Selçuk J. Appl. Math. Selçuk Journal of Vol. 13. No. 1. pp. 125-130, 2012 Applied Mathematics
Curves with Constant Curvature Ratios inL5 Esen ˙Iyigün
Uluda˘g University, Faculty of Arts and Science, Department of Mathematics, 16059, Bursa, Turkiye
e-mail: esen@ uludag.edu.tr
Received Date: March 25, 2012 Accepted Date: April 9, 2012
Abstract. In this study, we first give a relation between Frenet formulas and harmonic curvatures of a curve of osculating order 5 in L5. Also, we find a rela-tion between harmonic curvatures and ccr-curves of a curve in L5and also obtain some results. Finally, we calculate constant curvature ratios ε1
k2 k1 , ε2 k3 k2 , ε3 k4 k3 of the unit speed time-like curve in L5studied in [3].
Key words: Frenet curve of osculating order 5; Frenet curvatures, a general helix of rank 3; Ccr-curves; Harmonic curvatures.
2000 Mathematics Subject Classification: 53C40, 53C42. 1. Introduction
Let X = (x1, x2, x3, x4, x5) and Y = (y1, y2, y3, y4, y5) be two non-zero vectors in 5-dimensional Lorentz Minkowski space R5
1. We denote R51shortly by L5. For X, Y ∈ L5 hX, Y i = −x1y1+ 5 X i=2 xiyi is called the Lorentzian inner product. The couple©R5
1, h, i ª
is called Lorentzian space and briefly denoted by L5. Then a vector X in L5 is either called i) time-like if hX, Xi < 0, or ii) space-like if hX, Xi > 0 or X = 0, or iii) null (or light-like) vector if hX, Xi = 0, X 6= 0. Similarly, an arbitrary curve α = α(s) in L5can be locally space-like, time-like or null, if all of its velocity vectors α0
(s) are respectively space-like, time-like or null. Also, recall that the norm of a vector X is given by kXk =p|hX, Xi|. Therefore X is a unit vector if hX, Xi = ±1. Next, two vectors X, Y in L5 are said to be orthogonal if hX, Y i = 0. The velocity
of a curve α is given by°°°α0°°°. Thus, a space-like or a time-like α is said to be parametrized by the arclength function s, ifDX0, X0E= ±1, [1].
2. Basic Definitions of L5
Definition 1. Let M ⊂ L5, α : I −→ L5be a curve in L5 and k
1, k2, k3, k4 be the Frenet curvatures of α. Then for the unit tangent vector V1= α
0
(s), the ith e-curvature function mi, 1 ≤ i ≤ 5, is defined by
mi = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0 , i = 1 ε1ε2 k1 , i = 2 ∙ d dt(mi−1) + εi−2mi−2ki−2 ¸ εi ki−1 , 2 < i ≤ 5 ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ where εi= hVi, Vii = ±1.
Definition 2. Let α : I −→ L5 be a unit speed non-null curve in L5. The curve α is called a Frenet curve of osculating order 5, if its 5th order deriv-atives α0(s), α00(s), α000(s), α(iv)(s) and α(v)(s) are linearly independent and α0(s), α00(s), α000(s), α(iv)(s), α(v)(s) and α(vi)(s) are no longer linearly inde-pendent for all s ∈ I. For each Frenet curve of order 5, one can associate an orthonormal 5−frame V1, V2, V3, V4, V5along α (such that α
0
(s) = V1) called the Frenet frame and k1, k2, k3, k4: I −→ L5are called the Frenet curvatures. Then the Frenet formulas are defined in the usual way:
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ V10 V20 V30 V40 V50 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 ε2k1 0 0 0 −ε1k1 0 ε3k2 0 0 0 −ε2k2 0 ε4k3 0 0 0 −ε3k3 0 ε5k4 0 0 0 −ε4k4 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ V1 V2 V3 V4 V5 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
A non-null curve α : I −→ L5 is called a W − curve (or helix) of rank 5, if α is a Frenet curve of osculating order 5 and the Frenet curvatures ki, 1 ≤ i ≤ 4, are non-zero constants.
3. A General Helix of rank 3
Definition 3. Let α be a non-null curve of osculating order 5. The harmonic functions Hj: I −→ R , 0 ≤ j ≤ 3, defined by H0= 0, H1= k1 k2 ,
(1) Hj = © 5v1 ¡ H( j−1) ¢ + ε( j−2)H( j−2)kj ª εj k( j+1), 2 ≤ j ≤ 3
are called the harmonic curvatures of α, where k1, k2, k3, k4are Frenet curvatures of α which are not necessarily constant.
Definition 4. Let α be a non-null curve of osculating order 5. Then α is called a general helix of rank 3 if
(2)
3 X i=1
Hi2= c, where c 6= 0 is a real constant.
We now have the following result:
Corollary 1. If α is a general helix of rank 3, then H12+ µ H10ε2 k3 ¶2 +µ³H20 + ε1H1k3 ´ ε3 k4 ¶2 = c.
Proof. By the use of above definitions we obtain the proof. Proposition 1. Let α be a curve in L5 of osculating order 5, then
V10 = ε2k2H1V2, V20 = −ε1k2H1V1+ ε3 k1 H1 V3, V30 = −ε2 k1 H1 V2+ ε4ε2 H10 H2 V4, V40 = −ε3ε2 H10 H2 V3+ ε5ε3 Ã H2H 0 2+ ε1ε2H1H 0 1 H2H3 ! V5, and V50 = −ε4ε3 Ã H2H 0 2+ ε1ε2H1H 0 1 H2H3 ! V4, where H1, H2, H3 are harmonic curvatures of α.
Proof. By using the Frenet formulas and definitions of the harmonic curvatures, we get the result.
Proposition 2. [2] Let α be a non-null curve of osculating order 5, then (3) kr(t) = ε(r−2) Ãr−2 X i=1 Hi2 !0 2H(r−2)H(r−1) , 2 < r ≤ 3, where (Hi) 0
stands for differentiation with respect to the parameter t. Proof. From [2], we get the proof of the proposition.
By taking H3= 0, we get the following result:
Theorem 1. For a non-null curve α in L5, if α is a general helix of rank 3, then the 3rd harmonic curvature H
3is given by H3= ³ H20 + ε1H1k3 ´ ε3 k4 . 4. CCR-Curves inL5
Definition 5. A curve α : I −→ L5 is said to have constant curvature ratios (that is to say, it is a ccr-curve) if all the quotients εi
µ ki+1
ki ¶
are constant. Here ki, ki+1 are the Frenet curvatures of α and εi= hVi, Vii = ±1, 1 ≤ i ≤ 3. Corollary 2. i)For i=1, the ccr-curve is ε1
H1 . ii) For i=3, the ccr-curve is ε2H2H
0 2+ ε1H1H 0 1 H10H3 .
Proof. The proof can easily be seen by using the definitions of harmonic curvature and ccr-curve.
Corollary 3. Let α : I −→ L5 be a ccr-curve. If ε 1 µ k2 k1 ¶ = ε2 µ k3 k2 ¶ = ε3 µ k4 k3 ¶ = c, c is a constant, then ε1 µ k2 k1 ¶0 = ε2 µ k3 k2 ¶0 = ε3 µ k4 k3 ¶0 = 0.
Theorem 2. α is a ccr-curve in L5 if f 3 X i=1
εiHi2= constant.
Proof. By using the definitions of a general helix of rank 3 and a ccr-curve, one completes the proof of the theorem.
Now, we will calculate constant curvature ratios of a unit speed time-like curve in L5 studied in [3].
Example 1. Let us consider the following curve
α(s) =³√3 sinh s,√3 cosh s, sin s, s, cos s´. V1(s) = α
0
(s) =³√3 cosh s,√3 sinh s, cos s, 1, − sin s´
whereDα0(s), α0(s)E= −1. One can easily see that α(s) is a unit speed time-like curve. We express the following differentiations:
α00(s) =³√3 sinh s,√3 cosh s, − sin s, 0, − cos s´, α
000
(s) =³√3 cosh s,√3 sinh s, − cos s, 0, sin s´, α(ıv)(s) =³√3 sinh s,√3 cosh s, sin s, 0, cos s´, α(v)(s) =³√3 cosh s,√3 sinh s, cos s, 0, − sin s´. So, we have the first curvature as
° °
°α00(s)°°° = k1(s) = 2 = constant.
Moreover we can write second, third, fourth and fifth Frenet vectors of the curve, respectively, as V2(s) = ε2 Ã√ 3 2 sinh s, √ 3 2 cosh s, − 1 2sin s, 0, − 1 2cos s ! , V3(s) = 1 √ 14 ³
−3√3 cosh s, −3√3 sinh s, −5 cos s, −4, 5 sin s´,
V4(s) = μ µ
−12sinh s, −12cosh s, −32sin s, 0, −32cos s ¶ and V5(s) = μ Ã −√1 14cosh s, − 1 √ 14sinh s, r 3 14cos s, − r 2 7, − r 3 14sin s ! ,
where μ is taken ∓1 to make the determimant of {V1, V2, V3, V4, V5} (s) matrix +1. In addition, we can write second, third and fourth curvatures and harmonic curvature of α(s), respectively, as k2(s) = √ 14 = constant, k3(s) = r 3 14 = constant, k4(s) = r 2 7= constant, H1= 2 √ 14.
Now, we will calculate ccr-curves of α(s) in L5. If the vector V1 is time-like, then μ = 1, ε1= −1 and ε2= ε3= 1: ε1 k2 k1 = ε1 H1 = constant, ε2 k3 k2 = constant and ε3 k4 k3 = constant. Thus, α(s) is a ccr-curve in L5. References
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