Planar cubic G1 and quintic G2 Hermite interpolations via curvature variation minimization

Lu, Lingli; Jiang, Chengkai; Hu, Qianqian

doi:10.1016/j.cag.2017.07.007

Cited by 22 publications

(19 citation statements)

References 18 publications

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“…According to the literature [32], Since the curvature variation energy in Equation (19) is highly nonlinear, we can use some approximate forms to simplify the calculation. In [33,34], the curvature variation energy is expressed in the following form…”

Section: Shape Optimization Of Cubic T-bézier Curvesmentioning

confidence: 99%

Parametric Model for Kitchen Product Based on Cubic T-Bézier Curves with Symmetry

2020

Symmetry

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The parametric method of product design is a pivotal and practical technique in computer-aided design and manufacturing (CAD/CAM) and used in many manufacturing sectors. In this paper, we presented a novel parametric method to design a kitchen product in the residential environment, a kitchen cabinet, by using cubic T-Bézier curves with constraints of geometric continuities. First, we introduced a class of cubic T-Bézier curves with two shape parameters and derived the G1 and G2 continuity conditions of the cubic T-Bézier curves. Then, we constructed shape-controlled complex contour curves of the kitchen cabinet by using closed composite cubic T-Bézier curves. The shapes of the contour curves can be adjusted intuitively and predictably by altering the values of the shape parameters. Finally, we studied shape optimization and representation of ellipses for the contour curves of the kitchen cabinet by finding optimal shape parameters and applicable control points respectively. The provided modeling examples showed that our method in this paper can improve the design and scheme adjustment effectively in the conceptual design stage of kitchen products.

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Section: Shape Optimization Of Cubic T-bézier Curvesmentioning

confidence: 99%

Parametric Model for Kitchen Product Based on Cubic T-Bézier Curves with Symmetry

2020

Symmetry

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“…In [13] Jaklic and Zagar present a 4 th order accurate G 1 interpolation method which minimizes a functional approximating the curvature variation energy. Recently in [18], Lu et al introduced a scheme which computes G 1 cubic interpolants minimizing the true curvature variation energy through a constrained minimization problem. The results are concluded to be better than the approximate methods in [13], however this additional accuracy comes with a significant increase in computational cost.…”

Section: Introductionmentioning

confidence: 99%

Area-preserving geometric Hermite interpolation

McGregor

Nave

2019

Journal of Computational and Applied Mathematics

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In this paper we establish a framework for planar geometric interpolation with exact area preservation using cubic Bézier polynomials. We show there exists a family of such curves which are 5 th order accurate, one order higher than standard geometric cubic Hermite interpolation. We prove this result is valid when the curvature at the endpoints does not vanish, and in the case of vanishing curvature, the interpolation is 4 th order accurate. The method is computationally efficient and prescribes the parametrization speed at endpoints through an explicit formula based on the given data. Additional accuracy (i.e. same order but lower error constant) may be obtained through an iterative process to find optimal parametrization speeds which further reduces the error while still preserving the prescribed area exactly.

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“…Because the construction of fair (or smooth) curves is a fundamental problem in the filed of CAD and related application fields (see [1][2][3]), a natural idea is to find the optimal α 0 and α 1 for constructing the fair cubic G 1 Hermite interpolation curves. The general ways to construct fair curves are achieved by minimizing some energy functional representing the fairness (see [4]). The strain energy (bending energy) and curvature variation energy are two widely adopted metrics to describe the fairness of a planar curve, since curvature is the universal shape measure for curves (see [1,[5][6][7][8]).…”

Section: Introductionmentioning

confidence: 99%

“…The strain energy (bending energy) and curvature variation energy are two widely adopted metrics to describe the fairness of a planar curve, since curvature is the universal shape measure for curves (see [1,[5][6][7][8]). Recently, some works on determining suitable α 0 and α 1 of the planar cubic G 1 Hermite interpolation curve via strain energy minimization or curvature variation energy minimization have been proposed (see [4,[9][10][11][12]).…”

Section: Introductionmentioning

confidence: 99%

Length and curvature variation energy minimizing planar cubic G¹ Hermite interpolation curve

Zhang

2019

Journal of Taibah University for Science

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In this paper, we study how to construct a planar cubic G 1 Hermite interpolation curve with minimal length and curvature variation energy. This problem is solved by a bi-objective optimization model. We prove that there is a unique solution to this problem and give the expression of the solution. Some numerical examples show that the proposed method makes the curvature variation energy of the curve as small as possible when the length of the curve as short as possible.

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Planar cubic G1 and quintic G2 Hermite interpolations via curvature variation minimization

Cited by 22 publications

References 18 publications

Parametric Model for Kitchen Product Based on Cubic T-Bézier Curves with Symmetry

Parametric Model for Kitchen Product Based on Cubic T-Bézier Curves with Symmetry

Area-preserving geometric Hermite interpolation

Length and curvature variation energy minimizing planar cubic G¹ Hermite interpolation curve

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Planar cubic G1 and quintic G2 Hermite interpolations via curvature variation minimization

Cited by 22 publications

References 18 publications

Parametric Model for Kitchen Product Based on Cubic T-Bézier Curves with Symmetry

Parametric Model for Kitchen Product Based on Cubic T-Bézier Curves with Symmetry

Area-preserving geometric Hermite interpolation

Length and curvature variation energy minimizing planar cubic G1 Hermite interpolation curve

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Product

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Length and curvature variation energy minimizing planar cubic G¹ Hermite interpolation curve