The Structure of C1 Spline Spaces on Freudenthal Partitions

Hecklin, Gero; Nürnberger, Günther; Zeilfelder, Frank

doi:10.1137/040614980

Cited by 5 publications

(6 citation statements)

References 30 publications

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“…These spaces are usually extremely complex; however, the structure of the spaces over uniform type partitions (cf. Hangelbroek et al, 2004;Hecklin et al, 2006;Schumaker & Sorokina, 2004 is somewhat simplified, sometimes providing the possibility of using lowerdegree smooth splines. This is important from a practical point of view for the above mentioned applications.…”

Section: Introductionmentioning

confidence: 99%

Local quasi-interpolation by cubic C1 splines on type-6 tetrahedral partitions

Sorokina¹,

Zeilfelder²

2007

IMA Journal of Numerical Analysis

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We describe an approximating scheme based on cubic C 1 splines on type-6 tetrahedral partitions using data on volumetric grids. The quasi-interpolating piecewise polynomials are directly determined by setting their Bernstein-Bézier coefficients to appropriate combinations of the data values. Hence, each polynomial piece of the approximating spline is immediately available from local portions of the data, without using prescribed derivatives at any point of the domain. The locality of the method and the uniform boundedness of the associated operator provide an error bound, which shows that the approach can be used to approximate and reconstruct trivariate functions. Simultaneously, we show that the derivatives of the quasi-interpolating splines yield nearly optimal approximation order. Numerical tests with up to 17 × 10 6 data sites show that the method can be used for efficient approximation.

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Section: Introductionmentioning

confidence: 99%

Local quasi-interpolation by cubic C1 splines on type-6 tetrahedral partitions

Sorokina¹,

Zeilfelder²

2007

IMA Journal of Numerical Analysis

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“…It was shown in [13] that dim S 1 3 ( F ) = 12n 2 + 18n + 4. Furthermore, for type-6 tetrahedral partitions 6 (each cube Q is subdivided in 24 congruent tetrahedra which have a common vertex at the center of Q), it is known [12] that dim S 1 3 ( 6 ) = 6n 3 +24n 2 + 18n 2 + 4.…”

Section: Lemma 81 Fix F ∈ C(ω) and Let T Be A Tetrahedron In Supmentioning

confidence: 99%

“…the dimension result in [13]) that F is not a sufficiently fine triangulation to allow the construction of a Lagrange interpolating pair using C 1 cubic splines defined over F , at least not with the desirable properties listed in the Introduction. Instead, we define S over a refined partition obtained by splitting certain tetrahedra of F into subtetrahedra.…”

Section: §2 Freudenthal Partitionsmentioning

confidence: 99%

“…Trivariate spline spaces are much more difficult to analyze than bivariate spline spaces, where questions of dimension, stable local bases, and approximation power are already very challenging; see [17]. For results on dimension and the construction of bases, see [7], [8], [12], [13], [17]. For a survey on interpolation using bivariate splines, see [24].…”

Section: Lemma 81 Fix F ∈ C(ω) and Let T Be A Tetrahedron In Supmentioning

confidence: 99%

“…Since F is a Freudenthal partition, it is easy to see (cf. [13]) that the common vertices of any two tetrahedra T : …”

Section: Lemma 81 Fix F ∈ C(ω) and Let T Be A Tetrahedron In Supmentioning

confidence: 99%

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A local Lagrange interpolation method based on $C^{1}$ cubic splines on Freudenthal partitions

et al. 2007

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Abstract. A trivariate Lagrange interpolation method based on C 1 cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity. §1. Introductionbe a set of points in R 3 . In this paper we are interested in the following problem. Problem 1. Find a tetrahedral partition whose set of vertices includes V, an Ndimensional space S of trivariate splines defined on with a prescribed smoothness r, and a set of additional points {η i } N i=n+1 such that for every choice of the data, there is a unique spline s ∈ S satisfying (1.1)We call P :and S a Lagrange interpolation pair. It is easy to solve this problem using C 0 splines; see Remark 1. In this case no additional interpolation points are needed. However, the situation is much more complicated if we want to use C r splines with r ≥ 1. For r = 1 the problem was first solved in [29] for the special case where the points in V are the vertices of a cube partition. The method uses quintic splines, and provides full approximation power six for sufficiently smooth functions. In [23] we recently described two C 1 methods which solve the problem for arbitrary initial point sets V. The first method is based on quadratic splines, and requires that each tetrahedron be split into 24 subtetrahedra. The second method is based on cubic splines, and requires that each tetrahedron be split into 12 subtetrahedra. Both methods provide approximation order three for sufficiently smooth functions. For the quadratic spline case this is the optimal approximation order, but for the cubic case it is suboptimal.The aim of this paper is to describe a new method based on cubic C 1 splines which yields optimal approximation order four. To achieve this, we restrict ourselves

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Quasi-interpolation by quadratic -splines on truncated octahedral partitions

Rhein

Kalbe

2009

Computer Aided Geometric Design

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The Structure of C1 Spline Spaces on Freudenthal Partitions

Cited by 5 publications

References 30 publications

Local quasi-interpolation by cubic C1 splines on type-6 tetrahedral partitions

Local quasi-interpolation by cubic C1 splines on type-6 tetrahedral partitions

A local Lagrange interpolation method based on $C^{1}$ cubic splines on Freudenthal partitions

Quasi-interpolation by quadratic -splines on truncated octahedral partitions

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