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We propose new approaches to the investigation of tubular and canal surfaces, regarded as swept surfaces, we give a parametric representa-tion of the inverse of a canal surface and we suggest several applications of tubular surfaces in scientific vizualization.

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## Citations

... Then, the ESP is the path connecting P P P and Q Q Q, which has the minimum length and has to be through a given tubular space. The mathematical definition of tubular space is given as follows [35]: ...

We propose a novel algorithm to determine the Euclidean shortest path (ESP) from a given point (source) to another point (destination) inside a tubular space. The method is based on the observation data of a virtual particle (VP) assumed to move along this path. In the first step, the geometric properties of the shortest path inside the considered space are presented and proven. Utilizing these properties, the desired ESP can be segmented into three partitions depending on the visibility of the VP. Our algorithm will check which partition the VP belongs to and calculate the correct direction of its movement, and thus the shortest path will be traced. The proposed method is then compared to Dijkstra’s algorithm, considering different types of tubular spaces. In all cases, the solution provided by the proposed algorithm is smoother, shorter, and has a higher accuracy with a faster calculation speed than that obtained by Dijkstra’s method.
Keywords: Euclidean shortest path; tubular space; reactive algorithm; visibility; oriented drilling process; Dijkstra’s algorithm

... The authors [4] have studied some characterizatons of the tubular surfaces generated by non-null curves in Minkowski 3−space. Blaga [11] has presented a new approach to the tubular surfaces and provided CAD applications. Arslan et.al. ...

... Best known examples of them: Karacan and Yaylı [11] examined geodesics and some properties of tubular surfaces in Minkowski 3−space, Tuncer [10] studied about the linear Weingarten conditions for tubular surfaces by using their Gaussian and mean curvatures in Galilean and pseudo-Galilean 3−spaces and the authors examined the properties of the tube surfaces according to Bishop and Darboux frames in [4,5]. Blaga [2] handled the tubular surfaces as swept surfaces and put forward the new approaches to the investigation of tubular and canal surfaces. Moreover, he gave some applications of tubular surfaces in scientific visualization. ...

... By using the eq. 2 for the surface X (s, θ) and since r > 0, we obtain the condition in (i). ...

In this paper, we study tubular surfaces whose centers are semi-spherical indicatrices of a spatial curve using the alternative moving frame \( \left\{ N,C,W\right\} \). The first objective is to examine the basic properties of these surfaces. Then, we investigate the condition of the parameter curves on these surfaces to be asymptotic and geodesic curves. Finally, we present the graphs of some related examples using the Mathematica.

... Blaga [2] used tubular surfaces as swept surfaces and put forward new approaches for the investigation of tubular and canal surfaces in addition to showing applications of tubular surfaces in scientific visualization. ...

A basic goal of this paper is to investigate the tubular surface constructed by the spherical indicatrices of any spatial curve in the Euclidean 3-space. This kind of tubular surface is designed for the alternative moving frame (N.C.W.) in conjunction with finding a relationship between the tubular surfaces and their special curves, such as geodesic curves, asymptotic curves, and minimal curves. The minimal curve γ on a surface is defined by the property that its fundamental coefficients satisfy Eq. (3:7) along the curve γ. At the end of this article, we exemplify these curves on the tubular surfaces with their figures using the program Mathematica.

... The geometry of the Galilean space G 3 has been treated in detail in O. Roschl's habilitation in 1984 [17]. More about Galilean space and Pseudo-Galilean space may be found in [20,1,3,7,11,12,21]. The Galilean space G 3 is a Cayley-Klein space equipped with the projective metric of signature (0, 0, +, +), as in [21]. ...

... Some tube-like surfaces associated with focal curve of helices with r = 1, t = 0, Left: u ∈ [0, π], v ∈ [0,3 2 π], Middle: u ∈ [0,13 10 π], v ∈ [0, 2π] and Right: u ∈ [0, 2π], v ∈ [0, 2π]. ...

In this paper, we study inextensible flows of focal curves associated with tube-like surfaces in Galilean 3-space G3. We give some characterizations for curvature and torsion of focal curves associated with tube-like surfaces in Galilean 3-space G3. Furthermore, we show that if flow of this tube-like surface is inextensible then this surface is not developable as well as not minimal. Finally an example of tube-like surface is used to demonstrate our theoretical results and graphed.

... For example; Takashi Maekawa et al. [15] concerned with the question of when a pipe surface is nonsingular. More precisely, given a regular space curve α(s) they investigated the maximum R > 0, so that the pipe surface is nonsingular, whenever r < R. Blaga [3] handled the tubular surfaces as swept surfaces and he was put forward new approaches to the investigation of tubular and canal surfaces. Then, he gave several applications of tubular surfaces in scientific vizualization. ...

A canal surface is a surface constructed as the envelope of a family of spheres with the parametric (non constant radii) radii r(s) and a space curve \({\alpha (s)}\) called its center. If the radii of the generating spheres are constant r then the canal surface is called tubular surface such as right circular cylinder, torus, right circular cone, surface of revolution. The quaternions are number systems generalized by the complex numbers. Irish mathematician Hamilton (Lond Edinb Dublin Philos Mag J Sci 25(3):489–495, 1844) discovered them in 1843 and applied to mechanics in three-dimensional space. Many laws of curves and surfaces in differential geometry used quaternions such as quaternionic helices (Çöken and Tuna in Appl Math Comput 155:373–389, 2004; Gök et al. in Appl Clifford Algebras 21:707–719, 2011; Kahraman et al. in Appl Math Comput 218:6391–6400, 2012) and canal surfaces (Aslan and Yaylı in Adv Appl Clifford Algebras, 2015; Babaarslan and Yaylı in ISRN Geom 2012:Article ID 126358, p 8, 2012). This paper has two purposes. The first purpose is to form canal surfaces whose centers are spherical indicatrices of a spatial curve with a new idea in terms of alternative moving frame \({\{N,C,W\}}\) The second purpose is to obtain canal surfaces by using the quaternion product and matrix representation. Moreover, we give some related examples with their figures.

... Galilean space is the space of Galilean Relativity. More about Galilean space and pseudo-Galilean space may be found in [1][2][6][7]9,[11][12]. ...

At first, the definition of tubular surfaces in Galilean space is given. Then, differential properties of tubular surfaces are obtained. Consequently, we prove that tubular surfaces in Galilean space are Weingarten surfaces.

This paper deals with generalized tube surfaces (GTs) and their geometric properties in pseudo‐Galilean 3‐space. We classify these surfaces into two types. We firstly compute the first and second fundamental forms to investigate geometric properties of a GT. Then, we obtain the condition for such a surface to be minimal and present some results which express the conditions for which parameter curves on a GT are geodesics, asymptotics, or lines of curvature. Furthermore, we show how to form GTs by using split semi‐quaternions or their matrix representations. Finally, as an application, we introduce generalized magnetic flux tubes in pseudo‐Galilean 3‐space and obtain the local magnetic field components of such surfaces. The theory studied in the paper is supported by illustrated examples.

Fundamental Journal of Mathematics and Applications (FUJMA) is an international and peer-reviewed journal which publishes high quality papers on pure and applied mathematics. To be published in this journal, a paper must contain new ideas and be of interest to a wide range of readers.

In this study, tubular surfaces that play an important role in technological designs in various branches are examined for the case the base curve is not satisfying the fundamental theorem of the differential geometry. In order to give an alternative perspective to the researches on tubular surfaces, the modified orthogonal frame is used in this study. Firstly, the relationships between the Serret-Frenet frame and the modified orthogonal frame are summarized. Then the definitions of the tubular surfaces and some theorems and results are given. Moreover, the fundamental forms, the mean curvature and the Gaussian of the tubular surface are calculated according to the modified orthogonal frame. Finally, the properties of parameter curves of the tubular surface with modified orthogonal frame are expressed and the tubular surface, the center curve of which is eight curves, is drawn according to the Frenet frame and the modified orthogonal frame.