Mannheim Curves in 3-Dimensional Euclidean Space

Emre Öztürk

doi:10.47897/bilmes.818723

Abstract

In this paper, we consider the Mannheim curve and the slant helix together. We called this curve as a Mannheim slant helix shortly. First we calculate the (first) curvature 𝜿(𝒔), and the curvature of the tangent indicatrix of the Mannheim curve, in terms of the arc-lenght parameter of the curve. Also, we proved that if the Mannheim curve is also slant helix, i.e. if it is Mannheim slant helix, then the partner curve is general helix. Moreover, we show the striction curve of the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve, is the Mannheim partner curve. Finally, we show the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve is non-developable while the torsion of the Mannheim partner curve 𝝉(𝒔)≠±∞ for all s.

Highlights

Introduction and MaterialMethodThe Mannheim curve is firstly investigated by French mathematician Amédéé Mannheim (1831-1906) in 1878
The first curve is called as Mannheim curve and the second curve is called as Mannheim partner curve of the first one
The curvature of the tangent indicatrix of α is given by κT(s) = ±coshφ where φ is a linear function of arc-lenght parameter of the curve

Summary

Introduction

Introduction and MaterialMethodThe Mannheim curve is firstly investigated by French mathematician Amédéé Mannheim (1831-1906) in 1878. They stated the first derivative of the torsion of the partner curve depending on the curvatures of the Mannheim curve. Orbay and Kasap [2] gave the torsion of the partner curve in terms of the curvatures of the Mannheim curve by following: τ(t) κ(t) λτ(t) They defined the slant helix as a curve that its principal normal lines make a constant angle with a fixed direction and characterized the slant helices by following: Proposition 1.1 Let γ be a unit speed curve with κ(s) ≠0.

Results

Conclusion