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There is More than One Way to Frame a Curve

Taylor & Francis
The American Mathematical Monthly
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Richard L. Bishop at University of Illinois, Urbana-Champaign
Richard L. Bishop
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CALCULATION OF RPAF’S 13
References
[B] R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, vol. 82(1975), pp. 246–251
... Bishop frame was defined by Bishop. Thanks to this frame, Bishop frame is used as an alternative frame for situations where the Frenet frame cannot be defined (especially where the second derivative of the curve is zero) [12]. Hanson and Hui investigated Bishop frame as a quaternion. ...
... where ∠(N (s), N 1 (s)) = θ [12]. Let k 1 (s) and k 2 (s) be Bishop curvatures. ...
SPECIAL CANAL SURFACES WITH BISHOP FRAME AND QUATERNIONS
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  • May 2024
The purpose of this paper is to show that canal and tubular surfaces can be obtained by special curves. Also, we reckon the equations of these surfaces with Bishop frame and acquire some corollaries for special curves. Besides, these surfaces are acquired by quaternion and homothetic motion.
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... In the 3D space called Euclidean R3, the Frenet frame is a vital coordinate system that aligns with a smooth, continuous curve. This frame is composed of three distinct, perpendicular unit vector fields: the tangent vector (T), the normal vector (N), and the binormal vector (B) [90]. The tangent vector (T) signifies the direction in which the curve is moving, the normal vector (N) points towards the curve's principal normal, and the binormal vector (B) stands perpendicular to both T and N. ...
Deep Learning for Mandibular Canal Segmentation in Digital Dental Radiographs: A Systematic Literature Review
Article
Full-text available
  • Jan 2024
italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Context : Several recent studies in dentistry and maxillofacial imaging have concentrated on the mandibular canal (MC) segmentation in digital dental radiographs. In this research domain, deep learning approaches have demonstrated promising outcomes. Objective : This systematic literature review (SLR) aims to comprehensively analyze and synthesize the recent advancements in applying deep learning techniques for MC segmentation in digital dental radiographs. Method : This review encompasses studies published between 2018 and 2023, sourced from reputable databases, including PubMed, ScienceDirect, IEEE Xplore, and Google Scholar. Results : This study identified 30 primary research papers focusing on MC segmentation in digital dental radiographs. The review categorizes papers into two groups based on the digital dental radiograph type. The first group uses a 2D digital dental radiograph from a panoramic radiograph and 2D Cone Beam Computed Tomography (CBCT) scans. The second group uses 3D datasets from volumetric data from CBCT scans. Conclusion : The synthesized knowledge from this review is intended to guide researchers, dentists, and oral surgeons in leveraging deep learning advancements in MC segmentation for oral and maxillofacial surgery. Prior studies have faced challenges, including limited public datasets, variations in MC anatomy, time consumption and observer variability in MC annotation, complexities in deep learning models, and lack of practical implementation. To overcome these challenges, it is suggested that more public datasets be collaboratively created and shared within the research community, focusing on handling anatomy variability, improving digital dental radiograph quality, streamlining annotation processes through automated tools, and simplifying deep learning models for practical implementation.
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... Some of these alternative frames are the Bishop frame (rotation minimizing frame), the Flc (Frenet-like curve) frame, the {N, C, W} frame, etc. The Bishop frame [2] is also very suitable for engineering applications [10] because of being defined at points where even curvatures vanish. That's why many studies have been done by using this frame. ...
On some new frames along a space curve and integral curves with Darboux q-vector fields in $ \mathbb{E}^3
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  • Jan 2024
In this paper, to start we defined osculating q-frame, normal q-frame, and rectifying q-frame along a space curve in Euclidean 3-space $ \mathbb{E}^3 $ by using the Darboux vector field of the q-frame. We obtained the derivative equations of these new frames. Later, we defined some new integral curves of a space curve and called them $ \overline{\mathsf{d}}_o $-direction curve, $ \overline{\mathsf{d}}_n $-direction curve and $ \overline{\mathsf{d}}_r $-direction curve. Finally, we gave some theorems and results related with these curves.
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... The frame is usually denoted adapted if, at each parameter value, f 1 (t) coincides with the unit tangent t(t) = r ′ (t)/|r ′ (t)| to the curve, while f 2 (t), f 3 (t) span the normal plane orthogonal to f 1 (t). To avoid unnecessary rotations in the normal plane, among the family of adapted frames, we consider motions where the frame is rotation-minimizing (RMF) [1]. As well as the Frenet frame, defined by t, together with the normal and binormal unit vectors of the curve r, the RMF is completely determined by r, except for a constant rotation in the plane normal to the curve tangent. ...
Piecewise rational rotation-minimizing motions via data stream interpolation
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When a moving frame defined along a space curve is required to keep an axis aligned with the tangent direction of motion, the use of rotation-minimizing frames (RMF) avoids unnecessary rotations in the normal plane. The construction of rigid body motions using a specific subset of quintic curves with rational RMFs (RRMFs) is here considered. In particular, a novel geometric characterization of such subset enables the design of a local algorithm to interpolate an assigned stream of positions, together with an initial frame orientation. To achieve this, the translational part of the motion is described by a parametric $G^1$ spline curve whose segments are quintic RRMFs, with a globally continuous piecewise rational rotation-minimizing frame. A selection of numerical experiments illustrates the performances of the proposed method on synthetic and arbitrary data streams.
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... In order to deal with more general energies, both for the boundary curve and for the membrane, in Section 3 we use the framed curve approach introduced by Bishop [15]. We introduce a moving orthonormal frame {t, n, b} ∈ W 1,p ((0, 2π); SO(3)) with p > 1 which generates a curve r by integration and we define the energy in terms of the frame as ...
Elastic membranes spanning deformable curves
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We perform a variational analysis of an elastic membrane spanning a closed curve which may sustain bending and torsion. First, we deal with parametrized curves and linear elastic membranes proving the existence of equilibria and finding first‐order necessary conditions for minimizers computing the first variation. Second, we study a more general case, both for the boundary curve and for the membrane, using the framed curve approach. The infinite dimensional version of the Lagrange multipliers' method is applied to get the first‐order necessary conditions. Finally, a numerical approach is presented and employed in several numerical test cases.
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... Associated curves bring important geometric definitions to fields of differential geometry, physics, and mathematics in explanation of the behavior of curves and surfaces and in work of particle motion in ordinary space. Also, concepts such as magnetic curves, Bertrand curves, spherical curves, and involute-evolute curve pairs have been widely investigated in fields of differential geometry (do Carmo 1976;Bishop 1975;Bukcu and Karacan 2008;Bükcü and Karacan 2009a;Karacan and Bukcu 2007a;Bükcü and Karacan 2009b;Bukcu 2007b, 2008;Maluf and Faria 2008;Körpınar et al. 2021c). ...
Conformable modeling of normalization and recursional electromagnetic fields of spacelike magnetic curves
Article
Full-text available
  • Mar 2024
  • OPT QUANT ELECTRON
In this paper, we investigate spacelike magnetic curves according to Bishop frame. Firstly, we present conformable derivatives of Lorentz magnetic fields of these magnetic curves. Moreover, we calculate the conformable derivatives of the normalization and recursional electromagnetic vector fields. Finally, we give conformable energies of normalization and recursional electromagnetic fields related to spacelike magnetic curves.
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SMOF-Based Torsion Compensated Bishop Algorithm for Medical Shape Measurement
Article
  • Jun 2024
This paper proposes a novel algorithm for threedimensional shape reconstruction and torsion compensation using a spun multi-core optic fiber (SMOF). The torsion-compensation Bishop algorithm (TCBA) addresses torsion in SMOFs and enables accurate coordinate correction. Experimental validation is conducted by arranging the SMOF sensor in controlled helical shapes. The reconstruction accuracy is increased by 10.87%. Additionally, the mean error is reduced from 7.21 mm to 2.50 mm. The algorithm can be used for navigation in flexible robotics and minimally invasive surgery, where precise positioning and manipulation are crucial.
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A NEW APPROACH TO HYPERBOLIC SPINOR B-DARBOUX EQUATIONS
Article
  • Mar 2024
In this research, spinor descriptions of the curves in surfaces have been given according to the B-Darboux frame in Lorentzian 3-space E_1^3. The relations between B-Darboux and Darboux frames have been specified via their spinor conceptions which expressed in both timelike and spacelike surfaces, separately. Additionally, all these spinor representations have been portrayed in the view of the B-Darboux frame (via the curvatures) in Lorentzian 3-space. The findings are supported by some theorems and corollaries.
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ASSOCIATED CURVES ACCORDING TO BISHOP FRAME IN 4-DIMENSIONAL EUCLIDEAN SPACE
Article
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Many studies have been doneaccording to different frames of the theory of curves in Euclidean Space. Many scientists have studied frames such as the Frenet frame, Bishop frame, and Adapted frame in this theory. These frames help us in the characterization of curves.In this study, associated curves with the Frenet curve according to the Bishop frame in 4-dimensional Euclidean space are investigated. Direction and rectifying curves of the Frenet curve according to this frame are given.
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Bishop Frames of Salkowski Curves in E3
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In this study, alternative, type-1 Bishop, type-2 Bishop and N-Bishop frames of Salkowski curves in E3 are calculated. Moreover, curvatures, Darboux and pol vectors of these frames are found. Also, relationships between the Bishop frames, Darboux vectors and pole vectors are given.
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