CURVE COUPLES AND SPACELIKE FRENET PLANES IN MINKOWSKI 3-SPACE

Abstract

Abstract. In this study, we have investigated the possibility ofwhether any spacelike Frenet plane of a given space curve in Minkowski3-space E 31 also is any spacelike Frenet plane of another space curvein the same space. We have obtained some characterizations of agiven space curve by considering nine possible case. 1. IntroductionIn the theory of curves in Euclidean space, one of the importantand interesting problem is characterization of a regular curve. In thesolution of the problem, the curvature functions k 1 (or {) and k 2 (or ˝)of a regular curve have an e ective role. For example: if k 1 = 0 = k 2 ,then the curve is a geodesic or if k 1 = constant 6= 0 and k 2 = 0;thenthe curve is a circle with radius (1=k 1 ), etc. Thus we can determine theshape and size of a regular curve by using its curvatures.Another way in the solution of the problem is the relationship betweenthe Frenet vectors of the curves (see [6]). For instance Bertrand curves:In 1845, Saint Venant (see [6] and [8]) proposed the question whetherupon the surface generated by the principal normal of a curve, a secondcurve can exist which has for its principal normal the principal normalof the given curve. This question was answered by Bertrand in 1850 ina paper which he showed that a necessary and sucient condition forthe existence of such a second curve is that a linear relationship withconstant coecients shall exist between the rst and second curvaturesof the given original curve. In other word, if we denote rst and second