C. de. Boor. A Practical Guide to Splines. Springer-Verlag, Berlin, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....type stencils to de ne the discrete operator. At internal patch boundaries, however, some care must be taken. The general idea is to de ne ghost points near internal patch boundaries so that locally equispaced unknowns are available for use with a a regular stencil. In essence, simple B splines [4] are used along the boundaries to produce the needed ghost point values. The key point is that the resulting discretization enforces C continuity along the patch boundaries. continuity is di erent than what is normally required in the multigrid literature (which normally only imposes C ....

Boor, C.: A Practical Guide to Splines. Springer-Verlag, New York, 1978.

....splines introduced in next section. In Section 3 the spherical splines are defined in order to create a generalized cylinder of varying radius and a generalized plate with varying thickness. 2 Volumetric Spline Variational Approach Splines are well known in Computer Aided Geometric Design [3]. They have been widely used for modeling curves and surfaces that must pass exactly through individual points, so called interpolation splines. Let us first formulate the problem of interpolation with volumetric splines. Let Omega be an arbitrary shaped n dimensional domain containing a set of ....

C. De. Boor. A practical guide to splines. SpringerVerlag, 1978.

....choice does not depend on the geometry of domain or boundary conditions. In our case, we chose to approximate # by a linear combination of bicubic B splines over a uniform rectangular grid #50 # 50#, as illustrated in Figure 7. The attractive computational properties of B splines are well known [8]. As we discussed in the previous section, expression (24) can be also viewed as an operator B actingonthe unknown function #. B is clearly dependent on the geometry of the problem as represented by functions 1 and 2 and the boundary conditions q and T oil . If dependence of these variables ....

....of the basis functions, and not on the geometry of the domain. If f#g is a sequence of (global) multivariate polynomials, the resulting matrices are full and may quickly become ill conditioned [22] For B splines, it is well known that the matrices are banded and can be solved quickly and robustly [8]. 5Conclusion It is worth noting that modeling of motions and deformations is an active area of research in geometric modeling and computational geometry [3, 17, 6, 35] Specifying, controlling, and analyzing motions and deformations often lead to challenging problems that are not properly ....

Carl de Boor. A Practical Guide to Splines. Springer-Verlag, 1978.

....basis functions B j1 ; B jk j are computed with a k j vector of knots t j = t j1 ; t jk j ) from the support of each e ect modi er r ij . An appropriate choice is the widely used B spline basis with local support. For details and ecient algorithms for computing this basis see De Boor (1978), Eubank (1988) Schumaker (1993) or Dierckx (1993) and especially for natural splines Lyche and Schumaker (1973) or Lyche and Str m (1996) With the basis functions approach for each f j the predictor (4) of the BVCM is given as GLM i = z i x i1 B 1 (r i1 )c 1 : x ip B p (r ip )c ....

De Boor, C. (1978). A Practical Guide to Splines, Springer{Verlag, New York.

.... ; Calculate fitted values l = spl clim(x,y,w,p, 95) Calculate 95 confidence limits ptek = gspline.tkf ; Name of graphics file (Fig 2) title( GCV spline and 95 confidence limits ) Graph title yn = sin(x) 2 ; Original curve with no error plegstr = Sin[2] (no error) Legend labels 000Fitted curve 000Lower CL 000Upper CL ; let plwidth = 4 4 2 2 ; Set width of lines let pltype = 5 6 3 3 ; Set line patterns xy(x,yn yg l) Plot spline and confidence limits end ; ....

de Boor, C. (1978). A Practical Guide to Splines. Springer--Verlag, New York.

....of the bases centered one knot beyond the boundaries of the problem domain be included in the solution. This is implemented using what are typically termed phantom nodes or fictitious points [25] Working with only the Dirichlet boundary conditions, it is necessary to apply the not a knot [26] 16 boundary conditions to the assembled matrix M3 and vector b . The modification are as follows M3 11 = 3 Delta 3 Delta 2 ; M3 12 = K 1 Gamma K 2 3( Delta 2 Delta 3 ) Delta 3 ; M3 13 = 4K 1 Gamma 3K 2 Gamma 3 Delta 3 Delta 2 ; M3 14 = K 1 3K 2 Gamma ....

C. de Boor, A Practical Guide to Splines, Springer-Verlag, New York, pp. 55-56, 1978.

.... to notice that the local support property of the B spline basis functions implies that each control point has only a local influence: moving one control point d ij changes the surface only locally for all (u; v) 2 [u i ; u i 4 ] Theta [v j ; v j 4 ] For more details about B splines, see [5, 29]. 3.1 The fairing principle For the development of the fairing method we need first some quantitative measures of fairness. The fairing principle refers to Kjellander s beam model (Section 2) for surfaces [18] In terms of bicubic B spline surfaces it can be resumed as follows: Fairing ....

De Boor, C., A Practical Guide to Splines, Springer, New York, 1978.

....is not limited to piecewise linear polynomial approximations and its extention to higher degree polynomials is straight forward. There is, however, a question of the best basis. Many possibilities are available from design and approximation theory. Of these, splines and Hermite approximations [5] are generally not used because they offer more smoothness and or a larger support than needed or desired. Lagrange interpolation [2] and a hierarchical approximation in the spirit of Newton s divideddifference polynomials will be our choices. The piecewise linear hat function OE j (x) 8 ....

C. de Boor. A Practical Guide to Splines. Springer-Verlag, New York, 1978.

....for it to be codeable as a sequence of 1D functions. Hence, we are forced to introduce extra knots. Several methods have been suggested in the Spline literature, and we choose to sample the integral of the square root of the absolute curvature linearly in between knots as suggested by de Boor [12]. This implies that extra knots are introduced when there is much curvature, which conforms with the myopic view as described earlier. Hence the algorithm so far is as follows: 1. Calculate a scale space of the original image and find the contour of the blobs at each scale. 2. Calculate the ....

Carl de Boor. A Practical Guide to Splines. SpringerVerlag, 1978.

.... sufficiently generally in the calculations in [12] Using the upper bound from [12] we get jjAf Gamma Bf jj 1 c 0 jjAf Gamma Bf jj 2 c 0 jjAjj 2 jjf Gamma Bf jj 2 c 0 jjAjj 2 p T jjf Gamma Bf jj 1 : Now jjAf Gamma f jj 1 jjAf Gamma Bf jj 1 jjBf Gamma f jj 1 : Results in [8, 1] give jjBf Gamma f jj 1 h 3 8 jjf (3) jj 1 : Since T=h = N 1 we conclude 5 jjAf Gamma f jj 1 (3:71jjAjj 2 1) 8 p Th 5=2 jjf (3) jj 1 : Noting that jjAjj 2 jjP jj 2 and using the bound for jjP jj 2 from theorem 2.2, we get the bound asserted in the present ....

Carl de Boor; Practical guide to splines, Springer Verlag, New York (1978) 9

.... F (x i ) y i and the chosen boundary conditions at the ends of the input interval (where there is no joining polynomial and therefore two constraints are missing) lead to a tridiagonal set of linear equations for the parameters a ij , which can be solved by Gaussian elimination without pivoting [6]. Spline interpolation belongs to widely accepted techniques and can be found in any textbook on numerical analysis, see e.g. 17] for a more detailed presentation. What we want to stress here (and what is well known to mathematicians familiar with linear algebra, compare [27] as well as to ....

....interpolation belongs to widely accepted techniques and can be found in any textbook on numerical analysis, see e.g. 17] for a more detailed presentation. What we want to stress here (and what is well known to mathematicians familiar with linear algebra, compare [27] as well as to spline experts [6]) is that 11 4 3 2 1 0 1 2 3 4 0.2 0 0.2 0.4 0.6 0.8 1 F s b 4 3 2 1 0 1 2 3 4 0.2 0 0.2 0.4 0.6 0.8 1 F B b Figure 2.6: Base functions for cubic spline interpolation: function F s b (left) B spline F B b (right) splines form a linear space and can thus be fully ....

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Carl de Boor. A Practical Guide to Splines. Springer-Verlag, 1978.

....of the segmentation problem and the fitting problem becomes more readily apparent fitting a polycurve involves the estimation both of the positions of the knot points and of the parameters of the approximating segments. This is known as the optimal knot placement problem in spline fitting [28] and no efficient solution is currently known [117] B Splines Cubic B splines are among the most popular curve representations in computer graphics, being capable of modelling a large number of the curves occurring in computer aided design. As such, they satisfy the scope requirement for a ....

C. de Boor. A practical guide to splines. Springer-Verlag, New York, 1978.

....based on the wavelet expansion in the last subsection. 3.2. Biorthogonal multiresolution on R On R the piecewise polynomial splines of degree d Gamma 1 can be defined as follows. Denoting by [x 0 ; x d ]f the d th order divided difference at the points x 0 ; x d 2 R (see e.g. [15]) the (centered) cardinal B spline of order d is given by OE (d) x) d[0; 1; d] i Delta Gammax Gamma j d 2 kj d Gamma1 : where x l : maxf0; xg) l and bxc (dxe) is the largest (smallest) integer less (greater) than or equal to x. This scaling function is ....

C. de Boor. A practical guide to splines. Springer, 1978.

....2 =2 t : 4.22) e At M(t) Gamma1 = Gammat 3 =6 t 2 =2 Gammat 2 =2 t : 4.23) Then we have the polynomial spline with basis functions given by linear combinations of 1; t; t 2 ; t 3 . In the next section, we shall further show that this is the well known cubic spline [5]. From the above discussion, we see that we may encounter many kinds of splines by varying parameters fi and fl. Two general cases are 1.a. and 2.a. where we have full sized exponential or exponential trigonometric splines. Degeneration occurs when zero or multiple eigenvalues appear. The extremal ....

de Boor, C. (1978): A Practical Guide to Splines. Springer-Verlag

....to O N 3 i N 3 j N 3 which constitutes a substantial saving. Tensor products do not have to be implemented 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 underlying domain Omega 4.1 Shepard s Methods Shepard s method [76] may be the best ....

de Boor, C., A Practical Guide to Splines, Springer Verlag, New York, 1978.

....Thin plate splines [17] are an important example of radial basis functions which provide smooth approximations. In the one dimensional case, thinplate splines are the piecewise cubic smoothing splines. They are computationally very tractable as they can be represented with the local B spline basis [16]. However, explicit representations are also known in higher dimensions. In two dimensions, for example, one has f(x 1 , x 2 ) c 0 c 1 x 1 c 2 x 2 n X k=1 b k # (x 1 x (k) 1 ) 2 (x 2 x (k) 2 ) 2 10 where #(r 2 ) r 2 log(r 2 ) for higher dimensions ....

....is due to the fact that a greedy algorithm is used and the choice of basis functions may not be a global optimum. However, we think that more research into the computational performance of the MARS algorithm is required. The basic one dimensional functions used are truncated powers. It is known [16] that such functions lead to ill conditioned linear systems of equations. Furthermore, the evaluation at a certain point may require the computation of many terms and cancellation errors are likely to occur especially if highly local e#ects are modeled. Finally, the coe#cients of the basis ....

Carl de Boor. A Practical Guide to Splines. Springer, 1978.

....bounds of the (left side closed and right side open) value interval covered by the bucket: 8i 2 f1; 2; m Gamma 1g : high i = low i 1 low 1 = 1, highm = n 1. 1) Unlike histograms, we approximate the frequency in an interval by a linear function, resulting in a linear spline function [5, 6] over the m buckets. Because our interest is in compact representation of distributions and not in the features more advanced forms of splines offer (smoothness, differentiability) we have chosen simple linear spline functions for this task. In contrast to previous approaches to spline based ....

....i Gamma low i of bucket b i . 4.2 Optimal Partitioning of V We are now interested in a partitioning such that the overall error (formula 4) is minimized. When arbitrary partitionings and continuous splines of arbitrary degree are considered, this is known as the optimal knot placement problem [5], which due to its complexity is generally solved only approximatively by heuristic search algorithms (for a detailed discussion, see [6] In our case, however, only linear splines are used and only members of V are candidates for bucket boundaries. Since the value for each high i is ....

C. de Boor. A practical guide to splines. SpringerVerlag, 1978.

....Figure 1. The scaling function # and the associated wavelet #. We consider a smooth enough compactly supported scaling function whose first order derivative generates an admissible zero moment wavelet. To be precise, the scaling function # used to derive our results will be a cubic B spline (De Boor (1978)) whose support is the interval [ 1, 1] and whose integral is equal to 1. Mallat and Hwang (1992) have also adopted such a cubic B spline for extrema detection. The first order derivative # = d# dx of such a function generates a compactly supported admissible wavelet. Figure 1 displays the scaling ....

De Boor, C. (1978). A practical guide to splines. Springer-Verlag, New-York.

.... visual data about the environment, remote control) Various methods of motion generation Currently at the Institut National de Recherche en Informatique et en Automatique (INRIA) 655 avenue de l Europe, 38330 Montbonnot Saint Martin, France are developed and make use of spline interpolation [4], e.g. polynomial spline interpolation was considered in [5, 6] Obtaining the reference trajectories which are feasible for the servo systems of the robot was investigated in [7] Our paper focuses on the motion generation approach as well as the control architecture developed and implemented in ....

K. de Boor, A Practical Guide to Splines, Springer Verlag, New York, USA, 1978.

....= 3) is a common choice in practice. Model (5) is constructed as a linear combination of basis functions 1; x; x p ; x Gamma 1 ) p ; x Gamma K ) p . This basis is referred to as the power basis. A popular set of basis functions are the so called B splines (de Boor [9], p.108) Unlike the power basis splines, the B splines have compact support and are numerically more stable. However, their explicit expressions are more complicated. See Eubank [11] p.300 for further details. In what follows, we will write B j (x) j = 1; J for a set of (generic) basic ....

C. de Boor. A Practical Guide to Splines. Springer-Verlag, New York, 1978.

....x Gammak (n Gamma k) n Gamma ) nx(nx Gamma 1) nx Gamma k 1) 3.4) Proof. It is well known that the p n; are the B splines of degree n over the knot vector t 0 = Delta Delta Delta = t n = 0, t n 1 = Delta Delta Delta = t 2n 1 = 1. Specializing the de Boor Fix formula (see [2], p. 159) to this case one finds that the nth degree BB coefficients a n; of g satisfy a n; d X =0 ( Gamma1) n Gamma) n ( n)g ( n) 3.5) where ;n (x) x n Gamma (x Gamma 1) n : Expanding ;n (x) in powers of x we obtain ;n (x) 1 n X k=0 ( Gamma1) k ....

de Boor C., A Practical Guide to Splines, Springer-Verlag, New York, 1978.

....space gives us a mapping from a d simplex onto a piece of a d dimensional manifold. Here, we will concentrate on triangular patches (d = 2) where the domain is a triangle, and the manifold is a three dimensional surface patch. The algorithms presented here assume a canonical domain triangle: Delta(0; 0) 0; 1) 1; 0) Farin s book provides a more detailed introduction to triangular B ezier patches [Farin93] 4.1 de Casteljau Evaluation As with tensor product surfaces, the straightforward approach to evaluation of a triangular patch of this type is via the de Casteljau algorithm. The algorithm for ....

....the details of the technique used for storage of the control net for triangular patches. Na ively, one could store the control points in a three dimensional array, indexed by i. However, we know that i 0 , i 1 and i 2 are not independent, so we can store the net in a two dimensional array, indexed by i 0 and i 1 (for example) This approach is fastest, but is not most space efficient. The most space efficient technique is to store the control net in a linear array of size: Dim(n; d) Dim(n Gamma 1; d) Dim(n; d Gamma 1) where Dim(n; 0) Dim(0; d) 1 and Dim(n; d) 0 for n or d negative. ....

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C. de Boor, "A Practical Guide to Splines", Applied Mathematical Sciences, Volume 27, Springer-Verlag, New York.

....to the realization of the wavelet Galerkin scheme for the coupling. 2.1. Biorthogonal Multiresolution on R. On R piecewise polynomial functions of degree d 1 can be de ned as follows. Denoting by [x 0 ; x d ]f the d th order divided di erence at the points x 0 ; x d 2 R (see e.g. [11]) the (centered) cardinal B spline of order d is given by (d) x) d[0; 1; d] x j d 2 k d 1 : where x l : maxf0; xg) l and bxc (dxe) is the largest (smallest) integer less (greater) than or equal to x. This scaling function (d) is normalized ....

C. de Boor. A practical guide to splines. Springer, 1978.

.... of n m and for non negative weights w ij . The solutions here are polynomial splines. Problem (1. 2) reduces to the problem of best interpolation when = n m, and to the problem of smoothing when = This generalizes the standard problem of smoothing studied in [So64] R67] R71] [dB78], W90] D93] and [dB98] for example, to include the Hermite type functionals f 7 f (j 1) t i ) and possibly zero weights (the reciprocal of the weights appear in the characterizations given in those papers that include the weights) A third instance of (1.2) is when = f(i; 1)g and z ij ....

C. de Boor (1978), A Practical Guide to Splines, Springer Verlag (New York).

....control points. As a consequence, the robust and efficient computation of interpolating B spline curves and surfaces has been a fundamental task in geometric modeling. The mathematical description of the B spline interpolation problem can be found in many textbooks, tutorials and papers, such as [4], 5] 6] 7] 22] or others. The standard approach is to solve a sparse linear system of equations. For this purpose, we have to build and decompose a matrix, whose rows and columns are computed by evaluation of the B spline basis functions at discrete positions in parameter space. In the case ....

....space. In the case of cardinal B spline interpolation the knots are equally spaced. Depending on the order of the B spline, the resulting matrix is banded diagonal and respective algorithms perform correspondingly. That is, the entire interpolation step can be solved in O(N) i.e. linear time [4]. Our motivation for the research presented in this paper was to point out an alternative by treating the interpolation problem from a signal processing point of view. This enables to carry over and to extend some findings from the vast amount of research in that area for the benefit of ....

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C. de Boor. A Practical Guide to Splines. Springer Verlag, Berlin, 1978.

....the writing velocity and acceleration. The x,y components of velocity and acceleration were computed from the derivatives of the pen coordinates with respect to time v = v x ; v y ) x; y) a = a x ; a y ) v x ; v y ) The derivatives were calculated using cubic smoothing B splines [10]. The path velocity magnitude (speed) and path tangent angle of the pen motion are given in terms of the v x , v y components by v = v 2 x v 2 y ) 1=2 ; tan Gamma1 (v y =v x ) The features derived from these speed components were root mean square (rms) speed V, average ....

C. de. Boor. A practical guide to splines. Springer Verlag, 1978.

.... authors have discussed the notion of geometric continuity at a common point for two incident space curves[13, 15, 24, 25] Furthermore there are a plenty of references which use di erent continuity criteria and construct parametric B splines to approximate an ordered list of points (see for e.g. [9, 11, 25]) The frame continuity used in this paper is a simpler form of geometric continuity with the connection matrix being diagonal, and di ers from the well known Frenet frame continuity that has a lower triangular connection matrix. We also exhibit how Pad e approximation can be adapted to yield very ....

....for s = 0 and s = If either Q ni (s) has zeros in [0; or the error max s2[0; jjr(s) R(s)jj , we halve the . The approximation error is bounded in the following way: Since e i (s) r i (s)Q ni (s) P mi (s) O(s k 1 (s ) k 1 ) by the remainder formula of Hermite interpolation [9], we have e i (s) s(s ) k 1 (r i Q ni ) 0; 0 z k 1 ; z k 1 ; s] where f [t 0 ; t r ] stands for divided di erence of f on t 0 ; t r . Hence jr i (s) R i (s)j 2 2k 2 jD ki (s)j min s2[0; jQ ni (s)j (7.3) where D ki (s) r ....

de Boor, C., A Practical Guide To Splines, Springer{Verlag, New York, Heidelberg, Berlin, 1978.

....have been used successfully in the incompressible pipe flow simulation of Loulou (1996) and the compressible jet of Rao (1997) B splines have high resolving power, allow easy implementation of boundary conditions, and allow the use of stretched grids. More details on B splines may be found in: De Boor (1978); Kravchenko, Moin Moser (1996) and Shariff Moser (1998) Their use in the present work is described in Guarini (1998) In the wall normal direction, Giles (1989, 1990) second order non reflecting boundary conditions are used at the freestream boundary and adiabatic no slip boundary ....

De Boor, C. 1978 A practical guide to splines. Springer.

....pass through the midpoints of the vertical impedance jumps in each layer, and at the end points, the splines match the impedances of the adjoining half spaces. For cubic spline interpolation, we employed de Boor s routines cubspl and ppvalu, obtained through Netlib [16] and described in [15]. The final two degrees of freedom in the cubic spline interpolant were specified using the not a knot condition. That is, the jump in the third derivative at the ends of the first and the last interval was set to zero. In the next set of illustrations, we consider the nonreflective coatings ....

C. de Boor, A practical guide to splines, Springer-Verlag, New York, 1978.

....discussed in Akerlof [1] 81 We propose to use cubic regression splines to estimate the frequency as follows. For each trial frequency v, calculate the phases # j = wt j mod 1. Calculate fitted values for y j by modeling brightness as a cubic spline of phase, using the B spline basis of De Boor [15] for good numerical stability. Then define the frequency estimate to be the frequency that minimizes SS n (v) over v # (0, W] Some care must be taken in the choice of knots. There should be enough knots so that the function shape can be successfully approximated by the spline basis, but not so ....

....to the phase plots of two of the MACHO star data sets. Smoothing splines differ from regression splines by having knots at each of the data points and by estimating the parameters through minimization of the sum of the RSS and a term which penalizes roughness in the fitted curve; see de Boor [15], Ch. 14, for more details. The raw data and fitted curves are displayed in Figure 3.10. The data in the upper plot comes from the blue band of star 77017:379, a relatively bright cepheid variable star with a fundamental period of approximately 4.017 days. The fitted curve was produced by fitting ....

C. de Boor (1978) A Practical Guide to Splines. Springer-Verlag, New York.

....B spline quaternion curves. For other quaternion curves and more details, see [KKS94a] Given n 1 control points fp i g, the B spline curve P (t) of order k is defined by: P (t) n X i=0 p i B k i (t) where the base functions B k i (t) s are defined by the following recurrence relation [Boo78]: B 1 i (x) 8 : 1 if t i t t i 1 0 otherwise and B k i (t) t Gamma t i t i k Gamma1 Gamma t i B k Gamma1 i (t) t i k Gamma t t i k Gamma t i 1 B k Gamma1 i 1 (t) The basis functions are C k Gamma2 continuous piecewise polynomials of degree (k Gamma 1) They are C ....

C. Boor. A Practical Guide to Splines. Springer-Verlag, 1978.

....DOF that specified the deviation of the tip of the tail from the midline. The lateral displacement of the body was described by a cubic spline curve, which is widely used in geometric modeling of natural objects (Terzopoulos et al. 1987) The cubic spline was computed with a MATLAB function (Boor, 1978; Hearn Baker, 1997) The input points to the spline function were midline points for the non flexing anterior of the fish body and one point at the tip of the tail. The nodes of the polygonal model were then displaced such that the midline followed the spline curve. Initial tracking studies ....

Boor, C. de. (1978). A practical guide to splines. New York: Springer-Verlag.

....the transformation described for an arbitrary differentiable potential V . It computes the transformation from J = 0 (which is found by finding the minimum of the potential) up through the J corresponding to a given value of H . The basis functions B j and C j are both taken to be B Splines [3] in # J , whose knots t i are chosen to be t 0 = t k 1 =0 (23) t i k = 1 k 1 i k 1 X =i 1 # J (i) i = 0, n k 1 (24) t n = t n k 1 = # J (n 1) 25) as described on pp. 218 9 of [3] The code computes q m for m # M for a given integer M . We do not use the data ....

....a given value of H . The basis functions B j and C j are both taken to be B Splines [3] in # J , whose knots t i are chosen to be t 0 = t k 1 =0 (23) t i k = 1 k 1 i k 1 X =i 1 # J (i) i = 0, n k 1 (24) t n = t n k 1 = # J (n 1) 25) as described on pp. 218 9 of [3]. The code computes q m for m # M for a given integer M . We do not use the data for H (J (i ) as described above. We take as an example the potential V (q) 1 cos q.We know the transformation for this potential: J = 8 # H 2 K H 2 K H 2 E H 2 (26) ....

C. de Boor, A Practical Guide to Splines. New York: Springer-Verlag, 1978.

....or convex hulls. For space curves (curves in 3 D) there are O(n 3 log m) L# optimal algorithms [62] Asymptotic Approximation. In related work, McClure and de Boor analyzed the error when approximating a highly continuous function y(x) using piecewisepolynomials with variable knots [82, 21]. We discuss only the special case of piecewise linear approximations. They analyzed the asymptotic behavior of the L p error of approximation in the limit as m, the number of vertices (knots) of the approximation, goes to infinity. They showed that the asymptotic L p error with regular ....

Carl de Boor. A Practical Guide to Splines. Springer, Berlin, 1978.

.... to define the i th normalized cubic B splines, N 4;u;i (x) u i 4 Gamma u i ) u i ; u i 1 ; u i 2 ; u i 3 ; u i 4 ] x Gamma u) 3 ; i = 0; Delta Delta Delta ; 2m 1; 2:2) 2 where the standard notation of fourth order divided difference with respect to the variable u is used (see [1,6]. Similarly, let v = f0 = v 0 = v 3 v 4 = v 5 v 6 = v 7 : v 2n = v 2n 1 v 2n 2 = v 2n 5 = 1g (2:3) be another knot sequence that defines the normalized cubic B splines N 4;v;j (x) j = 0; Delta Delta Delta ; 2n 1. In this paper, we will consider C 1 bi cubic ....

C. de Boor, A Practical Guide to Splines, Springer Verlag, 1978.

....There is no such difficulty in the univariate case precisely because the decision in which knot interval a given argument lies is made once, and only then does the evaluation commence. In fact, the standard algorithm does not really evaluate a spline. Rather (as is pointed out, e.g. in [de Boor 1978: p. 136] it provides, for given argument, the value at that argument of the polynomial pointed to by the choice of index. The standard choice of the index only makes certain that the polynomial so selected is the one with which the given spline agrees on the knot interval which contains the ....

C. de Boor, 1978: A Practical Guide to Splines . Springer Verlag, New York.

....without any figures, in just seven pages. The relevant literature on (univariate) B splines up to about 1975 is summarized in [B76] which also contains hints of the most exciting developments concerning B splines since then: knot insertion and the multivariate B splines. The two books on splines, [B78] and [Schu81] which have appeared since 1975, cover B splines in the traditional way. As presentations of splines from the CAGD point of view, the survey article [BFK84] and the Killer B s [BBB85,87] are particularly recommended. I refer you to these references and to the original papers ....

C. de Boor (1978), A Practical Guide to Splines, Springer-Verlag, New York.

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C. de. Boor. A Practical Guide to Splines. Springer-Verlag, Berlin, 1978.

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BOOR C. D., A practical guide to splines, Springer-Verlag, 1978.

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De Boor, C. (1978) A Practical Guide to Splines. Springer-Verlag, New York.

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de Boor, C., A Practical Guide to Splines. Springer-Verlag, New York, 1978.

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C. de Boor. A Practical Guide to Splines. Springer-Verlag, New York, NY, US, 1978.

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Carl de Boor; Practical Guide to Splines, Springer Verlag, New York (1978)

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Carl de Boor; Practical Guide to Splines, Springer Verlag, New York (1978)

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de Boor, C. (1978). A Practical Guide to Splines. Springer, New York.

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de Boor, C. (1978). A Practical Guide to Splines. Springer, New York.

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de Boor, C., 1978. A Practical Guide to Splines. Springer, New York.

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de Boor, C. (1978). A Practical Guide to Splines. Springer, New York.

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de Boor, C. (1978). A Practical Guide to Splines. Springer, New York. EVENT HISTORY REGRESSION 25

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de Boor, Carl, (1978), A Practical Guide to Splines, Springer-Verlag, New York. 15

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