Intersection curve of two parametric surfaces in Euclidean n-space

Abstract

The aim of this paper is to study the differential geometric properties of the intersection curve of two parametric surfaces in Euclidean n-space. For this aim, we first present the mth order derivative formula of a curve lying on a parametric surface. Then, we obtain curvatures and Frenet vectors of the transversal intersection curve of two parametric surfaces in Euclidean n-space. We also provide computer code produced in MATLAB to simplify determining the coefficients relative to Frenet frame of higher order derivatives of a curve.

Implicit-implicit and parametric-implicit surface intersections in Euclidean n-space

“…The tangent vector fields of the surface M 2 are X u = 5u 4 , 1, 0, 0, 4u 3 , 3u 2 , X v = 0, 0, 1, 2v, 0, 0 . Then, since {X u , X v , e 1 , e 5 , e 6 } is linearly independent, we obtain the basis vectors of the normal space of M 2 as [11] N 4 , 0, 0, 320u 13 + 20u 7 , 240u 12 + 15u 6 . Also, we have…”

“…These intersection issues have been explored in 3-space using various techniques [2,10,13,19,21,22] and have been expanded and generalized to high dimensional spaces for the intersection of (n − 1) hypersurfaces in n-space [1, 3-9, 14, 15, 17, 20]. Recent research [11] has also investigated the parametricparametric surface intersection problem in n-dimensional Euclidean space. However, parametric-implicit and implicit-implicit intersection problems of two surfaces in n-space have not been studied.…”

“…While calculating the derivative vector β (r) given by ( 14), it is difficult to calculate the coefficient c 1 in high dimensional spaces. Therefore, this coefficient can be calculated using the MATLAB code given in [11].…”

“…It is obvious that the normal space of the surface M 1 is spanned by ∇G 1 , ∇G 2 , ..., ∇G n−2 . On the other hand, as expressed in [11], the normal space N 1 , N 2 , ..., N n−2 of the surface M 2 can be obtained by…”

“…By taking the dot product of both sides of V 1 = X u u ′ + X v v ′ with X u and X v , respectively, yields (see [11])…”