40
Şeyda Kılıçoğlu, Süleyman Şenyurt and Abdussamet Çalışkan
Proof Since the equations of the striction curves on Bertrandian Frenet ruled surfaces in terms of Frenet apparatus of curve α are c∗∗ 1 (s) =
∗∗ c∗∗ 3 (s) = α (s) = α (s) + λV2 (s)
the first and the second curvatures of the curve α∗∗ are given by k1∗∗ = k2∗∗ =
(λ2
1 1 . Also k2 k2∗∗ = 2 and 2 + β ) k2 (λ + β 2 )
βk1 − λk2 and (λ2 + β 2 ) k2
2 2 ∗∗ λ + β k k 2 1 ∗∗ V2 . V ∗∗ (s) = α + λ + c∗∗ 2 (s) = α (s) + ∗∗2 k1 + k2∗∗2 2 (βk1 − λk2 )2 + 1 Theorem 2.6 The tangent vector fields T1∗∗ , T2∗∗ and to Bertrandian Frenet ruled surface are given by
T1∗∗
1
0
T ∗∗ = a∗∗ 2 T3∗∗ 1
b∗∗ 0
0
2
T3∗∗ of striction curves belonging
V1∗∗
∗∗ c∗∗ V2 0 V3∗∗
where a∗∗ =
k ∗∗ 2 2 ′ , ∗∗ η c∗∗ 2 (s)
b∗∗ =
′ k1∗∗ ∗∗ η ′ , c∗∗ (s) 2
c∗∗ =
k1∗∗ k2∗∗ ′ and η ∗∗ = k1∗∗ 2 + k2∗∗ 2 . c∗∗ (s) 2
η ∗∗
Theorem 2.7 The product of tangent vector fields T1 ∗ , T2 ∗ , T3 ∗ and tangent vector fields T1 ∗∗ , T2 ∗∗ , T3 ∗∗ of striction curves on an involute and Bertrandian Frenet ruled surface respectively, are given by
0
T [T ∗ ] [T ∗∗ ] = A B 0
Ab∗∗ a∗∗ B + b∗∗ a∗ A + c∗∗ C b∗∗ A
0
B 0
where the coefficients are A=
q (λ2 + β 2 )(k1 2 + k2 2 ) , B = b∗ (−βk1 + λk2 ) + c∗ , C = b∗ + c∗ (−λk2 + βk1 ).
Proof Let [T ∗ ] = [A∗ ] [V ∗ ] and [T ∗∗ ] = [A∗∗ ] [V ∗∗ ] be given. By using the properties of a matrix following result can be obtained: