Intersection curves of hypersurfaces in

Abdel-All, Nassar H.; Badr, Sayed Abdel-Naeim; Soliman, M. A.; Hassan, Soad A.

doi:10.1016/j.cagd.2011.10.004

Cited by 14 publications

(7 citation statements)

References 9 publications

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“…Transversal intersections in E 4 were studied by [11,[13][14][15][16][17][18]. Non-transversal intersection of three hypersurfaces with the normal vectors N i occurs in two different cases:…”

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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“…Later on, they emphasized on the marching method [7] for solving problems which offer remarkable advantages, but still their approach is facing some problems because of complicated initial and boundary values. In the same way, an approach in [8] worked on solving differential geometry problems of hypersurfaces; also, they are doing this research [9] by increasing dimensions of surfaces which take more time for showing results. Extracting boundary and turning points of parametric surface intersection curves by the GK method are discussed in [10,11].…”

Section: Introductionmentioning

confidence: 99%

Optimal Intersection Curves for Surfaces

Gao¹,

Sarfraz

Irshad

et al. 2021

Journal of Mathematics

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In this article, an algorithm has been established to approximate parametric-parametric, explicit-implicit, and explicit-explicit surface intersection. Foremost, it extracts the characteristic points (boundary and turning points) from the sequence of intersection points and fits an optimal cubic spline curve to these points. Moreover, this paper utilizes genetic algorithm (GA) for optimization of shape parameters in the portrayal of cubic spline so that the error is minimal. The proposed algorithm is demonstrated with different types of surfaces to analyze its robustness and proficiency. In the end, all illustrations show the effectiveness of the algorithm which makes it more influential to resolve all complexities arises during intersection with a minimal error.

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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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Intersection curves of hypersurfaces in

Cited by 14 publications

References 9 publications

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

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

Optimal Intersection Curves for Surfaces

Intersection curve of two parametric surfaces in Euclidean n-space

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