve x = x() :I Q 3 E 4 1 can be written a ẋ() =α(), α() =κ()x() y(), β() =τ()x(), ẏ() = κ()α() τ()β(). (3.3) Definition 3.. The function κ() and τ() in(3.3) are called the (firt) cone curvature and cone torion (or econd cone curvature) of the curve x() in Q 3 E 4 1. The frame field {x(), α(), β(), y()} i called the cone Frenet frame of the curve x(). By (3.), (3.19) and (3.0) we have κ() = 1 ẍ, ẍ = 1 ρ ρ 1 θ + ρ 1 ρ = 1 [(log ρ) ] + (log ρ) 1 θ, (3.4) here θ = ( 1+ f g ) 1 ( ) f g. (3.5)

11 Vol. 59 (011) Repreentation of Cone Curve 447 From (3.), (3.3), uing x,... x =0and ẋ,... x = ẍ, ẍ, by a direct calculation we get τ =... x κα κx,... x κα κx κ =... x,... x 4κ =... x,... x ẍ, ẍ =4ρ ρ + ρ 4 ρ 4 +3ρ ρ θ 4ρ 3 ρ ρ 4ρ 1 ρ θ + θ 4 +ρ 1 ρ θ θ + θ ρ 4 ρ 4 θ 4 4ρ ρ ρ ρ θ +4ρ 3 ρ ρ +4ρ 1 ρ θ = ρ ρ θ +ρ 1 ρ θ θ + θ =(ρ 1 ρ θ + θ ) =(θ (log ρ) + θ ). (3.6) Alo by a direct calculation we have τ()β() =... x ()+ ẍ(), ẍ() ẋ()+... x (), ẍ() x(). (3.7) Propoition 3.1. Let x = x() :I Q 3 E 4 1 be a pacelike curve in Q 3 with arc length parameter and tructure function f() and g(). Then the cone curvature and torion of x are given by (3.4) and (3.6); the cone Frenet frame field of x i given by (3.13) and (3.7), repectively. Remark 4. From [1], for any aymptotic orthonormal frame {x, α, β, y} of the curve x : I Q 3 E 4 1 with x, x = y, y = x, α = x, β = y, α = y, β = α, β =0, x, y = α, α = β,β =1, the Frenet formula read ẋ() =α(), α() =κ()x()+λ()β() y(), β() =τ()x() λ()α(), ẏ() = κ()α() τ()β(). (3.8) We know that λ() 0 if and only if y() atifie (3.13). Therefore ome author called the frame, atifying (3.13), Cartan frame. We know that (3.13) and (3.14) are true in any dimenion. That i, we have the following concluion for the curve x : I Q n+1 E n+ 1. Propoition 3.. Aume that the curve x = x() :I Q n+1 E n+ 1 i a regular curve in Q n+1 with arc length parameter. Put y() = ẍ() 1 ẍ(), ẍ() x(),

12 448 H. Liu and Q. Meng Reult. Math. and the function α (),α 3 (),...,α n () pan{x, y, α 1 } uch that α 1 = ẋ and α i,α j = δ ij, 1 i, j n. Then Example 3.1. The curve ẋ() =α 1 (), α 1 () =κ 1 ()x() y(), α () =κ ()x(), α 3 () =κ 3 ()x(), α i () =κ i ()x(), α n 1 () =κ n 1 ()x(), α n () =κ n ()x(), ẏ() = n i=1 κ i()α i (). x() = ( 1,, 1, 1 ) +1 (3.9) (3.30) i pacelike in Q 3 with arc length parameter, cone curvature κ() 0, cone torion τ() 0, and tructure function g() = f() = 1 +1, +1, ρ() = +1. Example 3.. The curve ( x() = a, b, ( ) a + b inh, ( )) a + b coh a + b a + b (3.31) i pacelike in Q 3 with arc length parameter, cone curvature κ() = contant > 0, cone torion τ() 0, and tructure function f() = a ρ, g() = b ρ, ρ() = ( ) ( a + b (inh a +coh +b a )), +b where α + b 0. Example 3.3. The curve a ( ) x() =( b in, ( ) ) a b co,b,a (3.3) a b a b

13 Vol. 59 (011) Repreentation of Cone Curve 449 i pacelike in Q 3 with arc length parameter, cone curvature κ() = contant < 0, cone torion τ() 0, and tructure function f() = ( ( )) a b a+b in a, b g() = ( ( )) a b a+b co a, b where α b > 0. Example 3.4. The curve ρ() =a + b, x() =( a + b ) 1 (in a, co a, inh b, coh b) (3.33) i pacelike in Q 3 with arc length parameter, cone curvature κ() = contant, cone torion τ() = contant 0, and tructure function in a f() = inh b+coh b, co a g() = inh b+coh b, inh b+coh b ρ() = a, +b where ab Appendix Propoition 4.1. The differential equation f (t) = d f dt = ϕ(t)f(t), when ϕ(t) < 0, ha the olution ( ) ϕ co θ f(t) =a 1 exp in θ dt, (4.1) where a 1 i contant and θ atifie θ = ϕ + ϕ in θ co θ; ϕ when ϕ(t) > 0, ha the olution ( ) ϕ coh θ f(t) =a 1 exp inh θ dt, (4.) where a 1 i contant and θ atifie θ = ϕ + ϕ inh θ coh θ. ϕ

14 450 H. Liu and Q. Meng Reult. Math. Proof. (a) When ϕ() < 0, uing polar coordinate, we write { ρ in θ = f ϕ, ρ co θ = f (4.3). Then ρ in θ + ρ co θθ = ϕ ρ co θ + ϕ ρ in θ, (4.4) ϕ ϕ ρ co θ ρ in θθ = ϕ ρ in θ. (4.5) ϕ From the operation (4.4) co θ (4.5) in θ we have that i ρθ = ϕ ρ co θ + ϕ ϕ ρ in θ co θ ϕ ϕ ρ in θ ϕ = ρ ϕ co θ + ϕ ρ in θ + ϕ ρ in θ co θ, ϕ θ = ϕ + ϕ in θ co θ. ϕ Then the quotient of the equation of (4.3) yield (4.1). (b) When ϕ() > 0 and in caethat < 1, let f ϕ f Then { ρ inh θ = f ϕ, ρ coh θ = f. (4.6) ρ inh θ + ρ coh θθ = ϕρ coh θ + ϕ ρ inh θ, ϕ ϕ (4.7) ρ coh θ + ρ inh θθ = ϕ ρ inh θ. ϕ (4.8) From the operation (4.7) coh θ (4.8) inh θ we have ρθ = ϕρ coh θ + ϕ ρ inh θ coh θ ϕ ρ inh θ ϕ ϕ ϕ that i = ρ ϕ coh θ ϕρ inh θ + ϕ ρ inh θ coh θ, ϕ θ = ϕ + ϕ inh θ coh θ. ϕ Then the quotient of the equation of (4.6) yield (4.).

15 Vol. 59 (011) Repreentation of Cone Curve 451 Remark 5. In cae (b), if we may aume that f ϕ f > 1, { ρ coh θ = f ϕ, ρ inh θ = f, (4.9) and get a imilar concluion. If f ϕ f =1, by a direct calculation we have ϕ =0. Reference [1] Liu, H.: Curve in the lightlike cone. Contrib. Algebr. Geom. 45, (004) [] Liu, H.: Ruled urface with lightlike ruling in 3-Minkowkipace. J. Geom. Phy. 59, (009) [3] Liu, H.: Characterization of ruled urface with lightlike rulingin Minkowki 3-pace. Reult. Math. 56, (009) [4] Liu, H.: Pitch function of ruled urface and B-croll in Minkowki 3-pace. J. Geom. Phy. (009, NEU preprint, ubmitted) Huili Liu and Qingxian Meng Department of Mathematic Northeatern Univerity Shenyang People Republic of China liuhl@mail.neu.edu.cn; mengqingxian8@16.com URL: Received: July 7, 010. Revied: November 11, 010. Accepted: February, 011.

16 Reult. Math. 61 (01), c 01 Springer Bael AG /1/ publihed online May 1, 01 DOI / Reult in Mathematic ERRATUM Erratum to: Repreentation Formula of Curve in a Two- and Three-Dimenional Lightlike Cone Huili Liu and Qingxian Meng Erratum to: Reult. Math. (011) 59: DOI / y In the original publication of the article, Cae 3 in Theorem.4 ha been inadvertently mied out. The complete Theorem.4 i given below. Theorem.4. Let x : I Q E 3 1 be a pacelike curve in Q with arc length parameter and tructure function f(). If the curve x i a non planar helix, the tructure function f() atifie [(log f ) ] (log f ) =κ() =a( + b) (0.1) and can be written a (by an appropriate parameter tranformation) Cae 1. f() = c or f() = c, for c 0, ±1 anda = c 1; Cae. f() = c log log or f() = c, for c 0anda = 1; Cae 3. f() = c tan ( c log ) or f() = ( c tan 1 c log ), for c 0 and a +1= c. Supported by NSFC (No ); Joint Reearch of NSFC and NRF (No ). The online verion of the original article can be found under doi: / y.

17 44 H. Liu and Q. Meng Reult. Math. Huili Liu and Qingxian Meng Department of Mathematic Northeatern Univerity Shenyang People Republic of China URL: