Spacelike intersection curve of three spacelike hypersurfaces in E41

Abstract

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

Extended Darboux frame field in Minkowski space-time $mathbb{E}^4_1$

“…According to the transversal intersection of three hypersurfaces, T is the tangent vector of the intersection curve β . Using the similar calculations in [9], since the normal vectors of the hypersurfaces are orthogonal at the point β (s 0 ), for the first curvature of β at β (s 0 ) we find…”

Transversal Intersection Curves of Two Surfaces in Minkowski 3-Space

“…Some results on differential geometry of the intersection curves for the transversal intersections in Minkowski 3-space R 3 1 , and Minkowski 4-space R 4 1 , can be found in [6,22,16] and [13] respectively. Aléssio and Guadalupe [6] studied the transversal intersection curve of two parametric spacelike surfaces in R 3 1 considering parametric-parametric intersection problem.…”

“…Karaahmetoglu and Aydemir [16] studied intersection curve of two parametric spacelike and timelike surfaces in R 3 1 . Düldül and Ç alışkan [13] compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in 4-dimensional Minkowski space R 4 1 . Bonnor [10] described the geometry of null curves in a Minkowski space-time and he proved the fundamental existence and congruence theorems.…”

Differential geometry of non-transversal intersection curves of three implicit hypersurfaces in Euclidean 4-space

“…Transversal intersections in E 4 were studied by [11,[13][14][15][16][17][18]. Non-transversal intersection of three hypersurfaces with the normal vectors N i occurs in two different cases:…”