## Scattered Data Interpolation in Three or More Variables (1989)

Venue: | Mathematical Methods in Computer Aided Geometric Design |

Citations: | 50 - 0 self |

### BibTeX

@INPROCEEDINGS{Alfeld89scattereddata,

author = {Peter Alfeld},

title = {Scattered Data Interpolation in Three or More Variables},

booktitle = {Mathematical Methods in Computer Aided Geometric Design},

year = {1989},

pages = {1--33},

publisher = {Academic Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems. List of Contents 1. Introduction 2. Rendering of Trivariate Functions 3. Tensor Product Schemes 4. Point Schemes 4.1 Shepard's Methods 4.2 Radial Interpolants 4.2.1 Hardy Multiquadrics 4.2.2 Duchon Thin Plate Splines 5. Natural Neighbor Interpolation 6. k-dimensional Triangulations 7. Tetrahedral Schemes 7.1 Polynomial Schemes 7.2 Rational Schemes 8. Simplicial Schemes 8.1 Polynomial Schemes 8.2 Rational Schemes 8.3 A Transfinite Scheme 9. Multivariate Splines 10. Transfinite Hypercubal Methods 11. Derivative Generation 12. Interpolation on the sphere and other surfa...

### Citations

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Citation Context ...duct approximation. x4. Point Schemes The term Point Schemes refers to interpolation schemes that are not based on a tessellation of the underlying domain\Omega 4.1 Shepard's Methods Shepard's method =-=[76]-=- may be the best known among all scattered data interpolants in a general number of variables. In its simplest form, it is given by s(x) = N X i=1 w i (x)f(x i ) where w i (x) = kx 0 x i k 0p P N j=1 ... |

345 |
Introduction to Approximation Theory
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302 | Interpolation of scattered data: distance matrices and conditionally positive definite functions
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Citation Context ...P k m , then the choice ff i = 0, i = 1; 2; 1 1 1 ; N and p m = p will satisfy all requirements. Thus radial interpolants have degree of precision m. Recent discussions of radial schemes are in [35], =-=[57]-=-, and [62]. Micchelli's paper establishes in particular that for many radial schemes, including those described here, the interpolant exists and is unique for all data sets (provided only that the int... |

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Citation Context ...en the choice ff i = 0, i = 1; 2; 1 1 1 ; N and p m = p will satisfy all requirements. Thus radial interpolants have degree of precision m. Recent discussions of radial schemes are in [35], [57], and =-=[62]-=-. Micchelli's paper establishes in particular that for many radial schemes, including those described here, the interpolant exists and is unique for all data sets (provided only that the interpolation... |

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Citation Context ...is [37]. It surveys the properties of the method, and in particular analyzes its approximation order. Other papers on implementations, modifications, and tests of Shepard's method include [15], [17], =-=[40]-=-, [42], [49], and [56]. 4.2 Radial Interpolants The term radial is due to Rippa [68]. Radial interpolants are of the form s(x) = N X i=1 ff i g(kx 0 x i k) + pm (x) (26) where g is a given univariate ... |

200 |
Multiquadric equations of topography and other irregular surfaces
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Surface generated by moving least square methods
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Citation Context ...surveys the properties of the method, and in particular analyzes its approximation order. Other papers on implementations, modifications, and tests of Shepard's method include [15], [17], [40], [42], =-=[49]-=-, and [56]. 4.2 Radial Interpolants The term radial is due to Rippa [68]. Radial interpolants are of the form s(x) = N X i=1 ff i g(kx 0 x i k) + pm (x) (26) where g is a given univariate so-called ra... |

96 |
Piecewise quadratic approximations on triangles
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Citation Context ...nt is given in B'ezier-Bernstein Form. The MRC report [4] lists explicit expressions for the coefficients of the scheme. An alternative approach is described in [85]. The bivariate Powell-Sabin split =-=[63]-=- is applied to each facet of the tetrahedron and all points obtained in ` ' ` ' ` ' ` ' Multivariate Interpolation 17 this manner on the boundary of the tetrahedron are connected to a suitably chosen ... |

93 |
Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces
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- 1976
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Citation Context ...ics was gained by Buhmann [28], [29], and Jackson [47] who study the approximation properties of Multiquadrics. 4.2.2 Duchon Thin Plate Splines These interpolants were originally introduced by Duchon =-=[34]-=- for the case k = 2 as solutions of the variational principle Iss = IR k kD (s)k 2 dx = min; (29) where D = @ @ i 1 @ i 2 1 1 1 @ is: (i 1 ; i 2 ; 1 1 1 ; is) 2 [1; d] ;s= m+ 1 (30) over all admissibl... |

69 |
Some new mathematical methods for variational objective analysis using splines and cross-validation
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- 1980
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Citation Context ...polation scheme by choosing the dimension of the approximating space equal to the number of data. Used frequently, and related to some of the techniques described in this paper, are smoothing splines =-=[82]-=-, i.e., functions that minimize I(s) = N X i=1 0 s(x i ) 0 y i 1 2 + J(s) (5) wheresis a parameter and J(s) is a suitable semi-norm, measuring for example the strain energy in a clamped elastic plate ... |

67 |
A brief description of natural neighbor interpolation
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- 1980
(Show Context)
Citation Context ...called surface splines, with g(t) = t (20k) log t if k is even t (20k) if k is odd : (32) n o n o Multivariate Interpolation 13 x5. Natural Neighbor Interpolation This scheme was introduced by Sibson =-=[77]. It -=-is based on the Dirichlet (or Thiessen or Voronoi) Tessellation of \Omega This is the collection of tiles of the formsi = x 2 IR k : kx 0 x i kskx 0 x j k 8j = 1; 2; 1 1 1 ; N "\Omega (33) Obviou... |

34 |
Representation and approximation of surfaces
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- 1977
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Citation Context ...rd's method is [37]. It surveys the properties of the method, and in particular analyzes its approximation order. Other papers on implementations, modifications, and tests of Shepard's method include =-=[15]-=-, [17], [40], [42], [49], and [56]. 4.2 Radial Interpolants The term radial is due to Rippa [68]. Radial interpolants are of the form s(x) = N X i=1 ff i g(kx 0 x i k) + pm (x) (26) where g is a given... |

31 |
A trivariate Clough-Tocher scheme for tetrahedral data
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Citation Context ... to subdivide the triangle into subtriangles. Splitting a triangle about its centroid into three subtriangles and letting the polynomial degree be 3 gives rise to the widely used Clough-Tocher scheme =-=[3]-=-, [80], which requires only first order derivative data. A similar approach to the trivariate problem is taken in [3]. The tetrahedron is split about its centroid into 4 subtetrahedra, the interpolant... |

26 |
An n-dimensional Clough–Tocher interpolant
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Citation Context ...e simplex. A face that is the convex hull of precisely k vertices is also called a facet. A 1-face is also called an edge. Note that a -face of a simplex is itself a -dimensional simplex. (In [7] and =-=[84]-=-, the meanings of the terms facet and face are interchanged. On the other hand, the usage proposed here is also employed e.g., in [27] and [52]. There seem to be no linguistic reasons to prefer one us... |

21 |
On the dimension of spaces of piecewise polynomials in two variables, in
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Citation Context ... vertices x i 9 : (67) Then, for ds2r + 1, dim P r d = ae 0 + 3 X i=1 ae i a i (68) where ae 0 = dim IP 3 d = d + 3 3 ; (69) ae 1 = d + 2 3 0 3 r + 2 3 ; (70) ae 2 = d + 1 3 0 2 r + 1 3 0 2 r + 2 3 ; =-=(71)-=- and ae 3 = d 3 0 r 3 0 3 r + 1 3 : (72) For ds3 dim S 1 3smax ae 0 ; ae 0 + 3 X i=1si a i (73) wheres1 = 0 d(d 2 0 1) 1 =6; (74)s2 = 0 d(d 2 0 6d + 5) 1 =6; (75) Multivariate Interpolation 25 ands3 =... |

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Citation Context ... RGB) on certain test slides. Appropriate corrections can be determined for specific RGB values. Corrections for other RGB values can then be obtained by interpolation. 3. Implicitly Defined Surfaces =-=[73]-=-. Tom Sederberg has explored the design and representation of (two-dimensional) surfaces as the contour surfaces of trivariate piecewise polynomial functions. Such functions can be constructed by inte... |

17 |
Interpolation of data on the surface of a sphere
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- 1984
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Citation Context ...t instance of such a surface is a sphere. It is often unsatisfactory to project the surface, or different parts of it, into the plane. Instead, special methods have to be designed. Lawson [51], Renka =-=[65]-=-, and Nielson and Ramaraj ([60] and [64]) independently propose schemes based on a triangulation of the surface of a sphere. In all of these schemes, the triangulations (by segments of great circles, ... |

17 |
A multivariate Powell-Sabin interpolant
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Citation Context ...e up that eliminate them. The interpolant is given in B'ezier-Bernstein Form. The MRC report [4] lists explicit expressions for the coefficients of the scheme. An alternative approach is described in =-=[85]-=-. The bivariate Powell-Sabin split [63] is applied to each facet of the tetrahedron and all points obtained in ` ' ` ' ` ' ` ' Multivariate Interpolation 17 this manner on the boundary of the tetrahed... |

16 |
An Explicit Basis for C Quartic Bivariate Splines
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Citation Context ...variate splines is substantial and rapidly expanding. The subject was pioneered by Strang, [79], Morgan and Scott [54], Schumaker [67], [68]. The state of the art (as of this writing) is described in =-=[14]-=- and [46], and the references listed there. A few results for trivariate splines are given in [8]. Following is a summary and an update: Suppose for the moment that the 3-dimensional triangulation is ... |

16 |
A Method for Interpolating Scattered Data Based Upon a Minimum Norm Network
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Citation Context ...n of his earlier planar scheme ([66] and [32]). His paper also contains algorithmic details. Nielson and Ramaraj use an interpolant based upon a minimal norm network (described for the planar case in =-=[59]-=-). Their paper contains several pictorial illustrations of their 28 Peter Alfeld scheme. The surface can be rendered either as a shaded surface surrounding the sphere, or as a contour plot drawn on th... |

15 |
Surfaces in computer aided geometric design: A survey with new results, Computer Aided Geometric Design
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- 1985
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Citation Context ...of cross 8 Peter Alfeld sections where one of the variables is held constant, or, 2. display contour surfaces where the value of s(x) equals some constant. The latter approach is illustrated e.g., in =-=[16]-=-. Particular schemes of displaying contour surfaces of a function s are described in [61] and [74]. Petersen et al assume that s is piecewise polynomial (of any degree) in B'ezier-Bernstein form on a ... |

14 |
Bounds on the dimension of spaces of multivariate piecewise polynomials
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- 1984
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Citation Context ...1, dim P r d = ae 0 + 3 X i=1 ae i a i (68) where ae 0 = dim IP 3 d = d + 3 3 ; (69) ae 1 = d + 2 3 0 3 r + 2 3 ; (70) ae 2 = d + 1 3 0 2 r + 1 3 0 2 r + 2 3 ; (71) and ae 3 = d 3 0 r 3 0 3 r + 1 3 : =-=(72)-=- For ds3 dim S 1 3smax ae 0 ; ae 0 + 3 X i=1si a i (73) wheres1 = 0 d(d 2 0 1) 1 =6; (74)s2 = 0 d(d 2 0 6d + 5) 1 =6; (75) Multivariate Interpolation 25 ands3 = (d 3 0 12d 2 + 29d 0 18)=6: (76) The fo... |

13 |
On estimating partial derivatives for bivariate interpolation of scattered data. Rocky Mtn
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- 1984
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Citation Context ...facet of a simplex is either on the boundary of\Omega or else is a common facet of exactly two simplices. 4. Each simplex contains no points of X other than its vertices. 5. \Omega is homeomorphic to =-=[0; 1]-=- k . The last of the above conditions rules out degenerate triangulations consisting for example of two tetrahedra touching in just one vertex or edge. There are some significant differences between b... |

13 |
Piecewise polynomials and the finite element method
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- 1973
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Citation Context ...the way in which simplices are connected, but also on the precise location of the data sites. Knowledge of bivariate splines is substantial and rapidly expanding. The subject was pioneered by Strang, =-=[79]-=-, Morgan and Scott [54], Schumaker [67], [68]. The state of the art (as of this writing) is described in [14] and [46], and the references listed there. A few results for trivariate splines are given ... |

12 |
Properties of Shepard’s surfaces
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- 1983
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Citation Context ...ethod is [37]. It surveys the properties of the method, and in particular analyzes its approximation order. Other papers on implementations, modifications, and tests of Shepard's method include [15], =-=[17]-=-, [40], [42], [49], and [56]. 4.2 Radial Interpolants The term radial is due to Rippa [68]. Radial interpolants are of the form s(x) = N X i=1 ff i g(kx 0 x i k) + pm (x) (26) where g is a given univa... |

11 |
Surface Fitting with Scattered, Noisy Data on Euclidean D-Spaces and on the Sphere
- Wahba
- 1984
(Show Context)
Citation Context ...ontains several pictorial illustrations of their 28 Peter Alfeld scheme. The surface can be rendered either as a shaded surface surrounding the sphere, or as a contour plot drawn on the sphere. Wahba =-=[81]-=- describes an approach to approximation on the sphere that is based on periodic functions. Her technique can be specialized to interpolation. The paper also addresses the issues of noise in the data, ... |

10 |
Interpolation to arbitrary data on a surface
- Barnhill, Piper, et al.
- 1987
(Show Context)
Citation Context ...cialized to interpolation. The paper also addresses the issues of noise in the data, smoothing splines, and cross-validation of results. More general convex surfaces are considered by Barnhill et al (=-=[20]-=- and [21]) However, their main object of interest is the pressure on the wing of a particular aircraft. They propose a complex scheme, some of whose ingredients are: decomposition of the domain into t... |

9 |
Estimation of gradients from scattered data
- Stead
- 1984
(Show Context)
Citation Context ...ined by applying the technique to gradient values. 2. Sibson [77] describes a specific scheme designed (for an arbitrary number of variables) in the spirit of his natural neighbor technique. 3. Stead =-=[78]-=- compares several techniques, including: --- A Shepard based method where, for each data point, Shepard's interpolant is computed for the six closest sites (excluding the data site itself), and differ... |

7 |
Three- and four-dimensional surfaces
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- 1984
(Show Context)
Citation Context ...fferences between bivariate and higher dimensional triangulations. These include for example: 1. Specification of X and\Omega does not determine the number of simplices. Counter examples are given in =-=[19]-=- and [52]. 2. If k ? 3, possible triangulations may not be distinguishable by information about connectivity of points, i.e., by the specification of edges. For examples see [52]. 3. A standard techni... |

7 |
Efficient computer manipulation of tensor products
- Boor
- 1979
(Show Context)
Citation Context ...lemented in cardinal form, but in any case they can be reduced to univariate rather than truly multivariate interpolation problems. More detailed discussions of tensor product schemes are in [25] and =-=[26]-=-. See also [55] for a more abstract discussion of tensor product approximation. x4. Point Schemes The term Point Schemes refers to interpolation schemes that are not based on a tessellation of the und... |

7 |
A bibliography of multivariate approximation
- Franke, Schumaker
- 1986
(Show Context)
Citation Context ...bliographies should be consulted for further information. A recent extensive bibliography of 1107 entries on multivariate approximation and interpolation is [41]. Recent collections of papers include =-=[31]-=-, [36], and the five volumes of the CAGD journal that have appeared to date. A rich source of information and ideas is the Winter 1984 issue of the Rocky Mountain Journal of Mathematics which is entir... |

7 |
Properties of n-dimensional triangulations, Computer Aided Geometric Design 3
- Lawson
- 1986
(Show Context)
Citation Context ... a simplex is itself a -dimensional simplex. (In [7] and [84], the meanings of the terms facet and face are interchanged. On the other hand, the usage proposed here is also employed e.g., in [27] and =-=[52]-=-. There seem to be no linguistic reasons to prefer one usage over the other.) The location of a point x 2 IR k can be expressed uniquely in terms of barycentric coordinates b i with respect to a non-d... |

5 |
Adaptive contouring of a trivariate interpolant
- Petersen, Piper, et al.
- 1986
(Show Context)
Citation Context ...lay contour surfaces where the value of s(x) equals some constant. The latter approach is illustrated e.g., in [16]. Particular schemes of displaying contour surfaces of a function s are described in =-=[61]-=- and [74]. Petersen et al assume that s is piecewise polynomial (of any degree) in B'ezier-Bernstein form on a tetrahedral tessellation (but not necessarily a triangulation) of \Omega Their scheme com... |

5 |
Interpolation and smoothing of scattered data by radial basis functions
- Rippa
- 1984
(Show Context)
Citation Context ...roximation order. Other papers on implementations, modifications, and tests of Shepard's method include [15], [17], [40], [42], [49], and [56]. 4.2 Radial Interpolants The term radial is due to Rippa =-=[68]-=-. Radial interpolants are of the form s(x) = N X i=1 ff i g(kx 0 x i k) + pm (x) (26) where g is a given univariate so-called radial function, and pm 2 IP k m . The coefficients of s are determined by... |

5 |
Polynomial approximation on tetrahedrons in the finite element method
- Ženı́šek
- 1973
(Show Context)
Citation Context ...histicated machinery available for handling them. If a piecewise defined interpolant is to be polynomial on each tetrahedron and globally differentiable, then its polynomial degree must be at least 9 =-=[86]-=-. Rescorla [67] gives an explicit description of such a scheme. It requires 220 (i.e., the dimension of the space of trivariate nonic polynomials) data per tetrahedron. These include derivatives throu... |

4 |
Multistage trivariate surfaces
- Barnhill, Stead
- 1984
(Show Context)
Citation Context ...ltiply the basis functions w i by mollifying functionssi satisfyingsi (x i ) = 1 (20) ` ' ae 10 Peter Alfeld and having local support in some appropriate sense. For example, the Franke-Little weights =-=[22]-=- are given bysi (x) = 1 0 kx 0 x i k R i + (21) where x + = x if x ? 0 0 if xs0 ; (22) the R i are suitably chosen radii of circles around the data sites that constitute the support of the modified ba... |

4 |
Shepard's method of "metric interpolation" to bivariate and multivariate interpolation
- Gordon, Wixom
- 1978
(Show Context)
Citation Context ...]. It surveys the properties of the method, and in particular analyzes its approximation order. Other papers on implementations, modifications, and tests of Shepard's method include [15], [17], [40], =-=[42]-=-, [49], and [56]. 4.2 Radial Interpolants The term radial is due to Rippa [68]. Radial interpolants are of the form s(x) = N X i=1 ff i g(kx 0 x i k) + pm (x) (26) where g is a given univariate so-cal... |

4 |
Interpolation and display of scattered data over a sphere, M.S
- Ramaraj
- 1986
(Show Context)
Citation Context .... It is often unsatisfactory to project the surface, or different parts of it, into the plane. Instead, special methods have to be designed. Lawson [51], Renka [65], and Nielson and Ramaraj ([60] and =-=[64]-=-) independently propose schemes based on a triangulation of the surface of a sphere. In all of these schemes, the triangulations (by segments of great circles, rather than straight lines) are built by... |

3 |
A storage efficient method for construction of a Thiessen triangulation
- Cline, Renka
- 1984
(Show Context)
Citation Context ...triangulation procedure (yielding a generalization of the Delaunay triangulation), and a specific derivative generation scheme. Renka's scheme is a modification of his earlier planar scheme ([66] and =-=[32]-=-). His paper also contains algorithmic details. Nielson and Ramaraj use an interpolant based upon a minimal norm network (described for the planar case in [59]). Their paper contains several pictorial... |

2 |
A discrete C interpolant for tetrahedral data
- Alfeld
- 1984
(Show Context)
Citation Context ...l of flexibility can be gained by using rational functions. A particularly serviceable class of basic interpolation operators on triangles and tetrahedra are Barnhill-Birkhoff-Gordon (BBG) projectors =-=[2]-=-, [19]. Here we discuss their application to a tetrahedron t. We denote the edges of t by e ij = V j 0 V i ; i; j 2 f1; 2; 3; 4g ; i 6= j (40) where the V j are the vertices of t. There is a BBG proje... |

2 |
Derivative generation from multivariate scattered data by functional minimization. Computer Aided Geometric Design
- Alfeld
- 1985
(Show Context)
Citation Context ...vation. Its severe disadvantage is that in its unmodified form the derivative generation scheme is global rather than local. Some bivariate implementations and numerical examples are given in [5] and =-=[6]-=-. x12. Interpolation on the sphere and other surfaces A special---but important and widely encountered---problem arises when the data sites lie on a 2-dimensional surface embedded in IR 3 . Since we l... |

2 |
On multivariate vertex splines and applications
- Chui, Lai
- 1987
(Show Context)
Citation Context ...the polynomial degree is high, and the subspaces allow the construction of a minimally supported basis. A recently developed intermediate set of subspaces are those of supersplines and vertex splines =-=[30]-=-. These can be defined in any number of variables. A superspline is a spline s 2 S r d that is a specified number Rsr times differentiable at every vertex of the tessellation. For example, the finite ... |

2 |
Three-stage interpolation to scattered data
- Foley
- 1984
(Show Context)
Citation Context ...ne can obtain an interpolation scheme with an arbitrarily high degree of precision m, say, by letting Pf be Shepard's interpolant and Qf 2 IP k m be a Least Squares Approximation. 4. Delta Sums [38], =-=[39]-=-. A Boolean Sum can be formed only if the operator Q can be applied when f is represented solely by the given Multivariate Interpolation 11 data. This fact rules out, e..g., tensor product based opera... |

2 |
Interpolation to boundary data on the simplex
- Gregory
(Show Context)
Citation Context ...] for a more thorough discussion of perpendicular interpolants, including implementational details for k 2 f2; 3g and numerical examples. n o Multivariate Interpolation 23 8.3 A Transfinite Scheme In =-=[43]-=-, Gregory describes a transfinite simplicial scheme of arbitrary dimensionsk and arbitrary degree of smoothness m. The scheme is a convex combination of Boolean sums of Taylor type operators. It requi... |

2 |
A nodal basis for C piecewise polynomials in two variables
- Morgan, Scott
- 1975
(Show Context)
Citation Context ...e barycentric coordinates. The operators G m F produce Taylor interpolants to derivatives perpendicular to the base face F . They are defined by G m F (f)(x) = m X =0 b i ! P m0 F @ f @s F 0 BF (x) 1 =-=(58)-=- where 0 BF (x) 1 is the base point of the line of fixation with respect ot F through x, and s F = V i 0BF (V i ). Note that G m F is defined in terms of lower dimensional perpendicular interpolants. ... |

1 |
Multivariate Perpendicular Interpolation
- Alfeld
- 1985
(Show Context)
Citation Context ...ce of the simplex. A face that is the convex hull of precisely k vertices is also called a facet. A 1-face is also called an edge. Note that a -face of a simplex is itself a -dimensional simplex. (In =-=[7]-=- and [84], the meanings of the terms facet and face are interchanged. On the other hand, the usage proposed here is also employed e.g., in [27] and [52]. There seem to be no linguistic reasons to pref... |

1 |
A transfinite C interpolant over triangles, Rocky Mountain
- Alfeld, Barnhill
- 1984
(Show Context)
Citation Context ...usual by ffi ij = 1 if i = j 0 if i 6= j : (9) A frequently arising special case is given when M = N and w i (x j ) = ffi ij (10) Multivariate Interpolation 7 In that case, s(x) = N X i=1 y i w i (x) =-=(11)-=- and the interpolant is said to be in cardinal form. The generalization of a triangle to k-dimensional space is a k-simplex or just a simplex. A simplex S is the convex hull of k+1 points called the v... |

1 |
Polynomial Interpolation to Boundary Data
- Barnhill, Gregory
- 1975
(Show Context)
Citation Context ... two linear operators defined on a suitable function space, and such that the composition PQ is defined. The Boolean Sum of P and Q is defined by S = P 8Q = P +Q0 PQ (25) Barnhill and Gregory show in =-=[18]-=- that S has (at least) the interpolation properties of P and the precision properties of Q. Thus one can obtain an interpolation scheme with an arbitrarily high degree of precision m, say, by letting ... |