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

Abstract

The 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 non-transversal 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,b1,b2) and the curvatures (k1,k2,k3) of the non-transversal intersection curve.