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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