Geometric Interpretation of det(C^(3),C^(4),C^(5))=0 in E13

TL;DR

This paper explores the geometric properties of curves in Minkowski 3-space, characterizing null slant helices and tangent indicatrices through determinant conditions involving higher derivatives.

Contribution

It provides a novel determinant-based characterization of null slant helices and tangent indicatrices in Minkowski 3-space, extending previous geometric analyses.

Findings

Null slant helices characterized by det(C^(3),C^(4),C^(5))=0

Tangent indicatrix characterized by the same determinant condition

Results apply to both null and non-null curves with specific curvature conditions

Abstract

In this paper, we investigate the tangent indicatrix of the curve C with constant curvature. Tangent indicatrix of the curve C is characterized with det(C^(3),C^(4),C^(5))=0 in Minkowski 3-space E13. Moreover, we study null slant helices using the determinant approach and give the following characterization: A curve C is a null slant helix in E13 if and only if det(C^(3),C^(4),C^(5))=0. Then similar results are obtained for non-null curves with the condition k=1.