Bézier surfaces of minimal area: The Dirichlet approach

Monterde, J.

doi:10.1016/j.cagd.2003.07.009

Cited by 77 publications

(54 citation statements)

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“…The triangular Be´zier patch of minimal area 10 and the Plateau-Be´zier problem 6 are the special cases of the above Theorem 1.…”

Section: Theorem 1 the Toric Be´zier Patch B(u V)mentioning

confidence: 99%

“…5 The problem of Plateau-Be´zier, that is, to find the surface of minimal area among all Be´zier patches with given boundaries has been explored by Monterde with the Dirichlet energy functional (cf. Monterde 4,6 ). Nitsche 7 and Osserman 8 gave a comprehensive survey of the minimal surfaces.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, equality holds only if E = G and F = 0, that is, for isothermal surfaces. 6 Douglas proved an important property that both functionals have the same extremals among all functions. 8 For the toric Be´zier patches, both the functionals exit the minimums, but the extremal of Dirichlet functional we obtain is an approximation of the extremal of the area functional.…”

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confidence: 99%

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Approximation of minimal toric Bézier patch

Sun

Zhu

2016

Advances in Mechanical Engineering

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Toric Bézier patches are a multi-sided generalization of classical rational Bézier patches which are widely used in freeform surface modeling. In this article, we study the problem of finding the patch with minimal area among all possible toric Bézier patches with the given boundaries, that is, the Plateau-toric Bézier problem, and obtain an approximation of minimal toric Bézier patch. Some representative examples are given to verify our results.

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“…The triangular Be´zier patch of minimal area 10 and the Plateau-Be´zier problem 6 are the special cases of the above Theorem 1.…”

Section: Theorem 1 the Toric Be´zier Patch B(u V)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Approximation of minimal toric Bézier patch

Sun

Zhu

2016

Advances in Mechanical Engineering

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“…Our first steps were [12] and [13], where it was shown that, surprisingly, a unique harmonic Bézier surface is determined by prescribing only two boundary curves (not the four boundaries) of a tensor product Bézier 20 surface. The second step was to extend this result to the biharmonic case ( [14]) and to a class of more general 4-th order PDE ( [15]).…”

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confidence: 99%

Explicit Bézier control net of a PDE surface

Arnal

Monterde

2017

Computers & Mathematics with Applications

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The PDE under study here is a general fourth-order linear elliptic Partial Differential Equation. Having prescribed the boundary control points, we provide the explicit expression of the whole control net of the associated PDE Bézier surface.In other words, we obtain the explicit expressions of the interior control points as linear combinations of free boundary control points. The set of scalar coefficients of these combinations works like a mould for PDE surfaces. Thus, once this mould has been computed for a given degree, real-time manipulation of the resulting surfaces becomes possible by modifying the prescribed information.

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“…A simple change of sign transforms the harmonic conditions (see [1], [4]) into the 1D-wave conditions.…”

Section: The 1d-wave Equation In Terms Of the Control Pointsmentioning

confidence: 99%