Super-smooth cubic Powell–Sabin splines on three-directional triangulations: B-spline representation and subdivision

Grošelj, Jan; Speleers, Hendrik

doi:10.1016/j.cam.2020.113245

Cited by 10 publications

(5 citation statements)

References 53 publications

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“…Hence, for any triangle Δ of T all the coefficients in ( 24) are 0 and so s ≡ 0, i.e., M is a determining set. Moreover, the cardinality of M clearly equals the dimension of the space, see (19), and so M is a minimal determining set. Given a minimal determining set M for S 2 3 (T WS 3 ), suppose we assign values to all the coefficients corresponding to the domain points in it.…”

Section: Stable Bases For S S S 2 3 (T Ws 3 )mentioning

confidence: 99%

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Construction of $$C^2$$ Cubic Splines on Arbitrary Triangulations

2022

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In this paper, we address the problem of constructing $$C^2$$ C 2 cubic spline functions on a given arbitrary triangulation $${\mathcal {T}}$$ T . To this end, we endow every triangle of $${\mathcal {T}}$$ T with a Wang–Shi macro-structure. The $$C^2$$ C 2 cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in this space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of $$C^2$$ C 2 cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for $$C^2$$ C 2 joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang–Shi macro-structure is transparent to the user. Stable global bases for the full space of $$C^2$$ C 2 cubics on the Wang–Shi refined triangulation $${\mathcal {T}}$$ T are deduced from the local simplex spline basis by extending the concept of minimal determining sets.

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Section: Stable Bases For S S S 2 3 (T Ws 3 )mentioning

confidence: 99%

“…To achieve a global basis, one may apply the general framework of minimal determining sets via the local Bernstein basis; see [26]. Global B-spline bases have been constructed for C 1 PS spline spaces on triangulations [14,18,19,49], for PS spline spaces with higher smoothness [17,45,47], and for CT spline spaces [46].…”

Section: Introductionmentioning

confidence: 99%

Construction of $$C^2$$ Cubic Splines on Arbitrary Triangulations

2022

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“…Using similar arguments to those in Corollary 3.6 and Proposition 3.7, we get dim S r,s d (∆ • ) = ker(∂ 2 ). The Euler-Poincaré characteristic of the complex (17)…”

Section: Supersmooth Ideals At Edges and Verticesmentioning

confidence: 99%

“…On a general planar triangulation, the dimension of the space of C rcontinuous splines of polynomial degree at most d may depend on the geometry of the partition for small d. This is undesirable for finite elements as it complicates, for instance, the efficient construction of locally supported basis functions. However, enhanced supersmoothness can be employed to eliminate this geometric-dependence and yield more tractable spline spaces; e.g., see Speleers [34] and Groselj and Speleers [17]. Given this, developing an understanding or spline spaces with (enhanced) supersmoothness has both theoretical and practical relevance.…”

Section: Introductionmentioning

confidence: 99%

Algebraic Methods for Supersmooth Spline Spaces

Toshniwal,

Villamizar

2021

Preprint

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Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we address various challenges arising in the study of splines with enhanced mixed (super-)smoothness conditions at the vertices and across interior faces of the partition. Such supersmoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Using algebraic tools, a generalization of the Billera-Schenck-Stillman complex that includes the effect of additional smoothness constraints leads to a construction which requires the analysis of ideals generated by products of powers of linear forms in several variables. Specializing to the case of planar triangulations, a combinatorial lower bound on the dimension of splines with supersmoothness at the vertices is presented, and we also show that this lower bound gives the exact dimension in high degree. The methods are further illustrated with several examples.

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“…On a general planar triangulation, the dimension of the space of C r -continuous splines of polynomial degree at most d may depend on the geometry of the partition for small d. This is undesirable for finite elements as it complicates, for instance, the efficient construction of locally supported basis functions. However, enhanced supersmoothness can be employed to eliminate this geometric-dependence and yield more tractable spline spaces; e.g., see Speleers [34] and Grošelj and Speleers [17]. Given this, developing an understanding of spline spaces with (enhanced) supersmoothness has both theoretical and practical relevance.…”

Section: Introductionmentioning

confidence: 99%