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

Abstract

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.

Intersection curve of two parametric surfaces in Euclidean n-space

“…These studies have been extended to E 4 for the intersection of three parametric hypersurfaces [6,9,10,14] and to E 5 for the intersection of four parametric hypersurfaces [15]. Since surfaces can also be defined by their implicit equations, differential geometry of the intersection curve of two implicit surfaces has been studied in E 3 by [2,12,20,21] and of three implicit hypersurfaces has been studied in E 4 by [3,4,7,14,19]. There also exist some studies for the intersection curves of different type surfaces in E 3 [10,18,21] and in E 4 [1,8,10,14].…”

Transversal Intersection Curves of Two Surfaces in Minkowski 3-Space

“…A general reference including many topics in semi-Riemannian geometry is the classical book [19]. Differential geometry of the intersection curves in R 3 and R 4 can be found in [14,26,23,1,15,7,2,4,8,5,3].…”