ON CURVES IN THE LIGHTLIKE CONE 1. Introduction Lorentzian ...

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224 TWMS JOUR. PURE APPL. MATH., V.2, N.2, 2011Using a orthonormal frame on the curve x(s) and denoting by κ, τ, β and γ the curvature,the torsion, the principal normal and the binormal of the curve x(s) in E1 3 , we havex ′ = αα ′ = κx − y = κβ,where κ ≠ 0, < β, β >= ε = ±1, < α, β >= 0, < α, α >= 1, εκ < 0. Then we getChoosingwe obtainβ = ε κx − y √ −2εκ, ετγ =γ =κ = √ −2εκ,κ ′2 √ −2εκ (x + 1 y). (4)κ√−εκ2 (x + 1 y) (5)κτ = − 1 2 (κ′ ). (6)κTheorem 3.1. The curve x : I → Q 2 is a planar curve if and only if the cone curvature functionκ of the curve x(s) is constant [6].If the curve x : I → Q 2 ⊂ E 3 1is a planar cure, then we have the following Frenet formulasx ′ = α,α ′ = ε √ −2εκβ,β ′ = − √ −2εκα.(7)Definition 3.1. Let x : I → Q 2 be a curve with constant cone curvature κ and arc lengthparameter s, then the support function of x(s) with respect to a fixed point p 0 ∈ Q 2 is defined byDifferentiating (8) with respect to s and using (7), we haveρ(p 0 ) =< β, p 0 − x > . (8)ρ ′ (p 0 ) = − √ −2εκ < α, p 0 − x > (9)and we get a representation of x(s) in terms of the support function:x − p 0 =1√ −2εκρ ′ (p 0 )α − ερ(p 0 )β. (10)Theorem 3.2. Let x : I → Q 2 be a curve with constant cone curvature κ and arc lengthparameter s, then the support function ρ(p 0 ) of x(s) with respect to a fixed point p 0 ∈ Q 2 can bewritten as follows:(i) ρ(p 0 )(s) = c 1 s + c 2 ,for κ = 0, it is a part of circle;(ii) ρ(p 0 )(s) = c 1 cos √ −2κs + sin √ −2κs + √ 1−2κ,for ε = 1, κ < 0, it is an ellipse;(iii) ρ(p 0 )(s) = c 1 cosh √ 2κs + c 2 sinh √ 2κs − √ 12κ,for ε = −1, κ > 0, it is a hyperbola; where c 1 , c 2 ∈ E 3 1 .Proof. Differentiating (10), we get1√ −2εκρ ′′ (p 0 ) + ε √ −2εκρ(p 0 ) = 1. (11)
H.B. ÖZTEKIN, M. ERGÜT: ON CURVES IN THE LIGHTLIKE CONE 225Solving this equation (11), we obtain thatρ(p 0 )(s) = c 1 s + c 2 , κ = 0,ρ(p 0 )(s) = ρ(p 0 )(s) = c 1 cos √ −2κs + sin √ −2κs + 1 √ −2κ, ε = 1, κ < 0,ρ(p 0 )(s) = c 1 cosh √ 2κs + c 2 sinh √ 2κs − 1 √2κ, ε = −1, κ > 0,where c 1 , c 2 ∈ E 3 1 .□Definition 3.2. Let x : I → Q 2 be a curve with constant cone curvature κ and arc lengthparaameter s. Then the locus of the centre of curvature of a planar curve x(s) is called theevolute of the curve x(s) and given byE x (s) = x(s) +From (12), the evolute curve E x is not regular curve.1√ −2εκβ(s). (12)Definition 3.3. Let x : I → Q 2 be a curve with constant cone curvature κ and arc lengthparameter s. For a fixed value s 1 ∈ R, the involute of the curve x(s) is defined byIf we derivative of (13) with respect to s, we getI x,s1 (s) := x(s) − (s + s 1 )α(s). (13)I ′ x,s 1(s) = −ε(s + s 1 ) √ −2εκβ(s). (14)Thus the condition (s + s 1 ) ≠ 0 is equivalent to the regularity of the involute I x,s1 and wesuppose all involutes to be regular. If s 1 varies, one obtains a one-parameter family of involutes.If {α, β} is an orthonormal frame of x thenα I = −εsign(s + s 1 )ββ I = εsign(s + s 1 )α (15)defines an orthonormal frame of the involute I x,s1 , where < α I , α I >= ε and < β I , β I >= 1.Theorem 3.3. Let x : I → Q 2 be a curve with constant cone curvature κ and arc lengthparameter s. Then the involute I x,s1 of x(s) satisfies the following properties:(i) If {α, β} is an orthonormal frame of x(s) then{α I = −εsign(s + s 1 )β, β I = εsign(s + s 1 )α}is an orthonoral frame of I x,s1 .(ii) The cone curvature function κ I of the involute curve I x,s1 satisfies:εκ I = −2(s + s 1 ) 2 (16)and we insert the equation (13)√ −εI x,s1 (s) := x(s) − sign(s + s 1 ) α(s).2κ IProof. Considering (7) and (15), we have√ −2εκI =< α ′ I, β I >= − √ −2εκ,that means that κ = κ I .
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