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

Abstract

In this paper, a new type of quaternionic partner curves is defined as generalized quaternionic involute‐evolute curves or false(0,2false)‐quaternionic involute– false(1,3false)‐quaternionic evolute curves in the four‐dimensional Euclidean space. The relations between the Frenet frames and curvatures of the quaternionic involute‐evolute curve couple are introduced. Moreover, the necessary and sufficient conditions for a quaternionic curve to have a generalized involute are obtained.

A primer on the differential geometry of quaternionic curves

“…There are several former studies on this theme, [16][17][18][19][20] and their development is much more complicated than ours. In this section, we provide simple solutions for these solutions using a generalization to quaternionic curves that are analogous to the complex formulation for plane real curves.…”

“…To the best of our knowledge, this approach has never been reported, although two-dimensional curves within a real space have a simple parametrization with complex numbers, 15 and here we closely follow this approach. The example of evolutes and evolvents presented here shows that our proposal is much simpler than the current approach, [16][17][18][19][20] based in Bharathi and Nagaraj. 1 Before developing the differential geometric approach, in the next section, we present the most important features of quaternions that will be used in this article.…”

Involute–evolute and pedal–contrapedal curve pairs on S2

“…The curves constructed by the Frenet vectors of the given curves are important parts of the study. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] As a kind of well-known associated curves, involute-evolute curve pairs in different space forms were studied by many mathematicians. [5][6][7][8]10,16 Another kind of curves called pedal-contrapedal curves was also studied.…”

Special Associated Curves in Euclidean 3-Space

“…Helix, slant helix, plane curve, spherical curve, etc. are well-known instance of single special curves [1,9,10,13,18] and these curves, exclusively the helices, are used in many applications [2,7,8,15]. Additionally, special curves can be defined by careful Frenet planes.…”

“…Besides, special curve pairs are characterized by some relationships between their Frenet vectors or curvatures. Involute-evolute curves, Bertrand curves, Mannheim curves are admitted sample of curve pairs and studious by some mathematicians [3,[11][12][13]16,17].…”