Non-transversal intersection curves of hypersurfaces in Euclidean 4-space

Sayed Abdel-Naeim Badr ^[1] ; Nassar H. Abdel-All ^[1] ; Osmar Aléssio ^[3] ; Mustafa Düldül ^[2] ; Bahar Uyar Düldül ^[2]
1. [1] Assiut University
  
  Assiut University
  
  Egipto
2. [2] Yıldız Technical University
  
  Yıldız Technical University
  
  Turquía
3. [3] Triangulo Mineiro Federal University, Brazil
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Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 288, Nº 1 (November 2015), 2015, págs. 81-98
Idioma: inglés
DOI: 10.1016/j.cam.2015.03.054
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Resumen
- In this paper, we consider the problem of how to compute the Frenet apparatus of the non-transversal intersection curves (hyper-curves) of three hypersurfaces (given by their implicit–implicit–parametric and implicit–parametric–parametric equations) in Euclidean 4-space. The non-transversal intersection (in which the normal vectors of the intersecting hypersurfaces are linearly dependent) includes two different cases. In each case, on the contrary to transversal intersections, we face some difficulties even finding the tangential direction. To overcome such difficulties, we give some algorithms for finding all Frenet vectors and curvatures of the intersection curve. Finally, we demonstrate our methods by giving several examples.