An interpolating 4-point ternary non-stationary subdivision scheme with tension control

Beccari, Carolina Vittoria; Casciola, Giulio; Romani, Lucia

doi:10.1016/j.cagd.2007.02.001

Cited by 71 publications

(68 citation statements)

References 8 publications

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“…In 2007, Beccari et al [9] presented an interpolating 4-point ternary nonstationary scheme with tension control. They also offered a nonstationary uniform tension controlled interpolating 4-point scheme reproducing conics [10] in 2007.…”

Section: Introductionmentioning

confidence: 99%

A New 5-Point Ternary Interpolating Subdivision Scheme and Its Differentiability

Mustafa

Irum

Bari

2012

ISRN Computational Mathematics

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A new 5-point ternary interpolating scheme with a shape parameter is introduced. The resulting curve is C 2 for a certain range of parameters. The differentiable properties of the proposed scheme to extend its application in the generation of smooth curves are explored. Application of the proposed scheme is given to show its visual smoothness. The scheme is also extended to a 5-point tensor product ternary interpolating scheme, and its numerical examples are also included.

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Section: Introductionmentioning

confidence: 99%

A New 5-Point Ternary Interpolating Subdivision Scheme and Its Differentiability

Mustafa

Irum

Bari

2012

ISRN Computational Mathematics

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“…As discussed in Section 2.3 (see Remark 2.2), if compared with the corresponding order-3 NULIFS interpolant, the limit curve of the NULI 4-point scheme turns out to be tighter to the initial data polygon (in the sense of Remark 2.2). Although there are many proposals of stationary and non-stationary subdivision schemes whose refinement equations include shape parameters [1][2][3]12,17,19,30], the authors are not aware of any existing scheme whose parameters set has a behavior comparable to the NULI 4-point scheme. In fact, so far parameters have been introduced either to control the tension of the limit curve [2,3,30], to increase its smoothness [17,19] or to reproduce salient curves [1,3,12,30].…”

Section: Application Examples and Comparisonsmentioning

confidence: 99%

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

2011

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In this paper we present a family of Non-Uniform Local Interpolatory (NULI) subdivision schemes, derived from compactly supported interpolatory fundamental splines with non-uniform knots (NULIFS). For this spline family, the knot-partition is defined by a sequence of break points and by one additional knot, arbitrarily placed along each knot-interval. The resulting refinement algorithms are linear and turn out to contain a set of edge parameters that, when fixed to a value in the range [0,1], allow us to achieve special shape features by simply moving each auxiliary knot between the break points. Among all the members of this new family of schemes, we will then especially analyze the NULI 4-point refinement. This subdivision scheme has all the fundamental features of the quadratic fundamental spline basis it is originated from, namely compact support, C 1 smoothness, second order polynomials reproduction and approximation order 3. In addition the NULI 4-point subdivision algorithm has the possibility of setting consecutive edge parameters to simulate double and triple knots -that are not considered by the authors of the corresponding spline basis -thus allowing for limit curves with crease vertices, without using an ad hoc mask. Numerical examples and comparisons with other methods will be given to the aim of illustrating the performance of the NULI 4-point scheme in the case of highly non-uniform initial data.

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“…In the following some * E-mail: younis.pu@gmail.com literature, related to non-stationary subdivision schemes, has been discussed. In 2007, Beccari et al presented a non-stationary C 1 continuous interpolating 4-point uniform, tension controlled, scheme that reproduce conics [1] They also developed a 4-point ternary interpolating non-stationary subdivision scheme in the same year that can generate C 2 continuous limit curves showing considerable variation of shapes with a tension parameter [2]. Jena et al [8] introduced a 4-point binary interpolatory non-stationary C 1 subdivision scheme which was the generalization of four point stationary subdivision scheme developed by Dyn et al [11].…”

Section: Introductionmentioning

confidence: 99%

Ternary approximating non-stationary subdivision schemes for curve design

Siddiqi

Younis²

2014

Open Engineering

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Abstract:In this paper, an algorithm has been introduced to produce ternary 2m-point (for any integer m ≥ 1) approximating non-stationary subdivision schemes which can generate the linear spaces spanned by {1; cos(α.); sin(α.)}. The theory of asymptotic equivalence is being used to analyze the convergence and smoothness of the schemes. The proposed algorithm can be consider as the non-stationary counter part of the 2-point and 4-point existing ternary stationary approximating schemes, for different values of m. Moreover, the proposed algorithm has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines.

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An interpolating 4-point ternary non-stationary subdivision scheme with tension control

Cited by 71 publications

References 8 publications

A New 5-Point Ternary Interpolating Subdivision Scheme and Its Differentiability

A New 5-Point Ternary Interpolating Subdivision Scheme and Its Differentiability

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

Ternary approximating non-stationary subdivision schemes for curve design

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