Length and curvature variation energy minimizing planar cubic <i>G</i><sup>1</sup> Hermite interpolation curve

Li, Juncheng; Zhang, Li

doi:10.1080/16583655.2019.1703248

Cited by 5 publications

(3 citation statements)

References 14 publications

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“…Indeed, shape optimization of parametric curves has attracted more and more attention in recent years, and various objective functions have been proposed to optimize the shape of parametric curves. The internal energy of curves is a widely used objective function, which usually includes stretch energy, strain energy (also called bending energy), and curvature variation energy (often replaced by Jerk's energy), see [6][7][8][9][10][11][12][13]. The external energy of curves is another common objective functional for optimizing the shape of curves, see [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

The quartic Catmull–Rom spline with local adjustability and its shape optimization

Liu

2022

Adv Cont Discr Mod

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Parametric interpolatory curves play a vital part in geometric modeling. Cubic Catmull–Rom spline is a well-known tool for constructing parametric interpolatory curves, but it cannot be modified once its control points are fixed. We propose a novel quartic Catmull–Rom spline with free parameters to tackle this issue. The quartic Catmull–Rom spline owns shape adjustability based on inheriting the features of the cubic Catmull–Rom spline. Some modeling examples show that the shape of the quartic Catmull–Rom spline can realize both global adjustment and local adjustment by changing the free parameters. In addition, we give three schemes for optimizing the shape of the quartic Catmull–Rom spline, which can generate the spline with minimal internal energy, the shape-preserving spline, and the monotonicity-preserving spline. Numerical examples indicate that the proposed schemes are effective and the quartic Catmull–Rom spline is more practical than the cubic Catmull–Rom spline in data interpolation.

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Section: Introductionmentioning

confidence: 99%

The quartic Catmull–Rom spline with local adjustability and its shape optimization

Liu

2022

Adv Cont Discr Mod

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“…Various objective functions have been proposed to optimize the shape of parametric curves, among which the internal energy of curves is a widely used objective function. Generally, the internal energy of curves includes stretch energy, strain energy (also called bending energy), and curvature variation energy (often replaced by Jerk's energy), see [20][21][22][23][24][25][26][27][28]. This paper's second purpose is to optimize the shape of the new Hermite interpolation curve by minimizing the internal energy.…”

Section: Introductionmentioning

confidence: 99%

Cubic Trigonometric Hermite Interpolation Curve: Construction, Properties, and Shape Optimization

Liu

2022

Journal of Function Spaces

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Cubic Hermite interpolation curve plays a very important role in interpolation curves modeling, but it has three shortcomings including low continuity, difficult shape adjustment, and the inability to accurately represent some common engineering curves. We construct a cubic trigonometric Hermite interpolation curve to make up the three shortcomings of cubic Hermite interpolation curve once and for all. The cubic trigonometric Hermite interpolation curve not only inherits the features of cubic Hermite interpolation curve but also achieves C2 continuity, has local and global adjustability, and can accurately represent elliptical arc, circular arc, quadratic parabolic arc, cubic parabolic arc, and astroid arc that often appear in engineering. In addition, we give the schemes for optimizing the shape of the cubic trigonometric Hermite interpolation curve based on internal energy minimization. The schemes include optimizing the shape of planar curve and spatial curve. Some modeling examples show that the proposed schemes are effective and the cubic trigonometric Hermite interpolation curve is more practical than cubic Hermite interpolation curve.

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“…For constructing the smooth planar cubic Hermite curve, many researchers used the energy minimizations to select the two free parameters of the planar cubic Hermite curve. Among these methods, the strain energy minimization and curvature variation energy minimization are two commonly used methods (see Jaklič and Žagar, 2011a;Jaklič and Žagar, 2011b;Li and Zhang, 2020b;Li et al, 2012;Lu, 2015a;Lu et al, 2017c;Yong and Cheng, 2004). It may not be possible to evaluate which of the energy minimizations is better for constructing the smooth planar cubic Hermite curve, since there is no recognized way to describe the smoothness.…”

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confidence: 99%

Combined internal energy minimizing planar cubic Hermite curve

2020

JAMDSM

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The stretch energy, the strain energy and the curvature variation energy are three well known internal energies of the planar curve, they respectively represent length, curvature and variation of curvature that are the typical geometric characteristics of the curve (see Veltkamp and Wesselink, 1995;Xu et al, 2011a). Although there is no recognized way to describe the smoothness of a planar curve, a curve constructed by minimizing one of the internal energies is generally considered smooth. Some examples are the construction of energy-minimizing Bézier curve (

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

Cited by 5 publications

References 14 publications

The quartic Catmull–Rom spline with local adjustability and its shape optimization

The quartic Catmull–Rom spline with local adjustability and its shape optimization

Cubic Trigonometric Hermite Interpolation Curve: Construction, Properties, and Shape Optimization

Combined internal energy minimizing planar cubic Hermite curve

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

Cited by 5 publications

References 14 publications

The quartic Catmull–Rom spline with local adjustability and its shape optimization

The quartic Catmull–Rom spline with local adjustability and its shape optimization

Cubic Trigonometric Hermite Interpolation Curve: Construction, Properties, and Shape Optimization

Combined internal energy minimizing planar cubic Hermite curve

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Product

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