Abstract
The curve whose tangent and binormal indicatrices are slant helices is called a slant-slant helix.
 
 In this paper, we give a new characterization of a slant-slant helix and determine a vector differential equation of the third order satisfied by the derivative of principal normal vector fields of a regular curve. In terms of solution, we determine the parametric representation of the slant-slant helix from the intrinsic equations.
 
 Finally, we present some examples of slant-slant helices by means of intrinsic equations.
Highlights
We give a new characterization of a slant-slant helix and determine a vector differential equation of the third order satisfied by the derivative of principal normal vector fields of a regular curve
We determine the parametric representation of the slant-slant helix from the intrinsic equations
Helical structures are an important framework in differential geometry and it was heavily studied for a long time and is still studied
Summary
Helical structures are an important framework in differential geometry and it was heavily studied for a long time and is still studied. Takeuchi (2004) have defined a new special curve called slant helix for which the principal normal lines make a constant angle with a fixed straight line and they characterize a slant helix if and only if the geodesic curvature κ2 τ′. Yayli (2005), and they have proved that the spherical images of a slant helix are spherical helices. ILarslan (2010), characterize slant helices by certain differential equations verified for each one of spherical indicatrix in Euclidean 3-space. Ismail GÖk & Yusuf Yayli (2013), investigated a curve whose spherical images (the tangent indicatrice and binormal indicatrix) are slant helices and called it a slant-slant helix and have given some characterizations. Vol 11, No 5; 2019 fundamental existence and uniqueness theorem for space curves in Euclidean space E3 and constructed a vector differential equation to solve this problem in the case of a general helix. We present some new characterizations of a slant-slant helix and we give some illustrative examples
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