Triangular domain extension of algebraic trigonometric Bézier-like basis

Wei, Yongwei; Shen, Wanqiang; Wang, Guozhao

doi:10.1007/s11766-011-2672-z

Cited by 9 publications

(5 citation statements)

References 14 publications

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“…that were introduced in (Wei, Shen, Wang, 2011) and which degenerate at ∂Ω β to the univariate algebraic-trigonometric normalized B-basis presented in Example 2.4. Once again, for the sake of convenience, the β-dependent values of those double integrals (28) that are required for the minimization of the thin-plate spline energy ( 27) of order at most γ = 2, can be found in Appendix B.3.…”

Section: (C)→(a) Basis Functions Rmentioning

confidence: 99%

Nielson-type transfinite triangular interpolants by means of quadratic energy functional optimizations

Róth¹

2016

Preprint

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We generalize the transfinite triangular interpolant of (Nielson, 1987) in order to generate visually smooth (not necessarily polynomial) local interpolating quasi-optimal triangular spline surfaces. Given as input a triangular mesh stored in a half-edge data structure, at first we produce a local interpolating network of curves by optimizing quadratic energy functionals described along the arcs as weighted combinations of squared length variations of first and higher order derivatives, then by optimizing weighted combinations of first and higher order quadratic thin-platespline-like energies we locally interpolate each curvilinear face of the previous curve network with triangular patches that are usually only C 0 continuous along their common boundaries. In a following step, these local interpolating optimal triangular surface patches are used to construct quasi-optimal continuous vector fields of averaged unit normals along the joints, and finally we extend the G 1 continuous transfinite triangular interpolation scheme of (Nielson, 1987) by imposing further optimality constraints concerning the isoparametric lines of those groups of three side-vertex interpolants that have to be convexly blended in order to generate the final visually smooth local interpolating quasi-optimal triangular spline surface. While we describe the problem in a general context, we present examples in special polynomial, trigonometric, hyperbolic and algebraic-trigonometric vector spaces of functions that may be useful both in computer-aided geometric design and in computer graphics.

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Section: (C)→(a) Basis Functions Rmentioning

confidence: 99%

Nielson-type transfinite triangular interpolants by means of quadratic energy functional optimizations

Róth¹

2016

Preprint

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“…Therefore, we cannot get triangular surfaces with an adjustable shape through the method of tensor product. During the last years, some researchers have put many efforts on the establishments of new bases over triangular domain with shape parameters, see for example [8,43,48,49,51,53]. In [43], Shen and Wang proposed a kind of Bernstein-like basis with a shape parameter, which was a triangular domain extension of the p-Bézier basis of order 3 given in [39].…”

Section: Introductionmentioning

confidence: 99%

“…In [43], Shen and Wang proposed a kind of Bernstein-like basis with a shape parameter, which was a triangular domain extension of the p-Bézier basis of order 3 given in [39]. In [48], Wei, Shen and Wang extended the C-Bézier basis on the univariate domain given in [52] to a new Bézier-like basis on the triangular domain, which possess a shape parameter and can be used to generate some surfaces whose boundaries are arcs of ellipse. In [53], a kind of triangular Bernstein-Bézier-type patch with three exponential shape parameters was proposed.…”

Section: Introductionmentioning

confidence: 99%

New Trigonometric Basis Possessing Exponential Shape Parameters

Zhu

Han

2015

JCM

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Four new trigonometric Bernstein-like basis functions with two exponential shape parameters are constructed, based on which a class of trigonometric Bézier-like curves, analogous to the cubic Bézier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bézier-like curves is given. Any arc of an ellipse or a parabola can be represented exactly by using the trigonometric Bézier-like curves. The corresponding trigonometric Bernstein-like operator is presented and the spectral analysis shows that the trigonometric Bézier-like curves are closer to the given control polygon than the cubic Bézier curves. Based on the new proposed trigonometric Bernstein-like basis, a new class of trigonometric B-spline-like basis functions with two local exponential shape parameters is constructed. The totally positive property of the trigonometric B-spline-like basis is proved. For different values of the shape parameters, the associated trigonometric B-spline-like curves can be C 2 ∩ F C 3 continuous for a non-uniform knot vector, and C 3 or C 5 continuous for a uniform knot vector. A new class of trigonometric Bézier-like basis functions over triangular domain is also constructed. A de Casteljau-type algorithm for computing the associated trigonometric Bézier-like patch is developed. The conditions for G 1 continuous joining two trigonometric Bézier-like patches over triangular domain are deduced.

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“…Shen, G.-Z. Wang and Y.-W. Wei in recent papers [9,10] and [11], respectively. These special cases of triangular patches were obtained by certain constrained trivariate extensions of univariate normalized B-bases of the first and second order trigonometric and of the fourth order algebraic trigonometric vector spaces F α 2 = span {1, cos (t) , sin (t) : t ∈ [0, α]} , F α 4 = span {1, cos (t) , sin (t) , cos (2t) , sin (2t) : t ∈ [0, α]} and M α 4 = span {1, t, cos (t) , sin (t) : t ∈ [0, α]} , respectively, where α ∈ (0, π) is an arbitrarily fixed shape (or design) parameter.…”

Section: Introductionmentioning

confidence: 99%

A constructive approach to triangular trigonometric patches

Róth,

Juhász,

Kristály

2013

Preprint

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We construct a constrained trivariate extension of the univariate normalized B-basis of the vector space of trigonometric polynomials of arbitrary (finite) order n ∈ N defined on any compact interval [0, α], where α ∈ (0, π). Our triangular extension is a normalized linearly independent constrained trivariate trigonometric function system of dimension δ n = 3n (n + 1) + 1 that spans the same vector space of functions as the constrained trivariate extension of the canonical basis of truncated Fourier series of order n over [0, α]. Although the explicit general basis transformation is yet unknown, the coincidence of these vector spaces is proved by means of an appropriate equivalence relation. As a possible application of our triangular extension, we introduce the notion of (rational) triangular trigonometric patches of order n and of singularity free parametrization that could be used as control point based modeling tools in CAGD.

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Triangular domain extension of algebraic trigonometric Bézier-like basis

Cited by 9 publications

References 14 publications

Nielson-type transfinite triangular interpolants by means of quadratic energy functional optimizations

Nielson-type transfinite triangular interpolants by means of quadratic energy functional optimizations

New Trigonometric Basis Possessing Exponential Shape Parameters

A constructive approach to triangular trigonometric patches

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