On Special Kinds of Involute and Evolute Curves in 4-Dimensional Minkowski Space

Abstract

Recently, extensive research has been done on evolute curves in Minkowski space-time. However, the special characteristics of curves demand advanced level observations that are lacking in existing well-known literature. In this study, a special kind of generalized evolute and involute curve is considered in four-dimensional Minkowski space. We consider (1,3)-evolute curves with respect to the casual characteristics of the (1,3)-normal plane that are spanned by the principal normal and the second binormal of the vector fields and the (0,2)-evolute curve that is spanned by the tangent and first binormal of the given curve. We restrict our investigation of (1,3)-evolute curves to the (1,3)-normal plane in four-dimensional Minkowski space. This research contribution obtains a necessary and sufficient condition for the curve possessing the generalized evolute as well as the involute curve. Furthermore, the Cartan null curve is also discussed in detail.

Pedal Curves of Non-Lightlike Curves in Minkowski 3-Space

“…Since Minkowski space was proposed, it has been studied by researchers domestically (cf. [13][14][15][16][17]). Bakurová introduced pedal curves in Minkowski plane in [18].…”

On Involute-Evolute Curve Pair in Semi-Euclidean Space

“…Hanif et al presented a special kind of generalized involute and evolute curve pair in 4-dimensional Minkowski space by considering this curve pair in a different way. [9].…”

Generalized quaternionic involute‐evolute curves in the Euclidean four‐space E4

“…Moreover, Öztürk et al have studied involute‐evolute curves in ‐dimensional Euclidean space . Later, Hanif et al have given some important characterizations of generalized involute‐evolute curves in Euclidean space and Lorentzian space …”

“…13 Later, Hanif et al have given some important characterizations of generalized involute-evolute curves in Euclidean space E 4 and Lorentzian space E 4 1 . 14,15 In this paper, we define quaternionic involute-evolute curves in the four-dimensional Euclidean space. We consider the parallel planes spanned by Frenet quaternions of quaternionic curves and define generalized quaternionic involute-evolute curve couples, which will be called (0, 2) involute-(1, 3) evolute curve couple.…”