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

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)