Developable surfaces along frontal curves on embedded surfaces

Honda, Shun’ichi; Izumiya, Shyūichi; Takahashi, Masatomo

doi:10.1007/s00022-019-0485-z

Cited by 7 publications

(11 citation statements)

References 6 publications

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“…The approximation of a given surface by a flat surface (developable) along the arbitrary curve is called flat or developable approximation. An osculating developable is tangent to the surface along a curve and it gives a flat approximation of the surface along the curve [17]. The osculating cylinder (22) inherits this property and gives flat approximation of the surface along a helical geodesic.…”

Section: Generalized Cylinder With Geodesic Base Curvementioning

confidence: 99%

“…Approximating by an osculating cylinder (22) has nice geometrical properties: globally free of singularities, achieved along a geodesic path, some geometric quantities on both cylinder and the surface along the geodesic curve are congruent up to orientation (2.2), locally like approximating by the ribbon [28], and the cylinder is easy to modeling and reasoning from a mathematical or manufacturing viewpoint. For more details about the flat approximations of surfaces and hypersurfaces see [16,17,29].…”

Section: Generalized Cylinder With Geodesic Base Curvementioning

confidence: 99%

“…They showed that an osculating Darboux vector field has a constant direction if and only if the osculating developable surface is a generalized cylinder. An osculating developable surface is a ruled surface whose rulings are directed by the osculating Darboux vector field along the curve, it is tangent to the surface along the curve and it gives a flat approximation of the surface along the curve [17]. An osculating Darboux vector was studied in [18,19], it characterizes the helical geodesic on the osculating cylinder as shown in this paper.…”

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

Generating Generalized Cylinder with Geodesic Base Curve According to Darboux Frame

Althibany

2021

Journal of New Theory

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This paper aims to design a generalized cylinder with a geodesic base curve according to the Darboux frame in Euclidean 3-space. A generalized cylinder is a special ruled surface that is constructed by a continuous fixed motion of a generator line called the ruling along a given curve called the base curve. The necessary and sufficient conditions for the base curve to be geodesic are studied. The main results show that the generalized cylinder with a geodesic base curve is an osculating cylinder whose base curve is a helical geodesic, and the rulings are directed by the unit osculating Darboux vector.

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Section: Generalized Cylinder With Geodesic Base Curvementioning

confidence: 99%

Section: Generalized Cylinder With Geodesic Base Curvementioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Generating Generalized Cylinder with Geodesic Base Curve According to Darboux Frame

Althibany

2021

Journal of New Theory

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“…Luca et al [17] used two classic methods of approximate development to spherical surfaces. Honda et al [18] considered two types of developable surfaces along a frontal curve on an embedded surface in the Euclidean 3-space, and presented new invariants of the frontal curve which characterize singularities of the developable surfaces. Erdem et al [19] proposed an efficient algorithm for improving flattening result of triangular mesh surface patches having a convex shape, this method based on barycentric mapping technique, incorporated a dynamic virtual boundary, which considerably improved initial mapping result.…”

Section: Introductionmentioning

confidence: 99%

A free‐form surface flattening algorithm that minimizes geometric deformation energy

Zheng

Liu

Lou

et al. 2022

IET Image Processing

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Based on the analysis of the advantages and disadvantages of existing surface flattening algorithms, this paper proposes a free‐form surface flattening algorithm that minimizes geometric deformation by combining the advantages of the geometric flattening method and the mechanical energy flattening method to address the problems of many iterations, large changes in convergence, and weak visualization of deformation. The point cloud surface is meshed using the triangular slice search method, and the 3D surface mesh is wrapped around the surface using geometric mapping relationships. The initial correction and deformation analysis of the ring flatten graphic are carried out according to the average flattening error, and a global geometric deformation energy model is established to obtain the energy‐minimizing unfolding conditions and optimize the initial flattening graph by iterative optimization. The verification of the algorithm shows that the algorithm is general, has good robustness, high flattening accuracy, and visualizes the flattening deformation. The method is suitable for some design occasions of machining, such as the flattening calculation of automobile outer cladding parts and sheet metal parts. It is especially suitable for the flattening calculation of ring‐like parts, and is a good guide for the design calculation and stamping process of sheet‐metal‐like parts.

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“…There are many studies interested with many properties and some characterizations of the concepts ruled and developable surfaces in , e.g. [4][5][6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Developable Surfaces Generated by Using Moving Frame

2022

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In this paper, we define three different rotation-minimizing frames by rotating the moving frame around its coordinate axis vectors. Darboux vectors associated with these frames are obtained as special cases of the Darboux vector associated with the moving frame. Using these Darboux vectors and the moving frame we define six different developable surfaces. For each of these surfaces we give two invariants of curves on these surfaces to characterize their singularities. Moreover, we show that the base curves of these surfaces are contour generators with respect to an orthogonal projection or a central projection if and only if one of the invariants given for each surface is constantly equal to zero. Examples are provided to illustrate our theorems and results.

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Developable surfaces along frontal curves on embedded surfaces

Cited by 7 publications

References 6 publications

Generating Generalized Cylinder with Geodesic Base Curve According to Darboux Frame

Generating Generalized Cylinder with Geodesic Base Curve According to Darboux Frame

A free‐form surface flattening algorithm that minimizes geometric deformation energy

Developable Surfaces Generated by Using Moving Frame

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