Canal Surfaces with Quaternions

Aslan, Selahattin; Yaylı, Yusuf

doi:10.1007/s00006-015-0602-5

Cited by 34 publications

(18 citation statements)

References 4 publications

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“…For a unit real-quaternion q, we have q −1 =q. For more information about real-quaternions see [11,12,14,17,15,6]. The division algebra of real-quaternions H is isomorphic to the Clifford algebra C 0,2 spanned by the basis {1, e 1 , e 2 , e 1 e 2 } in 2-dimension by identifying the quaternionic units i, j, k with e 1 , e 2 , e 1 e 2 = e 3 in C 0,2 , respectively.…”

Section: Real-quaternionsmentioning

confidence: 99%

Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space

Bekar¹

2019

International Electronic Journal of Geometry

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In this paper, a one-to-one correspondence between the set of unit split semi-quaternions and unit tangent bundle of semi-Euclidean plane is given. It is shown that the set of unit split semiquaternions based on the group operation of multiplication is a Lie group. The Lie algebra of this group, consisting of the vector space matrix of the angular velocity vectors, is also considered. Planar rotations in Euclidean plane are expressed using split semi-quaternions. Some examples are given to illustrate the findings.

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Section: Real-quaternionsmentioning

confidence: 99%

Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space

Bekar¹

2019

International Electronic Journal of Geometry

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“…If q = 1, then the quaternion q is unitary and the unitary quaternion can be written in the trigonometric form as q = cos θ + v sin θ, where v ∈ R 3 and v = 1 (see [1], [2], [10]).…”

Section: Preliminariesmentioning

confidence: 99%

“…Canal surfaces (especially pipe surfaces) have been applied to many fields, such as the solid and the surface modeling for CAD/CAM, construction of blending surfaces, shape re-construction and so on. One can see the details for geometric and applied fields of canal surfaces in [1], [7], [13], [14], [15], [18] and etc.…”

Section: Introductionmentioning

confidence: 99%

Embankment Surfaces in Euclidean 3-Space and Their Visualizations

Kazan

Karadağ

2019

Comm. Math. & Appl.

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In the present paper, we obtain the parametric representation of an embankment surface and give an example for it. We define the notions of embankmentlike surfaces and tubembankmentlike surfaces. Furthermore, we create some embankmentlike and tubembankmentlike surface examples with the aid of different directrix and draw these directrix and surfaces. Also, we find the Gaussian, mean and second Gaussian curvatures of these surfaces and draw the Gaussian, mean and second Gaussian curvature functions' graphics and the variations of Gaussian, mean and second Gaussian curvatures on related surfaces with the aid of Mathematica.

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“…Many studies on quaternionic representations of surfaces have been achieved. [12][13][14][15][16][17] These studies have examined quaternionic expressions of surfaces such as constant slope and canal surfaces in Euclidean and Minkowski spaces.…”

Section: Introductionmentioning

confidence: 99%

Surface‐dependent growth kinematics in euclidean 3‐space

Öncül

Özdemi̇r

Gök

2020

Math Methods in App Sciences

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In this paper, we investigate the surface‐dependent growth model in Euclidean 3‐space. The surface‐dependent model is developed for modeling the kinematics of surface growth for objects that can be generated by the curves on the surface, such as parasites and plants. This paper includes two main purposes for this model. The first is to parameterize this model using quaternions and homothetic motions, while expressing matrix representations of the surface‐dependent growth model. The second one is to construct the surface‐dependent growth model by using the growth velocity components related to the Darboux frame at each point of the generating curve. Moreover, to support the theory studied in the paper, various examples are illustrated.

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Canal Surfaces with Quaternions

Cited by 34 publications

References 4 publications

Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space

Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space

Embankment Surfaces in Euclidean 3-Space and Their Visualizations

Surface‐dependent growth kinematics in euclidean 3‐space

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