In this paper, by considering a Frenet curve lying on an oriented hypersurface, we extend the Darboux frame field into Euclidean 4-space E 4 . Depending on the linear independency of the curvature vector with the hypersurface's normal, we obtain two cases for this extension. For each case, we obtain some geometrical meanings of new invariants along the curve on the hypersurface. We also give the relationships between the Frenet frame curvatures and Darboux frame curvatures in E 4 . Finally, we compute the expressions of the new invariants of a Frenet curve lying on an implicit hypersurface.

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Abstract. In this paper, we compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in four-dimensional Minkowski space E 4 1 . Introduction.The surface-surface intersection(SSI) is one of the basic problems in computational geometry. The main purpose here is to determine the intersection curve between the surfaces and to get information about the geometrical properties of the curve. Since the surfaces are mostly given by their parametric or implicit equations, three cases are valid for the SSI problems: parametric-parametric, implicit-implicit and parametricimplicit.There are two types of SSI problems: transversal or tangential. The intersection at the intersecting points is called transversal if the normal vectors of the surfaces are linearly independent, and is called tangential if the normal vectors of the surfaces are linearly dependent. The tangent vector of the intersection curve can be obtained easily by the vector product of the normal vectors of the surfaces in transversal intersection problems. Therefore, so many studies have recently been done about this type of problems. Hartmann [6], provides formulas for computing the curvatures of the intersection curves for all types of intersection problems in three-dimensional Euclidean space. Willmore [11], and using the implicit function theorem

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a b s t r a c tThe aim of this paper is to compute all the Frenet apparatus of non-transversal intersection curves (hyper-curves) of three implicit hypersurfaces in Euclidean 4-space. The tangential direction at a transversal intersection point can be computed easily, but at a nontransversal intersection point, it is difficult to calculate even the tangent vector. If three normal vectors are parallel at a point, the intersection is ''tangential intersection''; and if three normal vectors are not parallel but are linearly dependent at a point, we have ''almost tangential'' intersection at the intersection point. We give algorithms for each case to find the Frenet vectors (t, n, b 1 , b 2 ) and the curvatures (k 1 , k 2 , k 3 ) of the non-transversal intersection curve.

In this paper, we extend the Darboux frame field along a non-null curve lying on an orientable non-null hypersurface into Minkowski space-time E 4 1 in two cases which the curvature vector and the normal vector of the hypersurface are linearly independent or dependent. Then the normal curvature, the geodesic curvature(s), and the geodesic torsion(s) of the hypersurface are given when the curve lying on the hypersurface is an asymptotic or geodesic curve.

In this paper, considering the extended Darboux frame in Euclidean 4-space, we define some special Smarandache curves. We calculate the Frenet apparatus of these curves depending on the invariants of the extended Darboux frame of second kind.