Characterization of Rectifying and Sphere Curves in ℝ3

Abstract

Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extra condition on the constant curvature, Y. L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s work. This project, uses a different approach to characterize these rectifying curves. This paper investigates two problems. The first problem relates to figuring out what we can say about a unit speed curve with nonzero curvature if every rectifying plane of the curve passes through a fixed point in ℝ3. Secondly, some formulas of curvature and torsion for sphere curves are identified. KEYWORDS: Space Curve; Rectifying Curve; Curvature; Torsion; Rectifying Plane; Tangent Vector; Normal Vector; Binormal Vector