Representation and Approximation of Surfaces

Barnhill, Robert E.

doi:10.1016/b978-0-12-587260-7.50008-x

Cited by 154 publications

(60 citation statements)

References 11 publications

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“…As for the continuity class of the Shepard operator, and consequently the continuity class of the Shepard-Bernoulli operators, there is the following result [4]. …”

Section: Theorem 33 the Degree Of Exactness Of The Operator Smentioning

confidence: 99%

“…Remark 5.1. The combined Shepard operators are global (i.e., their values at each point x are affected by all the data); however, the global character of these interpolants can be avoided by substituting the weight functions A µ,i (x) with basis functions W µ,i (x) of an opportune class of differentiability C s 0 (R), 0 ≤ s ≤ ∞, with small compact support and which may depend on the local distribution of data points [4,23,24,26].…”

Section: An Application Of the Combined Shepard Operatorsmentioning

confidence: 99%

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Shepard--Bernoulli operators

Caira

Dell’Accio

2007

Math. Comp.

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Abstract. We introduce the Shepard-Bernoulli operator as a combination of the Shepard operator with a new univariate interpolation operator: the generalized Taylor polynomial. Some properties and the rate of convergence of the new combined operator are studied and compared with those given for classical combined Shepard operators. An application to the interpolation of discrete solutions of initial value problems is given.

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“…As for the continuity class of the Shepard operator, and consequently the continuity class of the Shepard-Bernoulli operators, there is the following result [4]. …”

Section: Theorem 33 the Degree Of Exactness Of The Operator Smentioning

confidence: 99%

Section: An Application Of the Combined Shepard Operatorsmentioning

confidence: 99%

Shepard--Bernoulli operators

Caira

Dell’Accio

2007

Math. Comp.

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“…as a tool for the finite element method [5], but for some years they have been used in the field of CAGD (Computer Aided Geometrie Design) in the area of scattered data interpolation [1], [7], [12].…”

Section: Introductionmentioning

confidence: 99%

An iterative Clough-Tocher interpolant

Farin¹,

Kashyap²

1992

ESAIM: M2AN

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“…However, as has been shown by McLain [7], Gordon and Wixom [4], Barnhill [1], and more recently Lancaster [5], least squares approximation ideas can be applied to generate interpolants by introducing the notion of moving least squares approximation together with appropriate singularities in the weights used in such approximations. This method includes the metric interpolation technique of Shepard [11].…”

mentioning

confidence: 99%

“…Evidently, the moving least squares approximant Gf, discussed in the previous sections, need not interpolate the data. However, an idea apparently due to Shepard [11] and discussed in some detail by Gordon and Wixom [4], Lancaster [6], and Barnhill [1] can be employed to ensure that Gf interpolates at some or all data points. The principle involved is to make w(*' become infinite at the data point zk if Gf is to interpolate there.…”

mentioning

confidence: 99%

Surfaces Generated by Moving Least Squares Methods

Lancaster¹,

Salkauskas²

1981

Mathematics of Computation

376

387

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Abstract. An analysis of moving least squares (m.l.s.) methods for smoothing and interpolating scattered data is presented. In particular, theorems are proved concerning the smoothness of interpolants and the description of m.l.s. processes as projection methods. Some properties of compositions of the m.l.s. projector, with projectors associated with finiteelement schemes, are also considered. The analysis is accompanied by examples of univariate and bivariate problems.1. Introduction. While the theory and practice of interpolation and approximation of functions of a single variable on the basis of a finite amount of information is well developed, the same is not true for functions of several variables. When the information consists of function values at the points of a rectangular grid, tensor product and blended interpolants based on univariate schemes can be employed. For irregularly distributed function-value data, the situation is much worse. Finiteelement techniques are of value, although, in order to produce C1 interpolants, more than just function-value information is required. As well, if the distribution of data is irregular, only triangular elements seem feasible. Their use has become more attractive in view of the recent development of fast and efficient triangulation algorithms. Nevertheless, the additional nodal information consisting of derivative data must somehow be concocted.Least squares approximation by polynomials is in widespread use and is, of course, not expected to produce an interpolant. However, as has been shown by McLain [7]

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Representation and Approximation of Surfaces

Cited by 154 publications

References 11 publications

Shepard--Bernoulli operators

Shepard--Bernoulli operators

An iterative Clough-Tocher interpolant

Surfaces Generated by Moving Least Squares Methods

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