On rectifying slant helices in Euclidean 3-space

TL;DR

This work addresses the problem of characterizing unit-speed rectifying slant helices in Euclidean 3-space and determining their position vectors. The authors derive explicit curvature and torsion formulas $\kappa(s)$ and $\tau(s)$ for such curves and reduce their construction to a second-order linear differential equation for $v=n'/\kappa$. Solving this yields a family of rectifying slant helices that lie on cones, supported by explicit parametric representations $\alpha(s)$ and cone relations, along with concrete examples on cones $8(x^2+y^2)=z^2$ and $99(x^2+y^2)=z^2$. The results provide a clear link between rectifying and slant-helix structure, enabling precise descriptions of these curves and their geometric embeddings with potential applications in spatial curve theory.

Abstract

In this paper, we study the position vectors of rectifying slant helices in $E^3$. First, we have found the general equations of the curvature and the torsion of rectifying slant helices. After that, we have constructed a second order linear differential equation and by solving the equation, we have obtained a family of rectifying slant helices which lie on cones.

Figures (4)

• Figure 1: Rectifying Slant Helix on $8\left(x^2+y^2\right)=z^2$
• Figure 2: Tangent, Normal,and Binormal indicatrix of $\alpha$ resp.
• Figure 3: Rectifying Slant Helix on $99\left(x^2+y^2\right)=z^2$
• Figure 4: Tangent, Normal,and Binormal indicatrix of $\beta$ resp.

Theorems & Definitions (13)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof
• Lemma 1