Bézier developable surfaces

Fernández–Jambrina, L.

doi:10.1016/j.cagd.2017.02.001

Cited by 8 publications

(7 citation statements)

References 15 publications

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“…As expected, when ν ≡ 0, no term along the projection direction appears and then σ ≡ 1. In this case, we recover the results for Bézier developable surfaces [19].…”

Section: Reparametrisation Of Rational Ruled Surfacessupporting

confidence: 78%

“…In [17,18] Bézier developable patches are constructed by applying affine transformations to the first cell of the control net of the patch. It is shown in [19] that this construction produces all Bézier developable surfaces with a polynomial edge of regression. This construction has been extended to spline developable surfaces [20,21] and to Bézier triangular surfaces [22].…”

Section: Introductionmentioning

confidence: 99%

“…such as control points, knots and weights. It would be interesting hence to extend Aumann's approach [17,19] from polynomial to rational developable surfaces in order to comply with the whole NURBS framework. The main advantage of this approach is the use of the elements which are used in CAD applications.…”

Section: Introductionmentioning

confidence: 99%

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Characterisation of rational and NURBS developable surfaces in Computer Aided Design

Fernandez-Jambrina

2020

Preprint

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In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions Λ, M , ν. Properties of developable surfaces are revised in this framework. In particular, a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions Λ, M , ν, which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative. It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant Λ, M , ν . The results are readily extended to rational spline developable surfaces.

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“…As expected, when ν ≡ 0, no term along the projection direction appears and then σ ≡ 1. In this case, we recover the results for Bézier developable surfaces [19].…”

Section: Reparametrisation Of Rational Ruled Surfacessupporting

confidence: 78%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Characterisation of rational and NURBS developable surfaces in Computer Aided Design

Fernandez-Jambrina

2020

Preprint

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“…Fernández-Jambrina proposed a linear algorithm to construct B-spline control nets for spline developable where five free parameters were applied to control vertices [12]. Developable surfaces were constructed with Bézier patches using Aumann's algorithm to form a polynomial edge of regression [13].…”

Section: Surface Flattenability Improvementmentioning

confidence: 99%

Improvement of flattenability using particle swarm optimizer for surface unfolding in bolus shaping

Peng

Ingleby

et al. 2020

SN Appl. Sci.

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Bolus, formed by a sheet of material, covers on tumors of patient skins to generate the desired dose distribution in a process of the high-energy radiotherapy. The existing methods of bolus shaping use a manual process to cut the commercial material into the shape, which can cause air gaps between the patient skin surface and bolus in the reduced effect of cancer care treatment. A method of automatic bolus shaping can improve the accuracy and reduce air gaps of bolus shaping, which uses a process of first segmenting 3D patient skin surface models into several patches and then unfolding the patches into 2D patterns for material cutting before folding the cut sheet into the bolus. However, air gaps between the bolus and patient skin surface cannot be directly evaluated to find the bolus accuracy. This paper proposes a method to improve model flattenability and evaluate the air gaps using a particle swarm optimizer (PSO). A 3D surface shape is unfolded only if the air gap and surface flattenability meet the requirement of accuracy. Otherwise, the surface will be segmented into several patches to improve the flattenability and reduce air gaps. The objective and constraint are identified to search for an optimal solution for the 3D surface with a high flattenability. A strategy of the dimension reduction is proposed for the selection of local nodes on a meshed surface to increase the searching efficiency of surface flattenability. A local node selection based PSO (L-PSO) method is developed to search for the optimal solution. The proposed method is verified in two case studies of bolus shaping.

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“…Finally, in [24] it is shown that the developable surfaces which can be constructed with Aumann's algorithm are the ones with a polynomial edge of regression. This poses an interesting problem.…”

Section: Introductionmentioning

confidence: 99%

Developable surface patches bounded by NURBS curves

Fernandez-Jambrina,

Perez-Arribas

2019

Preprint

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In this paper we construct developable surface patches which are bounded by two rational or NURBS curves, though the resulting patch is not a rational or NURBS surface in general. This is accomplished by reparameterizing one of the boundary curves. The reparameterization function is the solution of an algebraic equation. For the relevant case of cubic or cubic spline curves, this equation is quartic at most, quadratic if the curves are Bézier or splines and lie on parallel planes, and hence it may be solved either by standard analytical or numerical methods.

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Bézier developable surfaces

Cited by 8 publications

References 15 publications

Characterisation of rational and NURBS developable surfaces in Computer Aided Design

Characterisation of rational and NURBS developable surfaces in Computer Aided Design

Improvement of flattenability using particle swarm optimizer for surface unfolding in bolus shaping

Developable surface patches bounded by NURBS curves

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