A Triple Product of Vectors in Four-Space

Williams, Michael Z.; Stein, F. Max

doi:10.1080/0025570x.1964.11975525

Cited by 15 publications

(7 citation statements)

References 2 publications

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“…, , be vectors in Euclidean 4space , where , is the standart basis vectors of The vector product of three vector is given by [11] .…”

Section: Preliminaries Definition 21 Letmentioning

confidence: 99%

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Osculating Curves According to the Extended Darboux Frame in E4

Bektaș

2023

J. Sci. Arts

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In this paper, we define osculating curves according to the extended Darboux frame (ED-frame), in Euclidean 4-space. Then, according to Case 1 and Case 2 ED-frame fields, the necessary and sufficient condition for a unit speed curve lying on the hypersurface to be congruent to an osculating curve is obtained. For these necessary and sufficient conditions, the solutions of the differential equations have been investigated. Finally, an application has been made on the subject.

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“…, , be vectors in Euclidean 4space , where , is the standart basis vectors of The vector product of three vector is given by [11] .…”

Section: Preliminaries Definition 21 Letmentioning

confidence: 99%

“…Considering equation ( 6), the differentiable function can be expressed in terms of curvatures and as follows: (11) ff equations ( 9)-( 11) are written in equation ( 5) and the resulting expression is edited, we obtained that for Case…”

Section: Casementioning

confidence: 99%

Osculating Curves According to the Extended Darboux Frame in E4

Bektaș

2023

J. Sci. Arts

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“…(2. 1) (see details in [22]). Let M 3 be a hypersurface in Euclidean 4-space whose parametrization is given by…”

Section: Preliminariesmentioning

confidence: 99%

Translation Hypersurfaces in Euclidean 4-Spaces

AKKILIN

Yüce

2022

Punjab Univ. j. math.

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In this article, the translation hypersurfaces in Euclidean 4- space are defined as the sum of three curves with distinct parameters with unit speed, and non-planar. These curves are called the generator curves of the hypersurface. Utilizing the hypersurface theory in Euclidean 4-space, unit normal vector field, shape (Weingarten) operator matrix, fundamental forms, Gaussian curvature and mean curvature have been expressed for the translation hypersurfaces. Finally, the computational example is stated to efficiency of the theoretical results.

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“…a linear combination of vectors u and v, [7,10]. In an analogous manner in E 4 1 we can express the quintuple vector product of vectors u, v, w, x, and y as…”

Section: Preliminariesmentioning

confidence: 99%

Spacelike intersection curve of three spacelike hypersurfaces in E41

Düldül

Çalışkan

2013

Annales UMCS, Mathematica

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Abstract. In this paper, we compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in four-dimensional Minkowski space E 4 1 . Introduction.The surface-surface intersection(SSI) is one of the basic problems in computational geometry. The main purpose here is to determine the intersection curve between the surfaces and to get information about the geometrical properties of the curve. Since the surfaces are mostly given by their parametric or implicit equations, three cases are valid for the SSI problems: parametric-parametric, implicit-implicit and parametricimplicit.There are two types of SSI problems: transversal or tangential. The intersection at the intersecting points is called transversal if the normal vectors of the surfaces are linearly independent, and is called tangential if the normal vectors of the surfaces are linearly dependent. The tangent vector of the intersection curve can be obtained easily by the vector product of the normal vectors of the surfaces in transversal intersection problems. Therefore, so many studies have recently been done about this type of problems. Hartmann [6], provides formulas for computing the curvatures of the intersection curves for all types of intersection problems in three-dimensional Euclidean space. Willmore [11], and using the implicit function theorem

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A Triple Product of Vectors in Four-Space

Cited by 15 publications

References 2 publications

Osculating Curves According to the Extended Darboux Frame in E4

Osculating Curves According to the Extended Darboux Frame in E4

Translation Hypersurfaces in Euclidean 4-Spaces

Spacelike intersection curve of three spacelike hypersurfaces in E41

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