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

Badr, Sayed Abdel-Naeim; Abdel-All, Nassar H.; Aléssio, Osmar; Düldül, Mustafa; Düldül, Bahar Uyar

doi:10.1016/j.cam.2015.03.054

Cited by 7 publications

(3 citation statements)

References 24 publications

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“…Recently, we study the non-transversal intersection of parametric-parametric-parametric (PPP) hypersurfaces [19] and the non-transversal intersection of implicit-implicit-parametric (IIP), implicit-parametric-parametric (IPP) hypersurfaces [20] in E 4 (we review the previous studies given for intersection problems in these publications).…”

Section: Introductionmentioning

confidence: 99%

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

Aléssio

Düldül

et al. 2016

Journal of Computational and Applied Mathematics

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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.

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Section: Introductionmentioning

confidence: 99%

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

Aléssio

Düldül

et al. 2016

Journal of Computational and Applied Mathematics

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“…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]. On the other hand, by using the wedge product of two vectors in (n + 1)-dimensions, Goldman [12] derived a closed formula for the first curvature of the transversal intersection of n-implicit hypersurfaces in (n + 1)-dimensions.…”

Section: Introductionmentioning

confidence: 99%

Intersection curve of two parametric surfaces in Euclidean n-space

Düldül

Özçetin

2021

ACUTM

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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.

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“…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].…”

Section: Introductionmentioning

confidence: 99%