On Geometry of Ruled Surfaces Generated by the Spherical Indicatrices of a Regular Space Curve I

On geometry of ruled surfaces generated by the spherical indicatrices of a regular space curve II

“…This article is a continuation of what the author and A. Asiri did in [1]. In [1] the author and A. Asiri introduced two developable ruled surfaces Ω T N and Ω BN by taking the principal normal indicatrix of a regular space as a base curve for both surfaces and the tangent indicatrix and the binormal indicatrix as the director curves. In this article we take the versa of what has been taken in [1].…”

“…Let γ : I −→ R 3 be a regular curve with non-vanishing curvature. In [1] the author and A. Asiri introduce the developable surfaces Ω T N = T γ + uN γ and Ω BN = B γ + uN γ . The elementary classical differential geometry of these surfaces is investigated as well as their singularities in [1].…”

“…In [1] the author and A. Asiri introduce the developable surfaces Ω T N = T γ + uN γ and Ω BN = B γ + uN γ . The elementary classical differential geometry of these surfaces is investigated as well as their singularities in [1]. If we swap T γ and N γ in the first surface and swap B γ and N γ in the second surface we get the following surfaces Ω N T = N γ + uT γ and Ω N B = N γ + uB γ respectively.…”

“…In [1] the author and A. Asiri introduced two developable ruled surfaces Ω T N and Ω BN by taking the principal normal indicatrix of a regular space as a base curve for both surfaces and the tangent indicatrix and the binormal indicatrix as the director curves. In this article we take the versa of what has been taken in [1]. Precisely, we define two ruled surfaces Ω N T and Ω N B by taking the principal normal indicatrix of a regular space curve as the director curve for both surfaces and the tangent indicatrix and the binormal indicatrix as the base curves see section 3.…”

Position Vectors of the Natural Mate and Conjugate of a Space Curve

“…In differential geometry, curves and their Frenet frames play central roles for creating special surfaces (c.f [5][6][7][8][9][10]). The Frenet frame associated with a regular curve in E 3 , which is a moving frame along the curve, forms an orthonormal basis for the Euclidean space E 3 at each point of the given curve.…”