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


ALESSIO O., DÜLDÜL M., Duldul B. U., ABDEL-ALL N. H., BADR S. A.

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, vol.308, pp.20-38, 2016 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 308
  • Publication Date: 2016
  • Doi Number: 10.1016/j.cam.2016.05.011
  • Journal Name: JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.20-38
  • Keywords: Implicit-implicit-implicit intersection, Tangential intersection, Geometric properties, Non-transversal intersection, Implicit curves, GEODESIC TORSION, CURVATURE, SURFACES, FORMULAS, R-4
  • Yıldız Technical University Affiliated: Yes

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, b(1), b(2)) and the curvatures (k(1), k(2), k(3)) of the non-transversal intersection curve. (C) 2016 Elsevier B.V. All rights reserved.