A non-stationary subdivision scheme for generalizing trigonometric spline surfaces to arbitrary meshes

Jena, Mahendra Kumar; Shunmugaraj, P.; Das, P. C.

doi:10.1016/s0167-8396(03)00008-6

Cited by 30 publications

(12 citation statements)

References 12 publications

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“…We restrict our attention to them since they are the ones of practical relevance. Moreover, note that the assumption in (26) is nothing but the notion of regularity of the characteristic map ofS (see, e.g., [32] for details).…”

Section: Normal Continuity Analysis At the Limit Point Of An Extraordmentioning

confidence: 99%

“…In regular regions, the non-stationary scheme in [26] is described by the k-th level mask which contains the factor (1 + z 1 )(1 + z 2 ), thus satisfying assumption (ii) of Theorem 4.2. Differently, in irregular regions the refinement rules are given in terms of the k-th level matrix S k having the structure in (6)…”

Section: Generalized Trigonometric Spline Surfaces Of Ordermentioning

confidence: 99%

“…It also gives sufficient conditions for the limit surface to be normal continuous at the limit point of an extraordinary vertex/face. To the best of our knowledge, the only contributions in this domain are the works in [7,21,26], where specific schemes are considered.…”

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confidence: 99%

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Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

et al. 2019

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Convergence and normal continuity analysis of a bivariate non-stationary (leveldependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and normal continuity of any rotationally symmetric, non-stationary, subdivision scheme near an extraordinary vertex/face.Keywords Non-stationary subdivision · Extraordinary vertex/face · Convergence · Normal continuity Mathematics Subject Classification (2010) 26A15 · 68U07 IntroductionThis paper provides a general procedure to check convergence of non-stationary (leveldependent) subdivision schemes in the neighborhood of an extraordinary vertex/face. It also gives sufficient conditions for the limit surface to be normal continuous at the limit point of an extraordinary vertex/face. To the best of our knowledge, the only contributions in this domain are the works in [7,21,26], where specific schemes are considered.

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Section: Normal Continuity Analysis At the Limit Point Of An Extraordmentioning

confidence: 99%

Section: Generalized Trigonometric Spline Surfaces Of Ordermentioning

confidence: 99%

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Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

et al. 2019

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“…In the same space, nonuniform algebraic-trigonometric B-splines (NUAT splines) were constructed in [15]. A subdivision scheme on trigonometric spline was proposed in [2,3]. Later, hyperbolic splines were also extended to the case of nonuniform knot vector in [7].…”

Section: Introductionmentioning

confidence: 99%

Unified and extended form of three types of splines

Wang

Fang

2008

Journal of Computational and Applied Mathematics

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The three types refer to polynomial, trigonometric and hyperbolic splines. In this paper, we unify and extend them by a new kind of spline (UE-spline for short) defined over the space {cos t, sin t, 1, t, . . . , t l , . . .}, where l is an arbitrary nonnegative integer.is a frequency sequenceExisting splines, such as usual polynomial B-splines, CB-splines, HB-splines, NUAT splines, AH splines, FB-splines and the third form FB-splines etc., are all special cases of UE-splines. UE-splines inherit most properties of usual polynomial B-splines and enjoy some other advantageous properties for modelling. They can exactly represent classical conics, the catenary, the helix, and even the eight curve, a kind of snake-like curves etc.

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“…Spline schemes such as [9] aim at reproducing only trigonometric curves like a circle, elliptic, helix but not surfaces. Jena et al [11] presented a non-interpolatory scheme for tensor product bi-quadratic trigonometric spline surfaces.…”

Section: Introductionmentioning

confidence: 99%

A New Class of Non-stationary Interpolatory Subdivision Schemes Based on Exponential Polynomials

Choi

Lee

Yoon

et al. 2006

Geometric Modeling and Processing - GMP 2006

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Abstract. We present a new class of non-stationary, interpolatory subdivision schemes that can exactly reconstruct parametric surfaces including exponential polynomials. The subdivision rules in our scheme are interpolatory and are obtained using the property of reproducing exponential polynomials which constitute a shift-invariant space. It enables our scheme to exactly reproduce rotational features in surfaces which have trigonometric polynomials in their parametric equations. And the mask of our scheme converges to that of the polynomial-based scheme, so that the analytical smoothness of our scheme can be inferred from the smoothness of the polynomial based scheme.

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A non-stationary subdivision scheme for generalizing trigonometric spline surfaces to arbitrary meshes

Cited by 30 publications

References 12 publications

Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

Unified and extended form of three types of splines

A New Class of Non-stationary Interpolatory Subdivision Schemes Based on Exponential Polynomials

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