New special curves and their spherical indicatrices

From the view of differential geometry, a straight line is a geometric curve with the curvature κ(s) = 0. A plane curve is a family of geometric curves with torsion τ(s) = 0. Helix is a geometric curve with non-vanishing constant curvature κ and non-vanishing constant torsion τ [4]. The helix may be called a circular helix or W -curve [9]. It is known that straight line (κ(s) = 0) and circle (κ(s) = a, τ(s) = 0) are degenerate-helices examples [12]. In fact, circular helix is the simplest three-dimensional spirals [6]. A curve of constant slope or general helix in Euclidean 3-space E is defined by the property that the tangent makes a constant angle with a fixed straight line called the axis of the general helix. A classical result stated by Lancret in 1802 and first proved by de Saint Venant in 1845 (see [19] for details) says that: A necessary and sufficient condition that a curve be a general helix is that the function