9118:
488:
504:
496:
7897:
9169:
8784:
9138:
8644:
190:
305:
36:
135:
77:
7621:
9113:{\displaystyle {\begin{aligned}\mathbf {P} _{0}&={\frac {1}{6}}(\mathbf {b} _{0}+4\mathbf {b} _{1}+\mathbf {b} _{2}),\\\mathbf {P} _{1}&={\frac {1}{3}}(2\mathbf {b} _{1}+\mathbf {b} _{2}),\\\mathbf {P} _{2}&={\frac {1}{3}}(\mathbf {b} _{1}+2\mathbf {b} _{2}),\\\mathbf {P} _{3}&={\frac {1}{6}}(\mathbf {b} _{1}+4\mathbf {b} _{2}+\mathbf {b} _{3}).\end{aligned}}}
5801:
3887:
8239:
8337:
6223:(the last point of one curve coincides with the starting point of the next curve). Depending on the application, additional smoothness requirements (such as C1 or C2 continuity) may be added. C1 continuous curves have identical tangents at the breakpoint (where the two curves meet). C2 continuous curves have identical curvature at the breakpoint.
3306:
3611:
7892:{\displaystyle \mathbf {C} (t)={\frac {1}{6}}\;{\begin{bmatrix}t^{3}&t^{2}&t&1\end{bmatrix}}{\begin{bmatrix}-1&3&-3&1\\3&-6&3&0\\-3&0&3&0\\1&4&1&0\end{bmatrix}}{\begin{bmatrix}\mathbf {b} _{0}\\\mathbf {b} _{1}\\\mathbf {b} _{2}\\\mathbf {b} _{3}\end{bmatrix}}}
5226:
5554:
9195:
The knot vector is a sequence of parameter values that determines where and how the control points affect the NURBS curve. The number of knots is always equal to the number of control points plus curve degree plus one. Each time the parameter value enters a new knot span, a new control point becomes
6211:
mostly independent segments, whereas the B-spline with the same parameters smoothly transitions from subinterval to subinterval. To get something comparable from a Bézier curve, one would need to impose a smoothness condition on transitions between segments, resulting in some manner of Bézier spline
3622:
6456:
The main difficulty in applying this process is in determining the number of knots to use and where they should be placed. de Boor suggests various strategies to address this problem. For instance, the spacing between knots is decreased in proportion to the curvature (2nd derivative) of the data. A
8639:{\displaystyle \mathbf {C} (t)={\frac {1}{6}}{\biggl (}(-\mathbf {b} _{0}+3\mathbf {b} _{1}-3\mathbf {b} _{2}+\mathbf {b} _{3})t^{3}+(3\mathbf {b} _{0}-6\mathbf {b} _{1}+3\mathbf {b} _{2})t^{2}+(-3\mathbf {b} _{0}+3\mathbf {b} _{2})t+(\mathbf {b} _{0}+4\mathbf {b} _{1}+\mathbf {b} _{2}){\biggr )}}
7071:
6465:
curves has been investigated. Optimal spline functions of degrees 3–7 inclusive, based on symmetric arrangements of 5, 6, and 7 knots, have been computed and the method was applied for smoothing and differentiation of spectroscopic curves. In a comparable study, the two-dimensional version of the
7909:
1627:
A B-spline function is a combination of flexible bands that is controlled by a number of points that are called control points, creating smooth curves. These functions are used to create and manage complex shapes and surfaces using a number of points. B-spline function and Bézier functions are
897:
4980:
9161:. Like B-splines, they are defined by their order, and a knot vector, and a set of control points, but unlike simple B-splines, the control points each have a weight. When the weight is equal to 1, a NURBS is simply a B-spline and as such NURBS generalizes both B-splines and
3101:
1617:
10036:
2532:
6100:
is also a polynomial curve definable using a recursion from lower-degree curves of the same class and encoded in terms of control points, but a key difference is that all terms in the recursion for a Bézier curve segment have the same domain of definition (usually
2280:
1947:, the continuity of derivative order is reduced by 1 for each additional coincident knot. B-splines may share a subset of their knots, but two B-splines defined over exactly the same knots are identical. In other words, a B-spline is uniquely defined by its knots.
1326:
3361:
4066:
9179:
Like B-splines, NURBS control points determine the shape of the curve. Each point of the curve is computed by taking a weighted sum of a number of control points. The weight of each point varies according to the governing parameter. For a curve of degree
9362:
10151:
4991:
5796:{\displaystyle \mu _{k}=R_{k}(\mathbf {m} ;\mathbf {t} )=\int _{-\infty }^{\infty }x^{k}\cdot B_{i,n,{\textbf {norm}}}(x\mid t_{1}\dots t_{j})\,dx={\frac {\Gamma (k+1)\Gamma (m)}{\Gamma (m+k)}}\cdot D_{k}(\mathbf {m} ,\mathbf {t} )}
3882:{\displaystyle {\begin{aligned}&{\text{At }}x=1\colon \ B_{1}=B_{2}=0.5,\ {\frac {dB_{1}}{dx}}={\frac {dB_{2}}{dx}}=1.\\&{\text{At }}x=2\colon \ B_{2}=B_{3}=0.5,\ {\frac {dB_{2}}{dx}}={\frac {dB_{3}}{dx}}=-1.\end{aligned}}}
9175:
By evaluating a NURBS at various values of the parameters, the curve can be traced through space; likewise, by evaluating a NURBS surface at various values of the two parameters, the surface can be represented in
Cartesian space.
9641:
8234:{\displaystyle {\begin{aligned}B_{0}(t)&={\frac {1}{6}}(-t^{3}+3t^{2}-3t+1)\\B_{1}(t)&={\frac {1}{6}}(3t^{3}-6t^{2}+4)\\B_{2}(t)&={\frac {1}{6}}(-3t^{3}+3t^{2}+3t+1)\\B_{3}(t)&={\frac {1}{6}}t^{3}\end{aligned}}}
6842:
7294:
Because the B-splines are non-zero for just a finite number of knot intervals, if a single control point is moved, the corresponding change to the parametric curve is just over the parameter range of a small number knot
2022:
endpoints on each side, to give full support to the first and last B-spline, which affect the internal knot intervals. The values of the endpoints do not matter, usually the first or last internal knot is just repeated.
9797:
7290:
define a curve. If the control points are all transformed together in some way, such as being translated, rotated, scaled, or moved by any affine transformation, then the corresponding curve is transformed in the same
4231:
8329:
6401:
2142:
5399:
4677:
769:
4750:
5957:
3301:{\displaystyle {\begin{matrix}&&0\\&0&\\0&&B_{i-2,2}\\&B_{i-1,1}&\\B_{i,0}&&B_{i-1,2}\\&B_{i,1}&\\0&&B_{i,2}\\&0&\\&&0\end{matrix}}}
3627:
1369:
9808:
2822:
2288:
7259:
Working in reverse, a sequence of control points, knot values, and order of the B-spline define a parametric curve. This representation of a curve by control points has several useful properties:
9491:
8789:
7914:
6847:
3366:
2160:
2151:
for the spline function space, hence the name. This property follows from the fact that all pieces have the same continuity properties, within their individual range of support, at the knots.
1206:
448:. The places where the pieces meet are known as knots. The key property of spline functions is that they and their derivatives may be continuous, depending on the multiplicities of the knots.
2658:
984:
716:
3606:{\displaystyle {\begin{aligned}B_{1}&=x^{2}/2,&0&\leq x<1,\\B_{2}&=(-2x^{2}+6x-3)/2,&1&\leq x<2,\\B_{3}&=(3-x)^{2}/2,&2&\leq x<3.\end{aligned}}}
7227:
6831:
3898:
5519:
7354:
8776:
8745:
8714:
8683:
7613:
7582:
7551:
7520:
1794:
7402:
6685:
5445:
7445:
4354:
4292:
1866:
4536:
4631:
4388:
6057:
6015:
4561:
6620:
6451:
507:
Cardinal quartic B-spline with knot vector (0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5) and control points (0, 0, 0, 0, 1, 0, 0, 0, 0), and its first and second derivatives
6239:. When there is no theoretical basis for choosing a fitting function, the curve may be fitted with a spline function composed of a sum of B-splines, using the method of
3616:
These pieces are shown in the diagram. The continuity property of a quadratic spline function and its first derivative at the internal knots are illustrated, as follows
2947:
2577:
1109:
761:
584:
9205:
3353:
3062:
3023:
2901:
5476:
5993:
2984:
2862:
2725:
1169:
1067:
6181:
7483:
7288:
7254:
7155:
7128:
7101:
5546:
5221:{\displaystyle {\frac {d}{dx}}\sum _{i}\alpha _{i}B_{i,k}=\sum _{i=r-k+2}^{s-1}k{\frac {\alpha _{i}-\alpha _{i-1}}{t_{i+k}-t_{i}}}B_{i,k-1}\quad {\text{on}}\quad ,}
2692:
1748:
1361:
1136:
1034:
6772:
6743:
6714:
6570:
6521:
6086:
5266:
6209:
2020:
1994:
1925:
1895:
1721:
1675:
1198:
535:
446:
4705:
The term P-spline stands for "penalized B-spline". It refers to using the B-spline representation where the coefficients are determined partly by the data to be
6541:
6155:
6035:
5306:
5286:
2044:
1968:
1945:
1814:
1695:
1649:
1007:
644:
624:
604:
469:
413:
6131:
6235:, a set of data points is fitted with a curve defined by some mathematical function. For example, common types of curve fitting use a polynomial or a set of
1796:. The B-spline contributes only in the range between the first and last of these knots and is zero elsewhere. If each knot is separated by the same distance
9188:+1 intervals (knot spans) of the parameter space. Within those intervals, the weight changes according to a polynomial function (basis functions) of degree
7408:
A less desirable feature is that the parametric curve does not interpolate the control points. Usually the curve does not pass through the control points.
7066:{\displaystyle {\begin{aligned}X(t)&=\sum _{i}x_{i}B_{i,n}(t),\\Y(t)&=\sum _{i}y_{i}B_{i,n}(t),\\Z(t)&=\sum _{i}z_{i}B_{i,n}(t).\end{aligned}}}
10338:
Eilers, P. H. C. and Marx, B. D. (1996). Flexible smoothing with B-splines and penalties (with comments and rejoinder). Statistical
Science 11(2): 89–121.
9502:
10348:
Eilers, Paul H. C.; Marx, Brian D. (2003). "Multivariate calibration with temperature interaction using two-dimensional penalized signal regression".
9123:
A piecewise cubic B-spline is formed by a set of nodes and each four consecutive nodes define a cubic piece of the curve with the formulation above.
5239:
Univariate B-splines, i.e. B-splines where the knot positions lie in a single dimension, can be used to represent 1-d probability density functions
153:
9675:
499:
Cardinal cubic B-spline with knot vector (−2, −2, −2, −2, −1, 0, 1, 2, 2, 2, 2) and control points (0, 0, 0, 6, 0, 0, 0), and its first derivative
4093:
9192:. At the boundaries of the intervals, the basis functions go smoothly to zero, the smoothness being determined by the degree of the polynomial.
8247:
6137:
of the two terms in the B-spline recursion are different (the outermost subintervals are not common). This means that a Bézier curve of degree
10206:
Talebitooti, R.; Shojaeefard, M. H.; Yarmohammadisatri, Sadegh (2015). "Shape design optimization of cylindrical tank using b-spline curves".
6257:
4682:
Fast B-spline interpolation on a uniform sample domain can be done by iterative mean-filtering. Alternatively, a rectangle function equals
2052:
892:{\displaystyle B_{i,p}(t)={\begin{cases}{\text{non-zero}}&{\text{if }}t_{i}\leq t<t_{i+p+1},\\0&{\text{otherwise}}.\end{cases}}}
7404:, then the curve remains inside the bounding box of the control points. Also, in some sense, the curve broadly follows the control points.
5314:
4975:{\displaystyle {\frac {dB_{i,k}(x)}{dx}}=k\left({\frac {B_{i,k-1}(x)}{t_{i+k}-t_{i}}}-{\frac {B_{i+1,k-1}(x)}{t_{i+k+1}-t_{i+1}}}\right).}
10808:
4636:
1901:
at the knots. When all knots belonging to the B-spline are distinct, its derivatives are also continuous up to the derivative of degree
9141:
NURBS curve – polynomial curve defined in homogeneous coordinates (blue) and its projection on plane – rational curve (red)
5812:
4363:
A cardinal B-spline has uniformly spaced knots, therefore interpolation between the knots equals convolution with a smoothing kernel.
6836:
Because a B-splines form basis functions, each of the coordinate functions can be expressed as a linear sum of B-splines, so we have
4633:
gives first order interpolated B-spline values. Second-order B-spline interpolation is convolution with a rectangle function twice
475:
for spline functions of the same order defined over the same knots, meaning that all possible spline functions can be built from a
5231:
which shows that there is a simple relationship between the derivative of a spline function and the B-splines of degree one less.
1612:{\displaystyle B_{i,p}(t):={\dfrac {t-t_{i}}{t_{i+p}-t_{i}}}B_{i,p-1}(t)+{\dfrac {t_{i+p+1}-t}{t_{i+p+1}-t_{i+1}}}B_{i+1,p-1}(t).}
491:
Cardinal quadratic B-spline with knot vector (0, 0, 0, 1, 2, 3, 3, 3) and control points (0, 0, 1, 0, 0), and its first derivative
10031:{\displaystyle R_{i,j}(u,v)={\frac {N_{i,n}(u)N_{j,m}(v)w_{i,j}}{\sum _{p=1}^{k}\sum _{q=1}^{\ell }N_{p,n}(u)N_{q,m}(v)w_{p,q}}}}
2527:{\displaystyle B_{i,k}(x):={\frac {x-t_{i}}{t_{i+k}-t_{i}}}B_{i,k-1}(x)+{\frac {t_{i+k+1}-x}{t_{i+k+1}-t_{i+1}}}B_{i+1,k-1}(x).}
98:
85:
2733:
10764:
10745:
10711:
10685:
10662:
10582:
10482:
10182:
1868:) from its predecessor, the knot vector and the corresponding B-splines are called "uniform" (see cardinal B-spline below).
6059:
a vector with the respective knot multiplicities. One can therefore calculate any moment of a probability density function
3068:, B-splines are typically computed by algorithms that do not need to evaluate basis functions where they are zero, such as
352:
of B-splines of that degree. Cardinal B-splines have knots that are equidistant from each other. B-splines can be used for
2275:{\displaystyle B_{i,0}(x):={\begin{cases}1&{\text{if }}t_{i}\leq x<t_{i+1},\\0&{\text{otherwise}}.\end{cases}}}
1321:{\displaystyle B_{i,0}(t):={\begin{cases}1&{\text{if }}t_{i}\leq t<t_{i+1},\\0&{\text{otherwise}}.\end{cases}}}
9408:
10513:
10417:
Glüsenkamp, T. (2018). "Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data".
4694:
254:
10832:
10538:
Gans, Peter; Gill, J. Bernard (1984). "Smoothing and
Differentiation of Spectroscopic Curves Using Spline Functions".
2593:
908:
649:
291:
273:
226:
171:
116:
63:
10503:
9165:
and surfaces, the primary difference being the weighting of the control points which makes NURBS curves "rational".
10499:
1174:
B-splines can be constructed by means of the Cox–de Boor recursion formula. We start with the B-splines of degree
9157:, a powerful extension of B-splines is non-uniform rational B-splines (NURBS). NURBS are essentially B-splines in
4061:{\displaystyle {\frac {d^{2}B_{1}}{dx^{2}}}=1,\ {\frac {d^{2}B_{2}}{dx^{2}}}=-2,\ {\frac {d^{2}B_{3}}{dx^{2}}}=1.}
308:
Spline curve drawn as a weighted sum of B-splines with control points/control polygon, and marked component curves
4071:
Faster variants of the de Boor algorithm have been proposed, but they suffer from comparatively lower stability.
233:
10867:
7160:
6777:
211:
10774:
5481:
9132:
5308:, which each are area-normalized to unity (i.e. not directly evaluated using the standard de-Boor algorithm)
9150:
7301:
6220:
240:
7487:
10169:. Mathematics and Visualization. Berlin, Heidelberg: Springer Science & Business Media. p. 63.
8750:
8719:
8688:
8657:
7587:
7556:
7525:
7494:
1753:
7359:
6625:
5407:
7419:
6467:
6088:
represented by a sum of B-spline basis functions exactly, without resorting to numerical techniques.
4309:
4247:
1819:
371:, spline functions are constructed as linear combinations of B-splines with a set of control points.
357:
222:
207:
49:
6216:
4396:
2197:
1243:
806:
9399:
is a weight. The denominator is a normalizing factor that evaluates to one if all weights are one.
4690:. Therefore, cubic spline interpolation equals multiplying the signal in Fourier domain with sinc.
4566:
4369:
2154:
Expressions for the polynomial pieces can be derived by means of the Cox–de Boor recursion formula
17:
9357:{\displaystyle C(u)={\frac {\sum _{i=1}^{k}N_{i,n}(u)w_{i}P_{i}}{\sum _{i=1}^{k}N_{i,n}(u)w_{i}}}}
6040:
5998:
4544:
9158:
6575:
6429:
200:
2910:
2540:
1072:
724:
547:
10872:
10060:
4725:
3314:
3069:
3028:
2989:
2867:
392:
337:
10815:
5454:
10472:
9146:
6487:
6134:
5965:
3065:
2956:
2834:
2697:
1141:
1039:
364:
333:
10773:
Hovey, Chad (2022). Formulation and Python
Implementation of Bézier and B-Spline Geometry.
6457:
few applications have been published. For instance, the use of B-splines for fitting single
6453:
are the parameters to be determined. The knot values may be fixed or treated as parameters.
6160:
4679:; by iterative filtering with a rectangle function, higher-order interpolation is obtained.
10594:
10547:
10436:
7450:
7266:
7232:
7133:
7106:
7079:
6236:
5524:
2670:
1726:
1334:
1114:
1012:
345:
329:
10505:
The
Inventor Mentor: Programming Object-Oriented 3D Graphics with Open Inventor, Release 2
6748:
6719:
6690:
6546:
6497:
6062:
5242:
8:
6462:
6458:
6186:
4087:
are just shifted copies of each other. They can be obtained from the simpler definition.
2950:
1999:
1973:
1904:
1898:
1874:
1871:
For each finite knot interval where it is non-zero, a B-spline is a polynomial of degree
1700:
1654:
1177:
514:
425:
380:
379:
According to Gerald Farin, B-splines were explored as early as the nineteenth century by
10598:
10551:
10440:
10610:
10563:
10452:
10426:
10312:
10277:
6526:
6244:
6140:
6020:
5291:
5271:
4241:
2029:
1953:
1930:
1799:
1680:
1634:
992:
629:
609:
589:
476:
454:
398:
349:
317:
10837:
10361:
6104:
5548:, which in turn can be solved exactly via a contour integral and an iterative sum as
247:
10789:
10760:
10741:
10707:
10681:
10658:
10509:
10478:
10456:
10402:
10385:
10188:
10178:
9154:
6491:
368:
10614:
10567:
10219:
9636:{\displaystyle R_{i,n}(u)={\frac {N_{i,n}(u)w_{i}}{\sum _{j=1}^{k}N_{j,n}(u)w_{j}}}}
10848:
Uniform non rational B-Splines, Modelling curves in 2D space. Author:Stefan G. Beck
10725:
10602:
10555:
10444:
10397:
10357:
10316:
10304:
10281:
10269:
10215:
10170:
6475:
6212:(for which many control points would be determined by the smoothness requirement).
4710:
384:
10847:
10618:
149:
94:
10448:
721:
For a given sequence of knots, there is, up to a scaling factor, a unique spline
479:
of B-splines, and there is only one unique combination for each spline function.
388:
341:
10050:
9162:
8651:
6097:
10701:
10075:
9650:
6471:
4687:
3892:
The second derivative of a B-spline of degree 2 is discontinuous at the knots:
2148:
2046:
on a given set of knots can be expressed as a linear combination of B-splines:
2026:
The usefulness of B-splines lies in the fact that any spline function of order
472:
10792:
10704:
An
Introduction to Splines for Use in Computer Graphics and Geometric Modeling
10674:
10174:
10108:
de Boor gives FORTRAN routines for least-squares fitting of experimental data.
487:
10861:
10735:
10606:
10559:
10508:(1st ed.). Boston, MA, USA: Addison-Wesley Longman Publishing Co., Inc.
10192:
10164:
6240:
6232:
4706:
353:
55:
10260:
Lee, E. T. Y. (December 1982). "A Simplified B-Spline
Computation Routine".
9792:{\displaystyle S(u,v)=\sum _{i=1}^{k}\sum _{j=1}^{\ell }R_{i,j}(u,v)P_{i,j}}
503:
10842:
10721:
4366:
Example, if we want to interpolate three values in between B-spline nodes (
4226:{\displaystyle B_{i,n,t}(x)={\frac {x-t_{i}}{h}}n(\cdot -t_{i})_{+}^{n-1}.}
1970:-domain one is interested in. Since a single B-spline already extends over
495:
10099:
Strictly speaking, B-splines are usually defined as being left-continuous.
1950:
One distinguishes internal knots and end points. Internal knots cover the
8324:{\displaystyle \mathbf {C} (t)=\sum _{i=0}^{3}B_{i}(t)\,\mathbf {b} _{i}}
4718:
3079:-coded algorithm BSPLV, which generates values of the B-splines of order
313:
10308:
10273:
10205:
10055:
4714:
541:
419:
4724:
Two- and multidimensional P-spline approximations of data can use the
10797:
6396:{\displaystyle U=\sum _{{\text{all}}~x}\left\{W(x)\left\right\}^{2},}
2137:{\displaystyle S_{n,\mathbf {t} }(x)=\sum _{i}\alpha _{i}B_{i,n}(x).}
1750:, called knots or breakpoints, which must be in non-descending order
538:
416:
348:
partition. Any spline function of given degree can be expressed as a
10843:
TinySpline: Opensource C-library with bindings for various languages
9168:
5394:{\displaystyle p(x)=\sum _{i}c_{i}\cdot B_{i,n,{\textbf {norm}}}(x)}
4697:
for algebraic expressions for the cardinal B-splines of degree 1–4.
189:
10431:
10080:
10070:
10065:
9137:
6247:
for least-squares minimization is, for a spline function of degree
4672:{\displaystyle \mathbf {y} =\mathbf {x} *\mathbf {h} *\mathbf {h} }
1996:
knots, it follows that the internal knots need to be extended with
395:
in 1978 and is short for basis spline. A spline function of order
8650:
Since this is a cubic polynomial, we can also write it as a cubic
10852:
10470:
8331:. Expanding this, we can write the full polynomial form as below
3076:
304:
10727:
Curves and
Surfaces in Geometric Modeling: Theory and Algorithms
5952:{\displaystyle D_{k}={\frac {1}{k}}\sum \limits _{u=1}^{k}\left}
646:
where the pieces of polynomial meet are known as knots, denoted
10732:
This book is out of print and freely available from the author.
10295:
Lee, E. T. Y. (1986). "Comments on some B-spline algorithms".
6481:
10386:"B-splines, hypergeometric functions, and Dirichlet averages"
6572:
can be treated as two or three separate coordinate functions
6091:
2579:
is piecewise constant one or zero indicating which knot span
10702:
Richard H. Bartels; John C. Beatty; Brian A. Barsky (1987).
10583:"Time-Domain Filtering of Two-Dimensional Fluorescence Data"
6774:
are each spline functions, with a common set of knot values
3091:
is a linear combination of the pieces of B-splines of order
9653:
of two NURBS curves, thus using two independent parameters
7615:. It forms a polynomial of degree 3 that can be written as
4728:
of matrices to the minimization of calculation operations.
4683:
3087:. The following scheme illustrates how each piece of order
2268:
1314:
885:
10736:
Hartmut
Prautzsch; Wolfgang Boehm; Marco Paluszny (2002).
10165:
Hartmut
Prautzsch; Wolfgang Boehm; Marco Paluszny (2002).
10581:
Vicsek, Maria; Neal, Sharon L.; Warner, Isiah M. (1986).
6494:
applications, a spline curve is sometimes represented as
2907:
s are zero outside those respective ranges. For example,
1111:
becomes fixed. The knots in-between (and not including)
9184:, the influence of any control point is only nonzero in
4236:
The "placeholder" notation is used to indicate that the
4083:
between knots. The cardinal B-splines for a given order
2817:{\displaystyle {\frac {t_{i+k+1}-x}{t_{i+k+1}-t_{i+1}}}}
3311:
Application of the recursion formula with the knots at
3064:. However, because B-spline basis functions have local
2587:
is repeated). The recursion equation is in two parts:
7818:
7706:
7658:
3106:
9811:
9678:
9505:
9411:
9208:
8787:
8753:
8722:
8691:
8660:
8340:
8250:
7912:
7624:
7590:
7559:
7528:
7497:
7453:
7422:
7362:
7304:
7269:
7235:
7163:
7136:
7109:
7082:
6845:
6780:
6751:
6722:
6693:
6628:
6578:
6549:
6529:
6500:
6432:
6260:
6189:
6163:
6143:
6107:
6065:
6043:
6023:
6001:
5968:
5815:
5557:
5527:
5484:
5457:
5410:
5317:
5294:
5274:
5245:
4994:
4753:
4639:
4569:
4547:
4399:
4372:
4312:
4250:
4096:
3901:
3625:
3364:
3355:
gives the pieces of the uniform B-spline of order 3
3317:
3104:
3031:
2992:
2959:
2913:
2870:
2837:
2736:
2700:
2673:
2596:
2543:
2291:
2163:
2055:
2032:
2002:
1976:
1956:
1933:
1907:
1877:
1822:
1802:
1756:
1729:
1703:
1683:
1657:
1637:
1491:
1402:
1372:
1337:
1209:
1180:
1144:
1117:
1075:
1042:
1015:
995:
911:
772:
727:
652:
632:
612:
592:
550:
517:
457:
428:
401:
10471:
Eugene V. Shikin; Alexander I. Plis (14 July 1995).
9486:{\displaystyle C(u)=\sum _{i=1}^{k}R_{i,n}(u)P_{i}}
6470:and the spline method produced better results than
1628:applied extensively in shape optimization methods.
214:. Unsourced material may be challenged and removed.
144:
may be too technical for most readers to understand
10673:
10030:
9791:
9635:
9485:
9356:
9112:
8770:
8739:
8708:
8677:
8638:
8323:
8233:
7891:
7607:
7576:
7545:
7514:
7477:
7439:
7396:
7348:
7282:
7248:
7221:
7149:
7122:
7095:
7065:
6825:
6766:
6737:
6708:
6679:
6614:
6564:
6535:
6515:
6445:
6395:
6219:is a series of Bézier curves joined with at least
6203:
6175:
6149:
6125:
6080:
6051:
6029:
6009:
5987:
5951:
5795:
5540:
5513:
5470:
5439:
5393:
5300:
5280:
5260:
5234:
5220:
4974:
4671:
4625:
4555:
4530:
4382:
4348:
4286:
4225:
4060:
3881:
3605:
3347:
3300:
3056:
3017:
2978:
2941:
2895:
2856:
2816:
2719:
2686:
2652:
2571:
2526:
2274:
2136:
2038:
2014:
1988:
1962:
1939:
1927:. If the knots are coincident at a given value of
1919:
1889:
1860:
1808:
1788:
1742:
1715:
1689:
1669:
1643:
1611:
1355:
1320:
1192:
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1103:
1061:
1028:
1001:
978:
891:
755:
710:
638:
618:
598:
578:
529:
463:
440:
407:
10787:
9196:active, while an old control point is discarded.
8631:
8370:
10859:
10809:"B-splines of third order on a non-uniform grid"
10580:
2653:{\displaystyle {\frac {x-t_{i}}{t_{i+k}-t_{i}}}}
10759:. Springer. Chapter 7. B-Spline Approximation.
10350:Chemometrics and Intelligent Laboratory Systems
10129:Curves and surfaces for CAGD: a practical guide
979:{\displaystyle \sum _{i=0}^{m-p-1}B_{i,p}(t)=1}
711:{\displaystyle t_{0},t_{1},t_{2},\ldots ,t_{m}}
10730:. Morgan Kaufmann. Chapter 6. B-Spline Curves.
10464:
5521:can be written as Carlson's Dirichlet average
4079:A cardinal B-spline has a constant separation
10754:
10294:
1651:is a piecewise polynomial function of degree
10720:
10652:
10491:
6523:, a parametric curve of some real parameter
4740:is simply a function of B-splines of degree
10671:
9646:are known as the rational basis functions.
6482:Computer-aided design and computer graphics
5404:and with normalization constant constraint
1363:-degree B-splines are defined by recursion
64:Learn how and when to remove these messages
10416:
10347:
7652:
6092:Relationship to piecewise/composite Bézier
902:If we add the additional constraint that
10757:Curves and Surfaces for Computer Graphics
10740:. Springer Science & Business Media.
10430:
10401:
8308:
7903:This corresponds to B-spline polynomials
7222:{\displaystyle P_{i}=(x_{i},y_{i},z_{i})}
6826:{\displaystyle t_{1},t_{2},\ldots ,t_{n}}
5695:
4731:
292:Learn how and when to remove this message
274:Learn how and when to remove this message
172:Learn how and when to remove this message
156:, without removing the technical details.
117:Learn how and when to remove this message
10537:
10497:
9371:is the independent variable (instead of
9199:A NURBS curve takes the following form:
9136:
5514:{\displaystyle B_{i,n,{\textbf {norm}}}}
502:
494:
486:
303:
10383:
4736:The derivative of a B-spline of degree
1200:, i.e. piecewise constant polynomials.
14:
10860:
10341:
10199:
7256:are commonly known as control points.
10788:
7349:{\displaystyle \sum _{i}B_{i,n}(x)=1}
4695:Irwin–Hall distribution#Special cases
718:and sorted into nondecreasing order.
154:make it understandable to non-experts
4074:
3075:This relation leads directly to the
212:adding citations to reliable sources
183:
128:
70:
29:
10838:Interactive B-splines with JSXGraph
10640:Piegl and Tiller, chapter 4, sec. 4
10631:Piegl and Tiller, chapter 4, sec. 2
10259:
9649:A NURBS surface is obtained as the
8757:
8726:
8695:
8664:
7594:
7563:
7532:
7501:
5871:
5840:
5653:
5504:
5375:
4375:
24:
10695:
10672:Piegl, Les; Tiller, Wayne (1997).
9167:
8244:and the curve can be evaluated as
7411:
5740:
5726:
5708:
5616:
5611:
5288:B-spline basis functions of order
5268:. An example is a weighted sum of
25:
10884:
10781:
9402:It is customary to write this as
9379:is the number of control points,
9367:Here the notation is as follows.
8771:{\displaystyle {\textbf {P}}_{3}}
8740:{\displaystyle {\textbf {P}}_{2}}
8709:{\displaystyle {\textbf {P}}_{1}}
8678:{\displaystyle {\textbf {P}}_{0}}
7608:{\displaystyle {\textbf {b}}_{3}}
7577:{\displaystyle {\textbf {b}}_{2}}
7546:{\displaystyle {\textbf {b}}_{1}}
7515:{\displaystyle {\textbf {b}}_{0}}
6183:control points consists of about
1789:{\displaystyle t_{j}\leq t_{j+1}}
45:This article has multiple issues.
10474:Handbook on Splines for the User
9090:
9075:
9057:
9025:
9003:
8985:
8953:
8931:
8916:
8881:
8859:
8844:
8826:
8794:
8616:
8601:
8583:
8559:
8541:
8504:
8486:
8468:
8434:
8419:
8401:
8383:
8342:
8311:
8252:
7871:
7855:
7839:
7823:
7626:
7424:
7397:{\displaystyle B_{i,n}(x)\geq 0}
6680:{\displaystyle (x(t),y(t),z(t))}
6226:
6217:piecewise/composite Bézier curve
6045:
6003:
5786:
5778:
5593:
5585:
5440:{\displaystyle \sum _{i}c_{i}=1}
4665:
4657:
4649:
4641:
4571:
4549:
4500:
4467:
4440:
4413:
4401:
2068:
188:
133:
75:
34:
10806:
10634:
10625:
10574:
10531:
10522:
10410:
10390:Journal of Approximation Theory
10377:
10368:
10332:
10323:
10288:
10253:
10220:10.1016/j.compfluid.2014.12.004
10102:
9383:is a B-spline (used instead of
7485:is defined by four nodes (i.e.
7440:{\displaystyle \mathbf {C} (t)}
7157:can be combined to form points
5235:Moments of univariate B-splines
5185:
5179:
4349:{\displaystyle (t-x)_{+}^{n-1}}
4287:{\displaystyle (t-x)_{+}^{n-1}}
3025:and back to zero at and beyond
1861:{\displaystyle h=t_{j+1}-t_{j}}
1171:are called the internal knots.
374:
199:needs additional citations for
53:or discuss these issues on the
10853:B-Spline Editor by Shukant Pal
10833:bivariate B-spline from numpy
10738:Bézier and B-Spline Techniques
10244:
10235:
10226:
10167:Bézier and B-Spline Techniques
10158:
10143:
10134:
10121:
10093:
10006:
10000:
9981:
9975:
9896:
9890:
9871:
9865:
9840:
9828:
9770:
9758:
9694:
9682:
9617:
9611:
9559:
9553:
9528:
9522:
9470:
9464:
9421:
9415:
9338:
9332:
9270:
9264:
9218:
9212:
9100:
9052:
9013:
8980:
8941:
8908:
8869:
8821:
8626:
8578:
8569:
8530:
8514:
8460:
8444:
8375:
8352:
8346:
8305:
8299:
8262:
8256:
8197:
8191:
8174:
8124:
8104:
8098:
8081:
8043:
8023:
8017:
8000:
7953:
7933:
7927:
7636:
7630:
7472:
7460:
7434:
7428:
7385:
7379:
7337:
7331:
7216:
7177:
7053:
7047:
7001:
6995:
6982:
6976:
6930:
6924:
6911:
6905:
6859:
6853:
6761:
6755:
6732:
6726:
6703:
6697:
6674:
6671:
6665:
6656:
6650:
6641:
6635:
6629:
6609:
6606:
6600:
6591:
6585:
6579:
6559:
6553:
6510:
6504:
6371:
6365:
6317:
6311:
6300:
6294:
6120:
6108:
6075:
6069:
5790:
5774:
5755:
5743:
5735:
5729:
5723:
5711:
5692:
5660:
5597:
5581:
5388:
5382:
5327:
5321:
5255:
5249:
5212:
5186:
4915:
4909:
4838:
4832:
4782:
4776:
4709:, and partly by an additional
4620:
4578:
4531:{\displaystyle \mathbf {x} =.}
4522:
4408:
4390:), we can write the signal as
4326:
4313:
4264:
4251:
4200:
4180:
4177:
4159:
4125:
4119:
3558:
3545:
3486:
3452:
3342:
3318:
2936:
2930:
2566:
2560:
2518:
2512:
2401:
2395:
2314:
2308:
2186:
2180:
2128:
2122:
2080:
2074:
1603:
1597:
1484:
1478:
1395:
1389:
1350:
1338:
1232:
1226:
1098:
1092:
967:
961:
795:
789:
750:
744:
573:
567:
13:
1:
10528:de Boor, Chapter XIV, p. 235.
10362:10.1016/S0169-7439(03)00029-7
10115:
10041:as rational basis functions.
9133:Non-uniform rational B-spline
6017:represents a vector with the
4626:{\displaystyle \mathbf {h} =}
4383:{\displaystyle {\textbf {b}}}
1622:
1069:, then the scaling factor of
482:
10655:A Practical Guide to Splines
10403:10.1016/0021-9045(91)90006-V
9151:computer aided manufacturing
7447:with a normalized parameter
6052:{\displaystyle \mathbf {m} }
6010:{\displaystyle \mathbf {t} }
4556:{\displaystyle \mathbf {x} }
3095: − 1 to its left.
7:
10680:(2nd. ed.). Springer.
10477:. CRC Press. pp. 96–.
10152:Interpolating cubic splines
10044:
7229:in 3-d space. These points
6687:. The coordinate functions
6615:{\displaystyle (x(t),y(t))}
6446:{\displaystyle \alpha _{i}}
4700:
2147:B-splines play the role of
10:
10889:
10449:10.1140/epjp/i2018-12042-x
10131:. Morgan Kaufmann. p. 119.
9391:is the polynomial degree,
9130:
4563:with a rectangle function
4541:Convolution of the signal
2942:{\displaystyle B_{i,1}(x)}
2827:ramps from one to zero as
2663:ramps from zero to one as
2572:{\displaystyle B_{j,0}(x)}
1104:{\displaystyle B_{i,p}(t)}
756:{\displaystyle B_{i,p}(t)}
579:{\displaystyle B_{i,p}(t)}
332:function that has minimal
10175:10.1007/978-3-662-04919-8
6543:. In this case the curve
5478:of a normalized B-spline
4302:is to be taken by fixing
3348:{\displaystyle (0,1,2,3)}
3057:{\displaystyle x=t_{i+2}}
3018:{\displaystyle x=t_{i+1}}
2896:{\displaystyle t_{i+k+1}}
2583:is in (zero if knot span
391:"B-spline" was coined by
358:numerical differentiation
10607:10.1366/0003702864508773
10560:10.1366/0003702844555511
10498:Wernecke, Josie (1993).
10155:. Springer. p. 151.
10086:
9126:
7356:, and at all times each
6468:Savitzky–Golay filtering
6422:) is the datum value at
5471:{\displaystyle \mu _{k}}
336:with respect to a given
89:may need to be rewritten
9496:in which the functions
9395:is a control point and
9159:homogeneous coordinates
7416:A cubic B-spline curve
5988:{\displaystyle D_{0}=1}
2979:{\displaystyle x=t_{i}}
2857:{\displaystyle t_{i+1}}
2720:{\displaystyle t_{i+k}}
1164:{\displaystyle t_{m-p}}
1062:{\displaystyle t_{m-p}}
10755:David Salomon (2006).
10384:Carlson, B.C. (1991).
10149:Gary D. Knott (2000),
10032:
9958:
9937:
9793:
9741:
9720:
9637:
9594:
9487:
9447:
9358:
9315:
9247:
9172:
9142:
9114:
8772:
8741:
8710:
8679:
8640:
8325:
8288:
8235:
7893:
7609:
7578:
7547:
7516:
7479:
7441:
7398:
7350:
7284:
7250:
7223:
7151:
7124:
7097:
7067:
6827:
6768:
6739:
6710:
6681:
6616:
6566:
6537:
6517:
6447:
6397:
6205:
6177:
6176:{\displaystyle m\gg n}
6151:
6127:
6082:
6053:
6031:
6011:
5989:
5953:
5890:
5859:
5797:
5542:
5515:
5472:
5441:
5395:
5302:
5282:
5262:
5222:
5087:
4976:
4732:Derivative expressions
4726:face-splitting product
4673:
4627:
4557:
4532:
4384:
4350:
4288:
4227:
4062:
3883:
3607:
3349:
3302:
3058:
3019:
2980:
2943:
2897:
2858:
2818:
2721:
2688:
2654:
2573:
2528:
2276:
2138:
2040:
2016:
1990:
1964:
1941:
1921:
1891:
1862:
1810:
1790:
1744:
1717:
1691:
1671:
1645:
1613:
1357:
1322:
1194:
1165:
1132:
1105:
1063:
1030:
1003:
980:
944:
893:
757:
712:
640:
620:
600:
580:
531:
508:
500:
492:
465:
442:
409:
393:Isaac Jacob Schoenberg
360:of experimental data.
309:
10868:Splines (mathematics)
10653:Carl de Boor (1978).
10208:Computer & Fluids
10127:Farin, G. E. (2002).
10033:
9938:
9917:
9794:
9721:
9700:
9638:
9574:
9488:
9427:
9359:
9295:
9227:
9171:
9147:computer aided design
9140:
9115:
8773:
8742:
8711:
8680:
8654:with control points
8641:
8326:
8268:
8236:
7894:
7610:
7579:
7548:
7517:
7480:
7478:{\displaystyle t\in }
7442:
7399:
7351:
7285:
7283:{\displaystyle P_{i}}
7251:
7249:{\displaystyle P_{i}}
7224:
7152:
7150:{\displaystyle z_{i}}
7125:
7123:{\displaystyle y_{i}}
7098:
7096:{\displaystyle x_{i}}
7068:
6828:
6769:
6740:
6711:
6682:
6617:
6567:
6538:
6518:
6488:computer-aided design
6448:
6398:
6237:exponential functions
6206:
6178:
6152:
6128:
6083:
6054:
6032:
6012:
5990:
5954:
5870:
5839:
5798:
5543:
5541:{\displaystyle R_{k}}
5516:
5473:
5442:
5396:
5303:
5283:
5263:
5223:
5049:
4977:
4674:
4628:
4558:
4533:
4385:
4351:
4294:of the two variables
4289:
4228:
4063:
3884:
3608:
3350:
3303:
3059:
3020:
2981:
2944:
2898:
2859:
2819:
2722:
2689:
2687:{\displaystyle t_{i}}
2655:
2574:
2529:
2277:
2139:
2041:
2017:
1991:
1965:
1942:
1922:
1892:
1863:
1811:
1791:
1745:
1743:{\displaystyle t_{j}}
1718:
1697:. It is defined over
1692:
1672:
1646:
1614:
1358:
1356:{\displaystyle (p+1)}
1323:
1195:
1166:
1133:
1131:{\displaystyle t_{p}}
1106:
1064:
1031:
1029:{\displaystyle t_{p}}
1004:
981:
912:
894:
758:
713:
641:
621:
601:
581:
532:
506:
498:
490:
466:
443:
410:
365:computer-aided design
307:
10587:Applied Spectroscopy
10540:Applied Spectroscopy
9809:
9676:
9503:
9409:
9206:
8785:
8751:
8720:
8689:
8658:
8338:
8248:
7910:
7622:
7588:
7557:
7526:
7495:
7451:
7420:
7360:
7302:
7267:
7233:
7161:
7134:
7107:
7080:
6843:
6778:
6767:{\displaystyle z(t)}
6749:
6738:{\displaystyle y(t)}
6720:
6709:{\displaystyle x(t)}
6691:
6626:
6576:
6565:{\displaystyle C(t)}
6547:
6527:
6516:{\displaystyle C(t)}
6498:
6430:
6258:
6187:
6161:
6141:
6105:
6081:{\displaystyle p(x)}
6063:
6041:
6021:
5999:
5966:
5813:
5555:
5525:
5482:
5455:
5408:
5315:
5292:
5272:
5261:{\displaystyle p(x)}
5243:
4992:
4751:
4713:that aims to impose
4637:
4567:
4545:
4397:
4370:
4310:
4248:
4094:
3899:
3623:
3362:
3315:
3102:
3029:
2990:
2957:
2911:
2903:. The corresponding
2868:
2835:
2734:
2698:
2671:
2594:
2541:
2289:
2161:
2053:
2030:
2000:
1974:
1954:
1931:
1905:
1875:
1820:
1800:
1754:
1727:
1701:
1681:
1655:
1635:
1631:A B-spline of order
1370:
1335:
1207:
1178:
1142:
1115:
1073:
1040:
1013:
993:
909:
770:
725:
650:
630:
610:
590:
548:
515:
511:A B-spline of order
455:
426:
399:
208:improve this article
10706:. Morgan Kaufmann.
10657:. Springer-Verlag.
10599:1986ApSpe..40..542V
10552:1984ApSpe..38..370G
10441:2018EPJP..133..218G
10061:De Boor's algorithm
7263:The control points
6476:Chebyshev filtering
6426:. The coefficients
6414:) is a weight, and
6204:{\displaystyle m/n}
6037:knot positions and
5620:
4345:
4283:
4219:
3070:de Boor's algorithm
2953:that is zero below
2951:triangular function
2015:{\displaystyle n-1}
1989:{\displaystyle 1+n}
1920:{\displaystyle n-2}
1899:continuous function
1890:{\displaystyle n-1}
1716:{\displaystyle 1+n}
1670:{\displaystyle n-1}
1193:{\displaystyle p=0}
537:is a collection of
530:{\displaystyle p+1}
451:B-splines of order
441:{\displaystyle n-1}
422:function of degree
381:Nikolai Lobachevsky
10790:Weisstein, Eric W.
10309:10.1007/BF02240069
10274:10.1007/BF02246763
10028:
9789:
9633:
9483:
9354:
9173:
9143:
9110:
9108:
8768:
8737:
8706:
8675:
8636:
8321:
8231:
8229:
7889:
7883:
7807:
7695:
7605:
7574:
7543:
7512:
7475:
7437:
7394:
7346:
7314:
7280:
7246:
7219:
7147:
7120:
7093:
7063:
7061:
7020:
6949:
6878:
6823:
6764:
6735:
6706:
6677:
6612:
6562:
6533:
6513:
6443:
6393:
6332:
6284:
6245:objective function
6201:
6173:
6147:
6123:
6078:
6049:
6027:
6007:
5985:
5949:
5793:
5603:
5538:
5511:
5468:
5437:
5420:
5391:
5342:
5298:
5278:
5258:
5218:
5019:
4985:This implies that
4972:
4669:
4623:
4553:
4528:
4380:
4346:
4325:
4284:
4263:
4242:divided difference
4223:
4199:
4058:
3879:
3877:
3603:
3601:
3345:
3298:
3296:
3054:
3015:
2986:, ramps to one at
2976:
2939:
2893:
2854:
2814:
2717:
2684:
2650:
2569:
2524:
2272:
2267:
2134:
2095:
2036:
2012:
1986:
1960:
1937:
1917:
1897:. A B-spline is a
1887:
1858:
1806:
1786:
1740:
1713:
1687:
1667:
1641:
1609:
1567:
1454:
1353:
1318:
1313:
1190:
1161:
1128:
1101:
1059:
1026:
1009:between the knots
999:
976:
889:
884:
753:
708:
636:
616:
596:
576:
527:
509:
501:
493:
477:linear combination
461:
438:
405:
350:linear combination
318:numerical analysis
310:
10766:978-0-387-28452-1
10747:978-3-540-43761-1
10713:978-1-55860-400-1
10687:978-3-540-61545-3
10664:978-3-540-90356-7
10621:on June 23, 2017.
10484:978-0-8493-9404-1
10184:978-3-540-43761-1
10026:
9631:
9352:
9155:computer graphics
9050:
8978:
8906:
8819:
8759:
8728:
8697:
8666:
8366:
8215:
8122:
8041:
7951:
7650:
7596:
7565:
7534:
7503:
7305:
7011:
6940:
6869:
6536:{\displaystyle t}
6492:computer graphics
6323:
6279:
6275:
6267:
6150:{\displaystyle n}
6030:{\displaystyle j}
5837:
5759:
5655:
5506:
5411:
5377:
5333:
5301:{\displaystyle n}
5281:{\displaystyle i}
5183:
5155:
5010:
5008:
4962:
4873:
4794:
4377:
4356:as a function of
4154:
4075:Cardinal B-spline
4050:
4010:
3994:
3954:
3941:
3864:
3834:
3808:
3773:
3757:
3741:
3711:
3685:
3650:
3634:
2812:
2648:
2482:
2371:
2260:
2208:
2086:
2039:{\displaystyle n}
1963:{\displaystyle x}
1940:{\displaystyle x}
1809:{\displaystyle h}
1690:{\displaystyle x}
1644:{\displaystyle n}
1566:
1453:
1306:
1254:
1002:{\displaystyle t}
877:
819:
812:
639:{\displaystyle t}
619:{\displaystyle t}
599:{\displaystyle p}
464:{\displaystyle n}
408:{\displaystyle n}
369:computer graphics
302:
301:
294:
284:
283:
276:
258:
182:
181:
174:
127:
126:
119:
99:lead layout guide
68:
16:(Redirected from
10880:
10829:
10827:
10826:
10820:
10814:. Archived from
10813:
10803:
10802:
10770:
10751:
10731:
10717:
10691:
10679:
10668:
10641:
10638:
10632:
10629:
10623:
10622:
10617:. Archived from
10578:
10572:
10571:
10535:
10529:
10526:
10520:
10519:
10495:
10489:
10488:
10468:
10462:
10460:
10434:
10414:
10408:
10407:
10405:
10381:
10375:
10374:de Boor, p. 115.
10372:
10366:
10365:
10345:
10339:
10336:
10330:
10329:de Boor, p. 322.
10327:
10321:
10320:
10292:
10286:
10285:
10257:
10251:
10250:de Boor, p. 134.
10248:
10242:
10239:
10233:
10232:de Boor, p. 113.
10230:
10224:
10223:
10203:
10197:
10196:
10162:
10156:
10147:
10141:
10140:de Boor, p. 114.
10138:
10132:
10125:
10109:
10106:
10100:
10097:
10037:
10035:
10034:
10029:
10027:
10025:
10024:
10023:
9999:
9998:
9974:
9973:
9957:
9952:
9936:
9931:
9915:
9914:
9913:
9889:
9888:
9864:
9863:
9847:
9827:
9826:
9798:
9796:
9795:
9790:
9788:
9787:
9757:
9756:
9740:
9735:
9719:
9714:
9642:
9640:
9639:
9634:
9632:
9630:
9629:
9628:
9610:
9609:
9593:
9588:
9572:
9571:
9570:
9552:
9551:
9535:
9521:
9520:
9492:
9490:
9489:
9484:
9482:
9481:
9463:
9462:
9446:
9441:
9363:
9361:
9360:
9355:
9353:
9351:
9350:
9349:
9331:
9330:
9314:
9309:
9293:
9292:
9291:
9282:
9281:
9263:
9262:
9246:
9241:
9225:
9119:
9117:
9116:
9111:
9109:
9099:
9098:
9093:
9084:
9083:
9078:
9066:
9065:
9060:
9051:
9043:
9034:
9033:
9028:
9012:
9011:
9006:
8994:
8993:
8988:
8979:
8971:
8962:
8961:
8956:
8940:
8939:
8934:
8925:
8924:
8919:
8907:
8899:
8890:
8889:
8884:
8868:
8867:
8862:
8853:
8852:
8847:
8835:
8834:
8829:
8820:
8812:
8803:
8802:
8797:
8777:
8775:
8774:
8769:
8767:
8766:
8761:
8760:
8746:
8744:
8743:
8738:
8736:
8735:
8730:
8729:
8715:
8713:
8712:
8707:
8705:
8704:
8699:
8698:
8684:
8682:
8681:
8676:
8674:
8673:
8668:
8667:
8645:
8643:
8642:
8637:
8635:
8634:
8625:
8624:
8619:
8610:
8609:
8604:
8592:
8591:
8586:
8568:
8567:
8562:
8550:
8549:
8544:
8526:
8525:
8513:
8512:
8507:
8495:
8494:
8489:
8477:
8476:
8471:
8456:
8455:
8443:
8442:
8437:
8428:
8427:
8422:
8410:
8409:
8404:
8392:
8391:
8386:
8374:
8373:
8367:
8359:
8345:
8330:
8328:
8327:
8322:
8320:
8319:
8314:
8298:
8297:
8287:
8282:
8255:
8240:
8238:
8237:
8232:
8230:
8226:
8225:
8216:
8208:
8190:
8189:
8158:
8157:
8142:
8141:
8123:
8115:
8097:
8096:
8074:
8073:
8058:
8057:
8042:
8034:
8016:
8015:
7984:
7983:
7968:
7967:
7952:
7944:
7926:
7925:
7898:
7896:
7895:
7890:
7888:
7887:
7880:
7879:
7874:
7864:
7863:
7858:
7848:
7847:
7842:
7832:
7831:
7826:
7812:
7811:
7700:
7699:
7682:
7681:
7670:
7669:
7651:
7643:
7629:
7614:
7612:
7611:
7606:
7604:
7603:
7598:
7597:
7583:
7581:
7580:
7575:
7573:
7572:
7567:
7566:
7552:
7550:
7549:
7544:
7542:
7541:
7536:
7535:
7521:
7519:
7518:
7513:
7511:
7510:
7505:
7504:
7484:
7482:
7481:
7476:
7446:
7444:
7443:
7438:
7427:
7403:
7401:
7400:
7395:
7378:
7377:
7355:
7353:
7352:
7347:
7330:
7329:
7313:
7289:
7287:
7286:
7281:
7279:
7278:
7255:
7253:
7252:
7247:
7245:
7244:
7228:
7226:
7225:
7220:
7215:
7214:
7202:
7201:
7189:
7188:
7173:
7172:
7156:
7154:
7153:
7148:
7146:
7145:
7129:
7127:
7126:
7121:
7119:
7118:
7102:
7100:
7099:
7094:
7092:
7091:
7072:
7070:
7069:
7064:
7062:
7046:
7045:
7030:
7029:
7019:
6975:
6974:
6959:
6958:
6948:
6904:
6903:
6888:
6887:
6877:
6832:
6830:
6829:
6824:
6822:
6821:
6803:
6802:
6790:
6789:
6773:
6771:
6770:
6765:
6744:
6742:
6741:
6736:
6715:
6713:
6712:
6707:
6686:
6684:
6683:
6678:
6621:
6619:
6618:
6613:
6571:
6569:
6568:
6563:
6542:
6540:
6539:
6534:
6522:
6520:
6519:
6514:
6452:
6450:
6449:
6444:
6442:
6441:
6402:
6400:
6399:
6394:
6389:
6388:
6383:
6379:
6378:
6374:
6364:
6363:
6342:
6341:
6331:
6283:
6277:
6276:
6273:
6210:
6208:
6207:
6202:
6197:
6182:
6180:
6179:
6174:
6156:
6154:
6153:
6148:
6132:
6130:
6129:
6126:{\displaystyle }
6124:
6087:
6085:
6084:
6079:
6058:
6056:
6055:
6050:
6048:
6036:
6034:
6033:
6028:
6016:
6014:
6013:
6008:
6006:
5994:
5992:
5991:
5986:
5978:
5977:
5958:
5956:
5955:
5950:
5948:
5944:
5943:
5942:
5927:
5923:
5922:
5921:
5916:
5915:
5914:
5900:
5899:
5889:
5884:
5858:
5853:
5838:
5830:
5825:
5824:
5802:
5800:
5799:
5794:
5789:
5781:
5773:
5772:
5760:
5758:
5738:
5706:
5691:
5690:
5678:
5677:
5659:
5658:
5657:
5656:
5630:
5629:
5619:
5614:
5596:
5588:
5580:
5579:
5567:
5566:
5547:
5545:
5544:
5539:
5537:
5536:
5520:
5518:
5517:
5512:
5510:
5509:
5508:
5507:
5477:
5475:
5474:
5469:
5467:
5466:
5446:
5444:
5443:
5438:
5430:
5429:
5419:
5400:
5398:
5397:
5392:
5381:
5380:
5379:
5378:
5352:
5351:
5341:
5307:
5305:
5304:
5299:
5287:
5285:
5284:
5279:
5267:
5265:
5264:
5259:
5227:
5225:
5224:
5219:
5211:
5210:
5198:
5197:
5184:
5181:
5178:
5177:
5156:
5154:
5153:
5152:
5140:
5139:
5123:
5122:
5121:
5103:
5102:
5092:
5086:
5075:
5045:
5044:
5029:
5028:
5018:
5009:
5007:
4996:
4981:
4979:
4978:
4973:
4968:
4964:
4963:
4961:
4960:
4959:
4941:
4940:
4918:
4908:
4907:
4879:
4874:
4872:
4871:
4870:
4858:
4857:
4841:
4831:
4830:
4808:
4795:
4793:
4785:
4775:
4774:
4755:
4744: − 1:
4711:penalty function
4678:
4676:
4675:
4670:
4668:
4660:
4652:
4644:
4632:
4630:
4629:
4624:
4616:
4602:
4588:
4574:
4562:
4560:
4559:
4554:
4552:
4537:
4535:
4534:
4529:
4509:
4508:
4503:
4476:
4475:
4470:
4449:
4448:
4443:
4422:
4421:
4416:
4404:
4389:
4387:
4386:
4381:
4379:
4378:
4355:
4353:
4352:
4347:
4344:
4333:
4306:and considering
4293:
4291:
4290:
4285:
4282:
4271:
4244:of the function
4232:
4230:
4229:
4224:
4218:
4207:
4198:
4197:
4155:
4150:
4149:
4148:
4132:
4118:
4117:
4067:
4065:
4064:
4059:
4051:
4049:
4048:
4047:
4034:
4033:
4032:
4023:
4022:
4012:
4008:
3995:
3993:
3992:
3991:
3978:
3977:
3976:
3967:
3966:
3956:
3952:
3942:
3940:
3939:
3938:
3925:
3924:
3923:
3914:
3913:
3903:
3888:
3886:
3885:
3880:
3878:
3865:
3863:
3855:
3854:
3853:
3840:
3835:
3833:
3825:
3824:
3823:
3810:
3806:
3796:
3795:
3783:
3782:
3771:
3758:
3755:
3752:
3742:
3740:
3732:
3731:
3730:
3717:
3712:
3710:
3702:
3701:
3700:
3687:
3683:
3673:
3672:
3660:
3659:
3648:
3635:
3632:
3629:
3612:
3610:
3609:
3604:
3602:
3571:
3566:
3565:
3537:
3536:
3493:
3470:
3469:
3444:
3443:
3400:
3395:
3394:
3378:
3377:
3354:
3352:
3351:
3346:
3307:
3305:
3304:
3299:
3297:
3289:
3288:
3285:
3279:
3275:
3274:
3258:
3250:
3248:
3247:
3231:
3227:
3226:
3204:
3202:
3201:
3183:
3181:
3180:
3158:
3154:
3153:
3131:
3123:
3117:
3109:
3108:
3063:
3061:
3060:
3055:
3053:
3052:
3024:
3022:
3021:
3016:
3014:
3013:
2985:
2983:
2982:
2977:
2975:
2974:
2948:
2946:
2945:
2940:
2929:
2928:
2902:
2900:
2899:
2894:
2892:
2891:
2863:
2861:
2860:
2855:
2853:
2852:
2823:
2821:
2820:
2815:
2813:
2811:
2810:
2809:
2791:
2790:
2768:
2761:
2760:
2738:
2726:
2724:
2723:
2718:
2716:
2715:
2693:
2691:
2690:
2685:
2683:
2682:
2659:
2657:
2656:
2651:
2649:
2647:
2646:
2645:
2633:
2632:
2616:
2615:
2614:
2598:
2578:
2576:
2575:
2570:
2559:
2558:
2533:
2531:
2530:
2525:
2511:
2510:
2483:
2481:
2480:
2479:
2461:
2460:
2438:
2431:
2430:
2408:
2394:
2393:
2372:
2370:
2369:
2368:
2356:
2355:
2339:
2338:
2337:
2321:
2307:
2306:
2281:
2279:
2278:
2273:
2271:
2270:
2261:
2258:
2244:
2243:
2219:
2218:
2209:
2206:
2179:
2178:
2143:
2141:
2140:
2135:
2121:
2120:
2105:
2104:
2094:
2073:
2072:
2071:
2045:
2043:
2042:
2037:
2021:
2019:
2018:
2013:
1995:
1993:
1992:
1987:
1969:
1967:
1966:
1961:
1946:
1944:
1943:
1938:
1926:
1924:
1923:
1918:
1896:
1894:
1893:
1888:
1867:
1865:
1864:
1859:
1857:
1856:
1844:
1843:
1815:
1813:
1812:
1807:
1795:
1793:
1792:
1787:
1785:
1784:
1766:
1765:
1749:
1747:
1746:
1741:
1739:
1738:
1722:
1720:
1719:
1714:
1696:
1694:
1693:
1688:
1676:
1674:
1673:
1668:
1650:
1648:
1647:
1642:
1618:
1616:
1615:
1610:
1596:
1595:
1568:
1565:
1564:
1563:
1545:
1544:
1522:
1515:
1514:
1492:
1477:
1476:
1455:
1452:
1451:
1450:
1438:
1437:
1421:
1420:
1419:
1403:
1388:
1387:
1362:
1360:
1359:
1354:
1327:
1325:
1324:
1319:
1317:
1316:
1307:
1304:
1290:
1289:
1265:
1264:
1255:
1252:
1225:
1224:
1199:
1197:
1196:
1191:
1170:
1168:
1167:
1162:
1160:
1159:
1137:
1135:
1134:
1129:
1127:
1126:
1110:
1108:
1107:
1102:
1091:
1090:
1068:
1066:
1065:
1060:
1058:
1057:
1035:
1033:
1032:
1027:
1025:
1024:
1008:
1006:
1005:
1000:
985:
983:
982:
977:
960:
959:
943:
926:
898:
896:
895:
890:
888:
887:
878:
875:
861:
860:
830:
829:
820:
817:
813:
810:
788:
787:
762:
760:
759:
754:
743:
742:
717:
715:
714:
709:
707:
706:
688:
687:
675:
674:
662:
661:
645:
643:
642:
637:
626:. The values of
625:
623:
622:
617:
605:
603:
602:
597:
585:
583:
582:
577:
566:
565:
536:
534:
533:
528:
470:
468:
467:
462:
447:
445:
444:
439:
414:
412:
411:
406:
385:Kazan University
297:
290:
279:
272:
268:
265:
259:
257:
216:
192:
184:
177:
170:
166:
163:
157:
137:
136:
129:
122:
115:
111:
108:
102:
95:improve the lead
79:
78:
71:
60:
38:
37:
30:
21:
10888:
10887:
10883:
10882:
10881:
10879:
10878:
10877:
10858:
10857:
10824:
10822:
10818:
10811:
10807:Ruf, Johannes.
10784:
10767:
10748:
10714:
10698:
10696:Further reading
10688:
10665:
10644:
10639:
10635:
10630:
10626:
10579:
10575:
10536:
10532:
10527:
10523:
10516:
10496:
10492:
10485:
10469:
10465:
10415:
10411:
10382:
10378:
10373:
10369:
10346:
10342:
10337:
10333:
10328:
10324:
10293:
10289:
10258:
10254:
10249:
10245:
10241:de Boor, p 131.
10240:
10236:
10231:
10227:
10204:
10200:
10185:
10163:
10159:
10148:
10144:
10139:
10135:
10126:
10122:
10118:
10113:
10112:
10107:
10103:
10098:
10094:
10089:
10047:
10013:
10009:
9988:
9984:
9963:
9959:
9953:
9942:
9932:
9921:
9916:
9903:
9899:
9878:
9874:
9853:
9849:
9848:
9846:
9816:
9812:
9810:
9807:
9806:
9777:
9773:
9746:
9742:
9736:
9725:
9715:
9704:
9677:
9674:
9673:
9669:respectively):
9624:
9620:
9599:
9595:
9589:
9578:
9573:
9566:
9562:
9541:
9537:
9536:
9534:
9510:
9506:
9504:
9501:
9500:
9477:
9473:
9452:
9448:
9442:
9431:
9410:
9407:
9406:
9345:
9341:
9320:
9316:
9310:
9299:
9294:
9287:
9283:
9277:
9273:
9252:
9248:
9242:
9231:
9226:
9224:
9207:
9204:
9203:
9135:
9129:
9107:
9106:
9094:
9089:
9088:
9079:
9074:
9073:
9061:
9056:
9055:
9042:
9035:
9029:
9024:
9023:
9020:
9019:
9007:
9002:
9001:
8989:
8984:
8983:
8970:
8963:
8957:
8952:
8951:
8948:
8947:
8935:
8930:
8929:
8920:
8915:
8914:
8898:
8891:
8885:
8880:
8879:
8876:
8875:
8863:
8858:
8857:
8848:
8843:
8842:
8830:
8825:
8824:
8811:
8804:
8798:
8793:
8792:
8788:
8786:
8783:
8782:
8762:
8756:
8755:
8754:
8752:
8749:
8748:
8731:
8725:
8724:
8723:
8721:
8718:
8717:
8700:
8694:
8693:
8692:
8690:
8687:
8686:
8669:
8663:
8662:
8661:
8659:
8656:
8655:
8630:
8629:
8620:
8615:
8614:
8605:
8600:
8599:
8587:
8582:
8581:
8563:
8558:
8557:
8545:
8540:
8539:
8521:
8517:
8508:
8503:
8502:
8490:
8485:
8484:
8472:
8467:
8466:
8451:
8447:
8438:
8433:
8432:
8423:
8418:
8417:
8405:
8400:
8399:
8387:
8382:
8381:
8369:
8368:
8358:
8341:
8339:
8336:
8335:
8315:
8310:
8309:
8293:
8289:
8283:
8272:
8251:
8249:
8246:
8245:
8228:
8227:
8221:
8217:
8207:
8200:
8185:
8181:
8178:
8177:
8153:
8149:
8137:
8133:
8114:
8107:
8092:
8088:
8085:
8084:
8069:
8065:
8053:
8049:
8033:
8026:
8011:
8007:
8004:
8003:
7979:
7975:
7963:
7959:
7943:
7936:
7921:
7917:
7913:
7911:
7908:
7907:
7882:
7881:
7875:
7870:
7869:
7866:
7865:
7859:
7854:
7853:
7850:
7849:
7843:
7838:
7837:
7834:
7833:
7827:
7822:
7821:
7814:
7813:
7806:
7805:
7800:
7795:
7790:
7784:
7783:
7778:
7773:
7768:
7759:
7758:
7753:
7748:
7740:
7734:
7733:
7728:
7720:
7715:
7702:
7701:
7694:
7693:
7688:
7683:
7677:
7673:
7671:
7665:
7661:
7654:
7653:
7642:
7625:
7623:
7620:
7619:
7599:
7593:
7592:
7591:
7589:
7586:
7585:
7568:
7562:
7561:
7560:
7558:
7555:
7554:
7537:
7531:
7530:
7529:
7527:
7524:
7523:
7506:
7500:
7499:
7498:
7496:
7493:
7492:
7452:
7449:
7448:
7423:
7421:
7418:
7417:
7414:
7412:Cubic B-Splines
7367:
7363:
7361:
7358:
7357:
7319:
7315:
7309:
7303:
7300:
7299:
7274:
7270:
7268:
7265:
7264:
7240:
7236:
7234:
7231:
7230:
7210:
7206:
7197:
7193:
7184:
7180:
7168:
7164:
7162:
7159:
7158:
7141:
7137:
7135:
7132:
7131:
7114:
7110:
7108:
7105:
7104:
7087:
7083:
7081:
7078:
7077:
7060:
7059:
7035:
7031:
7025:
7021:
7015:
7004:
6989:
6988:
6964:
6960:
6954:
6950:
6944:
6933:
6918:
6917:
6893:
6889:
6883:
6879:
6873:
6862:
6846:
6844:
6841:
6840:
6817:
6813:
6798:
6794:
6785:
6781:
6779:
6776:
6775:
6750:
6747:
6746:
6721:
6718:
6717:
6692:
6689:
6688:
6627:
6624:
6623:
6577:
6574:
6573:
6548:
6545:
6544:
6528:
6525:
6524:
6499:
6496:
6495:
6484:
6437:
6433:
6431:
6428:
6427:
6384:
6347:
6343:
6337:
6333:
6327:
6307:
6303:
6290:
6286:
6285:
6272:
6271:
6259:
6256:
6255:
6229:
6193:
6188:
6185:
6184:
6162:
6159:
6158:
6142:
6139:
6138:
6133:), whereas the
6106:
6103:
6102:
6094:
6064:
6061:
6060:
6044:
6042:
6039:
6038:
6022:
6019:
6018:
6002:
6000:
5997:
5996:
5973:
5969:
5967:
5964:
5963:
5932:
5928:
5917:
5910:
5906:
5905:
5904:
5895:
5891:
5885:
5874:
5869:
5865:
5864:
5860:
5854:
5843:
5829:
5820:
5816:
5814:
5811:
5810:
5785:
5777:
5768:
5764:
5739:
5707:
5705:
5686:
5682:
5673:
5669:
5652:
5651:
5638:
5634:
5625:
5621:
5615:
5607:
5592:
5584:
5575:
5571:
5562:
5558:
5556:
5553:
5552:
5532:
5528:
5526:
5523:
5522:
5503:
5502:
5489:
5485:
5483:
5480:
5479:
5462:
5458:
5456:
5453:
5452:
5451:-th raw moment
5425:
5421:
5415:
5409:
5406:
5405:
5374:
5373:
5360:
5356:
5347:
5343:
5337:
5316:
5313:
5312:
5293:
5290:
5289:
5273:
5270:
5269:
5244:
5241:
5240:
5237:
5206:
5202:
5193:
5189:
5180:
5161:
5157:
5148:
5144:
5129:
5125:
5124:
5111:
5107:
5098:
5094:
5093:
5091:
5076:
5053:
5034:
5030:
5024:
5020:
5014:
5000:
4995:
4993:
4990:
4989:
4949:
4945:
4924:
4920:
4919:
4885:
4881:
4880:
4878:
4866:
4862:
4847:
4843:
4842:
4814:
4810:
4809:
4807:
4806:
4802:
4786:
4764:
4760:
4756:
4754:
4752:
4749:
4748:
4734:
4703:
4664:
4656:
4648:
4640:
4638:
4635:
4634:
4612:
4598:
4584:
4570:
4568:
4565:
4564:
4548:
4546:
4543:
4542:
4504:
4499:
4498:
4471:
4466:
4465:
4444:
4439:
4438:
4417:
4412:
4411:
4400:
4398:
4395:
4394:
4374:
4373:
4371:
4368:
4367:
4334:
4329:
4311:
4308:
4307:
4272:
4267:
4249:
4246:
4245:
4208:
4203:
4193:
4189:
4144:
4140:
4133:
4131:
4101:
4097:
4095:
4092:
4091:
4077:
4043:
4039:
4035:
4028:
4024:
4018:
4014:
4013:
4011:
3987:
3983:
3979:
3972:
3968:
3962:
3958:
3957:
3955:
3934:
3930:
3926:
3919:
3915:
3909:
3905:
3904:
3902:
3900:
3897:
3896:
3876:
3875:
3856:
3849:
3845:
3841:
3839:
3826:
3819:
3815:
3811:
3809:
3791:
3787:
3778:
3774:
3754:
3750:
3749:
3733:
3726:
3722:
3718:
3716:
3703:
3696:
3692:
3688:
3686:
3668:
3664:
3655:
3651:
3631:
3626:
3624:
3621:
3620:
3600:
3599:
3583:
3578:
3567:
3561:
3557:
3538:
3532:
3528:
3525:
3524:
3505:
3500:
3489:
3465:
3461:
3445:
3439:
3435:
3432:
3431:
3412:
3407:
3396:
3390:
3386:
3379:
3373:
3369:
3365:
3363:
3360:
3359:
3316:
3313:
3312:
3295:
3294:
3286:
3284:
3277:
3276:
3264:
3260:
3257:
3251:
3249:
3237:
3233:
3229:
3228:
3210:
3206:
3203:
3191:
3187:
3184:
3182:
3164:
3160:
3156:
3155:
3137:
3133:
3130:
3124:
3122:
3115:
3114:
3105:
3103:
3100:
3099:
3042:
3038:
3030:
3027:
3026:
3003:
2999:
2991:
2988:
2987:
2970:
2966:
2958:
2955:
2954:
2918:
2914:
2912:
2909:
2908:
2875:
2871:
2869:
2866:
2865:
2842:
2838:
2836:
2833:
2832:
2799:
2795:
2774:
2770:
2769:
2744:
2740:
2739:
2737:
2735:
2732:
2731:
2705:
2701:
2699:
2696:
2695:
2678:
2674:
2672:
2669:
2668:
2641:
2637:
2622:
2618:
2617:
2610:
2606:
2599:
2597:
2595:
2592:
2591:
2548:
2544:
2542:
2539:
2538:
2488:
2484:
2469:
2465:
2444:
2440:
2439:
2414:
2410:
2409:
2407:
2377:
2373:
2364:
2360:
2345:
2341:
2340:
2333:
2329:
2322:
2320:
2296:
2292:
2290:
2287:
2286:
2266:
2265:
2257:
2255:
2249:
2248:
2233:
2229:
2214:
2210:
2205:
2203:
2193:
2192:
2168:
2164:
2162:
2159:
2158:
2149:basis functions
2110:
2106:
2100:
2096:
2090:
2067:
2060:
2056:
2054:
2051:
2050:
2031:
2028:
2027:
2001:
1998:
1997:
1975:
1972:
1971:
1955:
1952:
1951:
1932:
1929:
1928:
1906:
1903:
1902:
1876:
1873:
1872:
1852:
1848:
1833:
1829:
1821:
1818:
1817:
1801:
1798:
1797:
1774:
1770:
1761:
1757:
1755:
1752:
1751:
1734:
1730:
1728:
1725:
1724:
1702:
1699:
1698:
1682:
1679:
1678:
1656:
1653:
1652:
1636:
1633:
1632:
1625:
1573:
1569:
1553:
1549:
1528:
1524:
1523:
1498:
1494:
1493:
1490:
1460:
1456:
1446:
1442:
1427:
1423:
1422:
1415:
1411:
1404:
1401:
1377:
1373:
1371:
1368:
1367:
1336:
1333:
1332:
1312:
1311:
1303:
1301:
1295:
1294:
1279:
1275:
1260:
1256:
1251:
1249:
1239:
1238:
1214:
1210:
1208:
1205:
1204:
1179:
1176:
1175:
1149:
1145:
1143:
1140:
1139:
1122:
1118:
1116:
1113:
1112:
1080:
1076:
1074:
1071:
1070:
1047:
1043:
1041:
1038:
1037:
1020:
1016:
1014:
1011:
1010:
994:
991:
990:
949:
945:
927:
916:
910:
907:
906:
883:
882:
874:
872:
866:
865:
844:
840:
825:
821:
816:
814:
809:
802:
801:
777:
773:
771:
768:
767:
732:
728:
726:
723:
722:
702:
698:
683:
679:
670:
666:
657:
653:
651:
648:
647:
631:
628:
627:
611:
608:
607:
591:
588:
587:
555:
551:
549:
546:
545:
516:
513:
512:
485:
473:basis functions
456:
453:
452:
427:
424:
423:
400:
397:
396:
387:in Russia. The
377:
298:
287:
286:
285:
280:
269:
263:
260:
217:
215:
205:
193:
178:
167:
161:
158:
150:help improve it
147:
138:
134:
123:
112:
106:
103:
92:
80:
76:
39:
35:
28:
27:Spline function
23:
22:
15:
12:
11:
5:
10886:
10876:
10875:
10870:
10856:
10855:
10850:
10845:
10840:
10835:
10830:
10804:
10783:
10782:External links
10780:
10779:
10778:
10775:SAND2022-7702C
10771:
10765:
10752:
10746:
10733:
10718:
10712:
10697:
10694:
10693:
10692:
10686:
10676:The NURBS Book
10669:
10663:
10643:
10642:
10633:
10624:
10593:(4): 542–548.
10573:
10546:(3): 370–376.
10530:
10521:
10515:978-0201624953
10514:
10490:
10483:
10463:
10409:
10396:(3): 311–325.
10376:
10367:
10356:(2): 159–174.
10340:
10331:
10322:
10303:(3): 229–238.
10287:
10268:(4): 365–371.
10252:
10243:
10234:
10225:
10198:
10183:
10157:
10142:
10133:
10119:
10117:
10114:
10111:
10110:
10101:
10091:
10090:
10088:
10085:
10084:
10083:
10078:
10076:Spline wavelet
10073:
10068:
10063:
10058:
10053:
10046:
10043:
10039:
10038:
10022:
10019:
10016:
10012:
10008:
10005:
10002:
9997:
9994:
9991:
9987:
9983:
9980:
9977:
9972:
9969:
9966:
9962:
9956:
9951:
9948:
9945:
9941:
9935:
9930:
9927:
9924:
9920:
9912:
9909:
9906:
9902:
9898:
9895:
9892:
9887:
9884:
9881:
9877:
9873:
9870:
9867:
9862:
9859:
9856:
9852:
9845:
9842:
9839:
9836:
9833:
9830:
9825:
9822:
9819:
9815:
9800:
9799:
9786:
9783:
9780:
9776:
9772:
9769:
9766:
9763:
9760:
9755:
9752:
9749:
9745:
9739:
9734:
9731:
9728:
9724:
9718:
9713:
9710:
9707:
9703:
9699:
9696:
9693:
9690:
9687:
9684:
9681:
9661:(with indices
9651:tensor product
9644:
9643:
9627:
9623:
9619:
9616:
9613:
9608:
9605:
9602:
9598:
9592:
9587:
9584:
9581:
9577:
9569:
9565:
9561:
9558:
9555:
9550:
9547:
9544:
9540:
9533:
9530:
9527:
9524:
9519:
9516:
9513:
9509:
9494:
9493:
9480:
9476:
9472:
9469:
9466:
9461:
9458:
9455:
9451:
9445:
9440:
9437:
9434:
9430:
9426:
9423:
9420:
9417:
9414:
9365:
9364:
9348:
9344:
9340:
9337:
9334:
9329:
9326:
9323:
9319:
9313:
9308:
9305:
9302:
9298:
9290:
9286:
9280:
9276:
9272:
9269:
9266:
9261:
9258:
9255:
9251:
9245:
9240:
9237:
9234:
9230:
9223:
9220:
9217:
9214:
9211:
9131:Main article:
9128:
9125:
9121:
9120:
9105:
9102:
9097:
9092:
9087:
9082:
9077:
9072:
9069:
9064:
9059:
9054:
9049:
9046:
9041:
9038:
9036:
9032:
9027:
9022:
9021:
9018:
9015:
9010:
9005:
9000:
8997:
8992:
8987:
8982:
8977:
8974:
8969:
8966:
8964:
8960:
8955:
8950:
8949:
8946:
8943:
8938:
8933:
8928:
8923:
8918:
8913:
8910:
8905:
8902:
8897:
8894:
8892:
8888:
8883:
8878:
8877:
8874:
8871:
8866:
8861:
8856:
8851:
8846:
8841:
8838:
8833:
8828:
8823:
8818:
8815:
8810:
8807:
8805:
8801:
8796:
8791:
8790:
8765:
8734:
8703:
8672:
8648:
8647:
8633:
8628:
8623:
8618:
8613:
8608:
8603:
8598:
8595:
8590:
8585:
8580:
8577:
8574:
8571:
8566:
8561:
8556:
8553:
8548:
8543:
8538:
8535:
8532:
8529:
8524:
8520:
8516:
8511:
8506:
8501:
8498:
8493:
8488:
8483:
8480:
8475:
8470:
8465:
8462:
8459:
8454:
8450:
8446:
8441:
8436:
8431:
8426:
8421:
8416:
8413:
8408:
8403:
8398:
8395:
8390:
8385:
8380:
8377:
8372:
8365:
8362:
8357:
8354:
8351:
8348:
8344:
8318:
8313:
8307:
8304:
8301:
8296:
8292:
8286:
8281:
8278:
8275:
8271:
8267:
8264:
8261:
8258:
8254:
8242:
8241:
8224:
8220:
8214:
8211:
8206:
8203:
8201:
8199:
8196:
8193:
8188:
8184:
8180:
8179:
8176:
8173:
8170:
8167:
8164:
8161:
8156:
8152:
8148:
8145:
8140:
8136:
8132:
8129:
8126:
8121:
8118:
8113:
8110:
8108:
8106:
8103:
8100:
8095:
8091:
8087:
8086:
8083:
8080:
8077:
8072:
8068:
8064:
8061:
8056:
8052:
8048:
8045:
8040:
8037:
8032:
8029:
8027:
8025:
8022:
8019:
8014:
8010:
8006:
8005:
8002:
7999:
7996:
7993:
7990:
7987:
7982:
7978:
7974:
7971:
7966:
7962:
7958:
7955:
7950:
7947:
7942:
7939:
7937:
7935:
7932:
7929:
7924:
7920:
7916:
7915:
7901:
7900:
7886:
7878:
7873:
7868:
7867:
7862:
7857:
7852:
7851:
7846:
7841:
7836:
7835:
7830:
7825:
7820:
7819:
7817:
7810:
7804:
7801:
7799:
7796:
7794:
7791:
7789:
7786:
7785:
7782:
7779:
7777:
7774:
7772:
7769:
7767:
7764:
7761:
7760:
7757:
7754:
7752:
7749:
7747:
7744:
7741:
7739:
7736:
7735:
7732:
7729:
7727:
7724:
7721:
7719:
7716:
7714:
7711:
7708:
7707:
7705:
7698:
7692:
7689:
7687:
7684:
7680:
7676:
7672:
7668:
7664:
7660:
7659:
7657:
7649:
7646:
7641:
7638:
7635:
7632:
7628:
7602:
7571:
7540:
7509:
7488:control points
7474:
7471:
7468:
7465:
7462:
7459:
7456:
7436:
7433:
7430:
7426:
7413:
7410:
7406:
7405:
7393:
7390:
7387:
7384:
7381:
7376:
7373:
7370:
7366:
7345:
7342:
7339:
7336:
7333:
7328:
7325:
7322:
7318:
7312:
7308:
7296:
7292:
7277:
7273:
7243:
7239:
7218:
7213:
7209:
7205:
7200:
7196:
7192:
7187:
7183:
7179:
7176:
7171:
7167:
7144:
7140:
7117:
7113:
7090:
7086:
7074:
7073:
7058:
7055:
7052:
7049:
7044:
7041:
7038:
7034:
7028:
7024:
7018:
7014:
7010:
7007:
7005:
7003:
7000:
6997:
6994:
6991:
6990:
6987:
6984:
6981:
6978:
6973:
6970:
6967:
6963:
6957:
6953:
6947:
6943:
6939:
6936:
6934:
6932:
6929:
6926:
6923:
6920:
6919:
6916:
6913:
6910:
6907:
6902:
6899:
6896:
6892:
6886:
6882:
6876:
6872:
6868:
6865:
6863:
6861:
6858:
6855:
6852:
6849:
6848:
6820:
6816:
6812:
6809:
6806:
6801:
6797:
6793:
6788:
6784:
6763:
6760:
6757:
6754:
6734:
6731:
6728:
6725:
6705:
6702:
6699:
6696:
6676:
6673:
6670:
6667:
6664:
6661:
6658:
6655:
6652:
6649:
6646:
6643:
6640:
6637:
6634:
6631:
6611:
6608:
6605:
6602:
6599:
6596:
6593:
6590:
6587:
6584:
6581:
6561:
6558:
6555:
6552:
6532:
6512:
6509:
6506:
6503:
6483:
6480:
6472:moving average
6440:
6436:
6404:
6403:
6392:
6387:
6382:
6377:
6373:
6370:
6367:
6362:
6359:
6356:
6353:
6350:
6346:
6340:
6336:
6330:
6326:
6322:
6319:
6316:
6313:
6310:
6306:
6302:
6299:
6296:
6293:
6289:
6282:
6270:
6266:
6263:
6228:
6225:
6200:
6196:
6192:
6172:
6169:
6166:
6146:
6122:
6119:
6116:
6113:
6110:
6093:
6090:
6077:
6074:
6071:
6068:
6047:
6026:
6005:
5984:
5981:
5976:
5972:
5960:
5959:
5947:
5941:
5938:
5935:
5931:
5926:
5920:
5913:
5909:
5903:
5898:
5894:
5888:
5883:
5880:
5877:
5873:
5868:
5863:
5857:
5852:
5849:
5846:
5842:
5836:
5833:
5828:
5823:
5819:
5804:
5803:
5792:
5788:
5784:
5780:
5776:
5771:
5767:
5763:
5757:
5754:
5751:
5748:
5745:
5742:
5737:
5734:
5731:
5728:
5725:
5722:
5719:
5716:
5713:
5710:
5704:
5701:
5698:
5694:
5689:
5685:
5681:
5676:
5672:
5668:
5665:
5662:
5650:
5647:
5644:
5641:
5637:
5633:
5628:
5624:
5618:
5613:
5610:
5606:
5602:
5599:
5595:
5591:
5587:
5583:
5578:
5574:
5570:
5565:
5561:
5535:
5531:
5501:
5498:
5495:
5492:
5488:
5465:
5461:
5436:
5433:
5428:
5424:
5418:
5414:
5402:
5401:
5390:
5387:
5384:
5372:
5369:
5366:
5363:
5359:
5355:
5350:
5346:
5340:
5336:
5332:
5329:
5326:
5323:
5320:
5297:
5277:
5257:
5254:
5251:
5248:
5236:
5233:
5229:
5228:
5217:
5214:
5209:
5205:
5201:
5196:
5192:
5188:
5176:
5173:
5170:
5167:
5164:
5160:
5151:
5147:
5143:
5138:
5135:
5132:
5128:
5120:
5117:
5114:
5110:
5106:
5101:
5097:
5090:
5085:
5082:
5079:
5074:
5071:
5068:
5065:
5062:
5059:
5056:
5052:
5048:
5043:
5040:
5037:
5033:
5027:
5023:
5017:
5013:
5006:
5003:
4999:
4983:
4982:
4971:
4967:
4958:
4955:
4952:
4948:
4944:
4939:
4936:
4933:
4930:
4927:
4923:
4917:
4914:
4911:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4884:
4877:
4869:
4865:
4861:
4856:
4853:
4850:
4846:
4840:
4837:
4834:
4829:
4826:
4823:
4820:
4817:
4813:
4805:
4801:
4798:
4792:
4789:
4784:
4781:
4778:
4773:
4770:
4767:
4763:
4759:
4733:
4730:
4702:
4699:
4688:Fourier domain
4667:
4663:
4659:
4655:
4651:
4647:
4643:
4622:
4619:
4615:
4611:
4608:
4605:
4601:
4597:
4594:
4591:
4587:
4583:
4580:
4577:
4573:
4551:
4539:
4538:
4527:
4524:
4521:
4518:
4515:
4512:
4507:
4502:
4497:
4494:
4491:
4488:
4485:
4482:
4479:
4474:
4469:
4464:
4461:
4458:
4455:
4452:
4447:
4442:
4437:
4434:
4431:
4428:
4425:
4420:
4415:
4410:
4407:
4403:
4343:
4340:
4337:
4332:
4328:
4324:
4321:
4318:
4315:
4281:
4278:
4275:
4270:
4266:
4262:
4259:
4256:
4253:
4234:
4233:
4222:
4217:
4214:
4211:
4206:
4202:
4196:
4192:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4153:
4147:
4143:
4139:
4136:
4130:
4127:
4124:
4121:
4116:
4113:
4110:
4107:
4104:
4100:
4076:
4073:
4069:
4068:
4057:
4054:
4046:
4042:
4038:
4031:
4027:
4021:
4017:
4007:
4004:
4001:
3998:
3990:
3986:
3982:
3975:
3971:
3965:
3961:
3951:
3948:
3945:
3937:
3933:
3929:
3922:
3918:
3912:
3908:
3890:
3889:
3874:
3871:
3868:
3862:
3859:
3852:
3848:
3844:
3838:
3832:
3829:
3822:
3818:
3814:
3805:
3802:
3799:
3794:
3790:
3786:
3781:
3777:
3770:
3767:
3764:
3761:
3753:
3751:
3748:
3745:
3739:
3736:
3729:
3725:
3721:
3715:
3709:
3706:
3699:
3695:
3691:
3682:
3679:
3676:
3671:
3667:
3663:
3658:
3654:
3647:
3644:
3641:
3638:
3630:
3628:
3614:
3613:
3598:
3595:
3592:
3589:
3586:
3584:
3582:
3579:
3577:
3574:
3570:
3564:
3560:
3556:
3553:
3550:
3547:
3544:
3541:
3539:
3535:
3531:
3527:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3506:
3504:
3501:
3499:
3496:
3492:
3488:
3485:
3482:
3479:
3476:
3473:
3468:
3464:
3460:
3457:
3454:
3451:
3448:
3446:
3442:
3438:
3434:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3413:
3411:
3408:
3406:
3403:
3399:
3393:
3389:
3385:
3382:
3380:
3376:
3372:
3368:
3367:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3309:
3308:
3293:
3290:
3287:
3283:
3280:
3278:
3273:
3270:
3267:
3263:
3259:
3256:
3253:
3252:
3246:
3243:
3240:
3236:
3232:
3230:
3225:
3222:
3219:
3216:
3213:
3209:
3205:
3200:
3197:
3194:
3190:
3186:
3185:
3179:
3176:
3173:
3170:
3167:
3163:
3159:
3157:
3152:
3149:
3146:
3143:
3140:
3136:
3132:
3129:
3126:
3125:
3121:
3118:
3116:
3113:
3110:
3107:
3051:
3048:
3045:
3041:
3037:
3034:
3012:
3009:
3006:
3002:
2998:
2995:
2973:
2969:
2965:
2962:
2938:
2935:
2932:
2927:
2924:
2921:
2917:
2890:
2887:
2884:
2881:
2878:
2874:
2851:
2848:
2845:
2841:
2825:
2824:
2808:
2805:
2802:
2798:
2794:
2789:
2786:
2783:
2780:
2777:
2773:
2767:
2764:
2759:
2756:
2753:
2750:
2747:
2743:
2714:
2711:
2708:
2704:
2681:
2677:
2661:
2660:
2644:
2640:
2636:
2631:
2628:
2625:
2621:
2613:
2609:
2605:
2602:
2568:
2565:
2562:
2557:
2554:
2551:
2547:
2535:
2534:
2523:
2520:
2517:
2514:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2487:
2478:
2475:
2472:
2468:
2464:
2459:
2456:
2453:
2450:
2447:
2443:
2437:
2434:
2429:
2426:
2423:
2420:
2417:
2413:
2406:
2403:
2400:
2397:
2392:
2389:
2386:
2383:
2380:
2376:
2367:
2363:
2359:
2354:
2351:
2348:
2344:
2336:
2332:
2328:
2325:
2319:
2316:
2313:
2310:
2305:
2302:
2299:
2295:
2283:
2282:
2269:
2264:
2256:
2254:
2251:
2250:
2247:
2242:
2239:
2236:
2232:
2228:
2225:
2222:
2217:
2213:
2204:
2202:
2199:
2198:
2196:
2191:
2188:
2185:
2182:
2177:
2174:
2171:
2167:
2145:
2144:
2133:
2130:
2127:
2124:
2119:
2116:
2113:
2109:
2103:
2099:
2093:
2089:
2085:
2082:
2079:
2076:
2070:
2066:
2063:
2059:
2035:
2011:
2008:
2005:
1985:
1982:
1979:
1959:
1936:
1916:
1913:
1910:
1886:
1883:
1880:
1855:
1851:
1847:
1842:
1839:
1836:
1832:
1828:
1825:
1805:
1783:
1780:
1777:
1773:
1769:
1764:
1760:
1737:
1733:
1712:
1709:
1706:
1686:
1677:in a variable
1666:
1663:
1660:
1640:
1624:
1621:
1620:
1619:
1608:
1605:
1602:
1599:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1572:
1562:
1559:
1556:
1552:
1548:
1543:
1540:
1537:
1534:
1531:
1527:
1521:
1518:
1513:
1510:
1507:
1504:
1501:
1497:
1489:
1486:
1483:
1480:
1475:
1472:
1469:
1466:
1463:
1459:
1449:
1445:
1441:
1436:
1433:
1430:
1426:
1418:
1414:
1410:
1407:
1400:
1397:
1394:
1391:
1386:
1383:
1380:
1376:
1352:
1349:
1346:
1343:
1340:
1329:
1328:
1315:
1310:
1302:
1300:
1297:
1296:
1293:
1288:
1285:
1282:
1278:
1274:
1271:
1268:
1263:
1259:
1250:
1248:
1245:
1244:
1242:
1237:
1234:
1231:
1228:
1223:
1220:
1217:
1213:
1189:
1186:
1183:
1158:
1155:
1152:
1148:
1125:
1121:
1100:
1097:
1094:
1089:
1086:
1083:
1079:
1056:
1053:
1050:
1046:
1023:
1019:
998:
987:
986:
975:
972:
969:
966:
963:
958:
955:
952:
948:
942:
939:
936:
933:
930:
925:
922:
919:
915:
900:
899:
886:
881:
873:
871:
868:
867:
864:
859:
856:
853:
850:
847:
843:
839:
836:
833:
828:
824:
815:
808:
807:
805:
800:
797:
794:
791:
786:
783:
780:
776:
752:
749:
746:
741:
738:
735:
731:
705:
701:
697:
694:
691:
686:
682:
678:
673:
669:
665:
660:
656:
635:
615:
606:in a variable
595:
575:
572:
569:
564:
561:
558:
554:
526:
523:
520:
484:
481:
460:
437:
434:
431:
404:
376:
373:
300:
299:
282:
281:
196:
194:
187:
180:
179:
141:
139:
132:
125:
124:
84:The article's
83:
81:
74:
69:
43:
42:
40:
33:
26:
9:
6:
4:
3:
2:
10885:
10874:
10873:Interpolation
10871:
10869:
10866:
10865:
10863:
10854:
10851:
10849:
10846:
10844:
10841:
10839:
10836:
10834:
10831:
10821:on 2013-11-06
10817:
10810:
10805:
10800:
10799:
10794:
10791:
10786:
10785:
10777:. (153 pages)
10776:
10772:
10768:
10762:
10758:
10753:
10749:
10743:
10739:
10734:
10729:
10728:
10723:
10719:
10715:
10709:
10705:
10700:
10699:
10689:
10683:
10678:
10677:
10670:
10666:
10660:
10656:
10651:
10650:
10649:
10648:
10637:
10628:
10620:
10616:
10612:
10608:
10604:
10600:
10596:
10592:
10588:
10584:
10577:
10569:
10565:
10561:
10557:
10553:
10549:
10545:
10541:
10534:
10525:
10517:
10511:
10507:
10506:
10501:
10494:
10486:
10480:
10476:
10475:
10467:
10458:
10454:
10450:
10446:
10442:
10438:
10433:
10428:
10424:
10420:
10413:
10404:
10399:
10395:
10391:
10387:
10380:
10371:
10363:
10359:
10355:
10351:
10344:
10335:
10326:
10318:
10314:
10310:
10306:
10302:
10298:
10291:
10283:
10279:
10275:
10271:
10267:
10263:
10256:
10247:
10238:
10229:
10221:
10217:
10213:
10209:
10202:
10194:
10190:
10186:
10180:
10176:
10172:
10168:
10161:
10154:
10153:
10146:
10137:
10130:
10124:
10120:
10105:
10096:
10092:
10082:
10079:
10077:
10074:
10072:
10069:
10067:
10064:
10062:
10059:
10057:
10054:
10052:
10049:
10048:
10042:
10020:
10017:
10014:
10010:
10003:
9995:
9992:
9989:
9985:
9978:
9970:
9967:
9964:
9960:
9954:
9949:
9946:
9943:
9939:
9933:
9928:
9925:
9922:
9918:
9910:
9907:
9904:
9900:
9893:
9885:
9882:
9879:
9875:
9868:
9860:
9857:
9854:
9850:
9843:
9837:
9834:
9831:
9823:
9820:
9817:
9813:
9805:
9804:
9803:
9784:
9781:
9778:
9774:
9767:
9764:
9761:
9753:
9750:
9747:
9743:
9737:
9732:
9729:
9726:
9722:
9716:
9711:
9708:
9705:
9701:
9697:
9691:
9688:
9685:
9679:
9672:
9671:
9670:
9668:
9664:
9660:
9656:
9652:
9647:
9625:
9621:
9614:
9606:
9603:
9600:
9596:
9590:
9585:
9582:
9579:
9575:
9567:
9563:
9556:
9548:
9545:
9542:
9538:
9531:
9525:
9517:
9514:
9511:
9507:
9499:
9498:
9497:
9478:
9474:
9467:
9459:
9456:
9453:
9449:
9443:
9438:
9435:
9432:
9428:
9424:
9418:
9412:
9405:
9404:
9403:
9400:
9398:
9394:
9390:
9386:
9382:
9378:
9374:
9370:
9346:
9342:
9335:
9327:
9324:
9321:
9317:
9311:
9306:
9303:
9300:
9296:
9288:
9284:
9278:
9274:
9267:
9259:
9256:
9253:
9249:
9243:
9238:
9235:
9232:
9228:
9221:
9215:
9209:
9202:
9201:
9200:
9197:
9193:
9191:
9187:
9183:
9177:
9170:
9166:
9164:
9163:Bézier curves
9160:
9156:
9152:
9148:
9139:
9134:
9124:
9103:
9095:
9085:
9080:
9070:
9067:
9062:
9047:
9044:
9039:
9037:
9030:
9016:
9008:
8998:
8995:
8990:
8975:
8972:
8967:
8965:
8958:
8944:
8936:
8926:
8921:
8911:
8903:
8900:
8895:
8893:
8886:
8872:
8864:
8854:
8849:
8839:
8836:
8831:
8816:
8813:
8808:
8806:
8799:
8781:
8780:
8779:
8763:
8732:
8701:
8670:
8653:
8621:
8611:
8606:
8596:
8593:
8588:
8575:
8572:
8564:
8554:
8551:
8546:
8536:
8533:
8527:
8522:
8518:
8509:
8499:
8496:
8491:
8481:
8478:
8473:
8463:
8457:
8452:
8448:
8439:
8429:
8424:
8414:
8411:
8406:
8396:
8393:
8388:
8378:
8363:
8360:
8355:
8349:
8334:
8333:
8332:
8316:
8302:
8294:
8290:
8284:
8279:
8276:
8273:
8269:
8265:
8259:
8222:
8218:
8212:
8209:
8204:
8202:
8194:
8186:
8182:
8171:
8168:
8165:
8162:
8159:
8154:
8150:
8146:
8143:
8138:
8134:
8130:
8127:
8119:
8116:
8111:
8109:
8101:
8093:
8089:
8078:
8075:
8070:
8066:
8062:
8059:
8054:
8050:
8046:
8038:
8035:
8030:
8028:
8020:
8012:
8008:
7997:
7994:
7991:
7988:
7985:
7980:
7976:
7972:
7969:
7964:
7960:
7956:
7948:
7945:
7940:
7938:
7930:
7922:
7918:
7906:
7905:
7904:
7884:
7876:
7860:
7844:
7828:
7815:
7808:
7802:
7797:
7792:
7787:
7780:
7775:
7770:
7765:
7762:
7755:
7750:
7745:
7742:
7737:
7730:
7725:
7722:
7717:
7712:
7709:
7703:
7696:
7690:
7685:
7678:
7674:
7666:
7662:
7655:
7647:
7644:
7639:
7633:
7618:
7617:
7616:
7600:
7569:
7538:
7507:
7490:
7489:
7469:
7466:
7463:
7457:
7454:
7431:
7409:
7391:
7388:
7382:
7374:
7371:
7368:
7364:
7343:
7340:
7334:
7326:
7323:
7320:
7316:
7310:
7306:
7297:
7293:
7275:
7271:
7262:
7261:
7260:
7257:
7241:
7237:
7211:
7207:
7203:
7198:
7194:
7190:
7185:
7181:
7174:
7169:
7165:
7142:
7138:
7115:
7111:
7088:
7084:
7056:
7050:
7042:
7039:
7036:
7032:
7026:
7022:
7016:
7012:
7008:
7006:
6998:
6992:
6985:
6979:
6971:
6968:
6965:
6961:
6955:
6951:
6945:
6941:
6937:
6935:
6927:
6921:
6914:
6908:
6900:
6897:
6894:
6890:
6884:
6880:
6874:
6870:
6866:
6864:
6856:
6850:
6839:
6838:
6837:
6834:
6818:
6814:
6810:
6807:
6804:
6799:
6795:
6791:
6786:
6782:
6758:
6752:
6729:
6723:
6700:
6694:
6668:
6662:
6659:
6653:
6647:
6644:
6638:
6632:
6603:
6597:
6594:
6588:
6582:
6556:
6550:
6530:
6507:
6501:
6493:
6489:
6479:
6477:
6473:
6469:
6464:
6460:
6454:
6438:
6434:
6425:
6421:
6417:
6413:
6409:
6390:
6385:
6380:
6375:
6368:
6360:
6357:
6354:
6351:
6348:
6344:
6338:
6334:
6328:
6324:
6320:
6314:
6308:
6304:
6297:
6291:
6287:
6280:
6268:
6264:
6261:
6254:
6253:
6252:
6250:
6246:
6242:
6241:least squares
6238:
6234:
6233:curve fitting
6227:Curve fitting
6224:
6222:
6221:C0 continuity
6218:
6213:
6198:
6194:
6190:
6170:
6167:
6164:
6144:
6136:
6117:
6114:
6111:
6099:
6089:
6072:
6066:
6024:
5982:
5979:
5974:
5970:
5945:
5939:
5936:
5933:
5929:
5924:
5918:
5911:
5907:
5901:
5896:
5892:
5886:
5881:
5878:
5875:
5866:
5861:
5855:
5850:
5847:
5844:
5834:
5831:
5826:
5821:
5817:
5809:
5808:
5807:
5782:
5769:
5765:
5761:
5752:
5749:
5746:
5732:
5720:
5717:
5714:
5702:
5699:
5696:
5687:
5683:
5679:
5674:
5670:
5666:
5663:
5648:
5645:
5642:
5639:
5635:
5631:
5626:
5622:
5608:
5604:
5600:
5589:
5576:
5572:
5568:
5563:
5559:
5551:
5550:
5549:
5533:
5529:
5499:
5496:
5493:
5490:
5486:
5463:
5459:
5450:
5434:
5431:
5426:
5422:
5416:
5412:
5385:
5370:
5367:
5364:
5361:
5357:
5353:
5348:
5344:
5338:
5334:
5330:
5324:
5318:
5311:
5310:
5309:
5295:
5275:
5252:
5246:
5232:
5215:
5207:
5203:
5199:
5194:
5190:
5174:
5171:
5168:
5165:
5162:
5158:
5149:
5145:
5141:
5136:
5133:
5130:
5126:
5118:
5115:
5112:
5108:
5104:
5099:
5095:
5088:
5083:
5080:
5077:
5072:
5069:
5066:
5063:
5060:
5057:
5054:
5050:
5046:
5041:
5038:
5035:
5031:
5025:
5021:
5015:
5011:
5004:
5001:
4997:
4988:
4987:
4986:
4969:
4965:
4956:
4953:
4950:
4946:
4942:
4937:
4934:
4931:
4928:
4925:
4921:
4912:
4904:
4901:
4898:
4895:
4892:
4889:
4886:
4882:
4875:
4867:
4863:
4859:
4854:
4851:
4848:
4844:
4835:
4827:
4824:
4821:
4818:
4815:
4811:
4803:
4799:
4796:
4790:
4787:
4779:
4771:
4768:
4765:
4761:
4757:
4747:
4746:
4745:
4743:
4739:
4729:
4727:
4722:
4720:
4716:
4712:
4708:
4698:
4696:
4691:
4689:
4685:
4680:
4661:
4653:
4645:
4617:
4613:
4609:
4606:
4603:
4599:
4595:
4592:
4589:
4585:
4581:
4575:
4525:
4519:
4516:
4513:
4510:
4505:
4495:
4492:
4489:
4486:
4483:
4480:
4477:
4472:
4462:
4459:
4456:
4453:
4450:
4445:
4435:
4432:
4429:
4426:
4423:
4418:
4405:
4393:
4392:
4391:
4364:
4361:
4359:
4341:
4338:
4335:
4330:
4322:
4319:
4316:
4305:
4301:
4297:
4279:
4276:
4273:
4268:
4260:
4257:
4254:
4243:
4239:
4220:
4215:
4212:
4209:
4204:
4194:
4190:
4186:
4183:
4174:
4171:
4168:
4165:
4162:
4156:
4151:
4145:
4141:
4137:
4134:
4128:
4122:
4114:
4111:
4108:
4105:
4102:
4098:
4090:
4089:
4088:
4086:
4082:
4072:
4055:
4052:
4044:
4040:
4036:
4029:
4025:
4019:
4015:
4005:
4002:
3999:
3996:
3988:
3984:
3980:
3973:
3969:
3963:
3959:
3949:
3946:
3943:
3935:
3931:
3927:
3920:
3916:
3910:
3906:
3895:
3894:
3893:
3872:
3869:
3866:
3860:
3857:
3850:
3846:
3842:
3836:
3830:
3827:
3820:
3816:
3812:
3803:
3800:
3797:
3792:
3788:
3784:
3779:
3775:
3768:
3765:
3762:
3759:
3746:
3743:
3737:
3734:
3727:
3723:
3719:
3713:
3707:
3704:
3697:
3693:
3689:
3680:
3677:
3674:
3669:
3665:
3661:
3656:
3652:
3645:
3642:
3639:
3636:
3619:
3618:
3617:
3596:
3593:
3590:
3587:
3585:
3580:
3575:
3572:
3568:
3562:
3554:
3551:
3548:
3542:
3540:
3533:
3529:
3521:
3518:
3515:
3512:
3509:
3507:
3502:
3497:
3494:
3490:
3483:
3480:
3477:
3474:
3471:
3466:
3462:
3458:
3455:
3449:
3447:
3440:
3436:
3428:
3425:
3422:
3419:
3416:
3414:
3409:
3404:
3401:
3397:
3391:
3387:
3383:
3381:
3374:
3370:
3358:
3357:
3356:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3291:
3281:
3271:
3268:
3265:
3261:
3254:
3244:
3241:
3238:
3234:
3223:
3220:
3217:
3214:
3211:
3207:
3198:
3195:
3192:
3188:
3177:
3174:
3171:
3168:
3165:
3161:
3150:
3147:
3144:
3141:
3138:
3134:
3127:
3119:
3111:
3098:
3097:
3096:
3094:
3090:
3086:
3082:
3078:
3073:
3071:
3067:
3049:
3046:
3043:
3039:
3035:
3032:
3010:
3007:
3004:
3000:
2996:
2993:
2971:
2967:
2963:
2960:
2952:
2933:
2925:
2922:
2919:
2915:
2906:
2888:
2885:
2882:
2879:
2876:
2872:
2849:
2846:
2843:
2839:
2830:
2806:
2803:
2800:
2796:
2792:
2787:
2784:
2781:
2778:
2775:
2771:
2765:
2762:
2757:
2754:
2751:
2748:
2745:
2741:
2730:
2729:
2728:
2712:
2709:
2706:
2702:
2679:
2675:
2666:
2642:
2638:
2634:
2629:
2626:
2623:
2619:
2611:
2607:
2603:
2600:
2590:
2589:
2588:
2586:
2582:
2563:
2555:
2552:
2549:
2545:
2521:
2515:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2485:
2476:
2473:
2470:
2466:
2462:
2457:
2454:
2451:
2448:
2445:
2441:
2435:
2432:
2427:
2424:
2421:
2418:
2415:
2411:
2404:
2398:
2390:
2387:
2384:
2381:
2378:
2374:
2365:
2361:
2357:
2352:
2349:
2346:
2342:
2334:
2330:
2326:
2323:
2317:
2311:
2303:
2300:
2297:
2293:
2285:
2284:
2262:
2252:
2245:
2240:
2237:
2234:
2230:
2226:
2223:
2220:
2215:
2211:
2200:
2194:
2189:
2183:
2175:
2172:
2169:
2165:
2157:
2156:
2155:
2152:
2150:
2131:
2125:
2117:
2114:
2111:
2107:
2101:
2097:
2091:
2087:
2083:
2077:
2064:
2061:
2057:
2049:
2048:
2047:
2033:
2024:
2009:
2006:
2003:
1983:
1980:
1977:
1957:
1948:
1934:
1914:
1911:
1908:
1900:
1884:
1881:
1878:
1869:
1853:
1849:
1845:
1840:
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354:curve-fitting
351:
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225: –
224:
220:
219:Find sources:
213:
209:
203:
202:
197:This article
195:
191:
186:
185:
176:
173:
165:
155:
151:
145:
142:This article
140:
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100:
97:and read the
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73:
72:
67:
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58:
57:
52:
51:
46:
41:
32:
31:
19:
10823:. Retrieved
10816:the original
10796:
10756:
10737:
10726:
10722:Jean Gallier
10703:
10675:
10654:
10646:
10645:
10636:
10627:
10619:the original
10590:
10586:
10576:
10543:
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10524:
10504:
10493:
10473:
10466:
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10418:
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10211:
10207:
10201:
10166:
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10150:
10145:
10136:
10128:
10123:
10104:
10095:
10051:Bézier curve
10040:
9801:
9666:
9662:
9658:
9654:
9648:
9645:
9495:
9401:
9396:
9392:
9388:
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9368:
9366:
9198:
9194:
9189:
9185:
9181:
9178:
9174:
9144:
9122:
8778:, such that
8652:Bézier curve
8649:
8243:
7902:
7486:
7415:
7407:
7258:
7076:The weights
7075:
6835:
6485:
6455:
6423:
6419:
6415:
6411:
6407:
6405:
6248:
6243:. Thus, the
6230:
6214:
6098:Bézier curve
6095:
5961:
5805:
5448:
5403:
5238:
5230:
4984:
4741:
4737:
4735:
4723:
4704:
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2025:
1949:
1870:
1630:
1626:
1330:
1173:
988:
901:
720:
510:
450:
378:
375:Introduction
362:
326:basis spline
325:
321:
316:subfield of
314:mathematical
311:
288:
270:
264:January 2022
261:
251:
244:
237:
230:
218:
206:Please help
201:verification
198:
168:
162:January 2022
159:
143:
113:
104:
93:Please help
88:
86:lead section
61:
54:
48:
47:Please help
44:
10647:Works cited
10214:: 100–112.
6231:Usually in
4719:overfitting
1331:The higher
763:satisfying
107:August 2014
10862:Categories
10825:2012-05-02
10793:"B-Spline"
10432:1712.01293
10425:(6): 218.
10116:References
10056:Box spline
7295:intervals.
6459:Lorentzian
4715:smoothness
2831:goes from
2667:goes from
1723:locations
1623:Properties
586:of degree
544:functions
542:polynomial
483:Definition
420:polynomial
342:smoothness
234:newspapers
223:"B-spline"
50:improve it
10798:MathWorld
10457:125665629
10297:Computing
10262:Computing
10193:851370272
9955:ℓ
9940:∑
9919:∑
9738:ℓ
9723:∑
9702:∑
9576:∑
9429:∑
9297:∑
9229:∑
8534:−
8479:−
8412:−
8379:−
8270:∑
8128:−
8060:−
7986:−
7957:−
7763:−
7743:−
7723:−
7710:−
7458:∈
7389:≥
7307:∑
7013:∑
6942:∑
6871:∑
6808:…
6435:α
6335:α
6325:∑
6321:−
6269:∑
6168:≫
6157:given by
5937:−
5902:⋅
5872:∑
5841:∑
5762:⋅
5741:Γ
5727:Γ
5709:Γ
5680:…
5667:∣
5632:⋅
5617:∞
5612:∞
5609:−
5605:∫
5560:μ
5460:μ
5413:∑
5354:⋅
5335:∑
5172:−
5142:−
5116:−
5109:α
5105:−
5096:α
5081:−
5064:−
5051:∑
5022:α
5012:∑
4943:−
4902:−
4876:−
4860:−
4825:−
4717:to avoid
4662:∗
4654:∗
4493:…
4339:−
4320:−
4277:−
4258:−
4213:−
4187:−
4184:⋅
4169:…
4138:−
4000:−
3870:−
3769::
3646::
3588:≤
3552:−
3510:≤
3481:−
3456:−
3417:≤
3215:−
3169:−
3142:−
2793:−
2763:−
2635:−
2604:−
2537:That is,
2505:−
2463:−
2433:−
2388:−
2358:−
2327:−
2259:otherwise
2221:≤
2098:α
2088:∑
2007:−
1912:−
1882:−
1846:−
1768:≤
1662:−
1590:−
1547:−
1517:−
1471:−
1440:−
1409:−
1305:otherwise
1267:≤
1154:−
1052:−
938:−
932:−
914:∑
876:otherwise
832:≤
693:…
539:piecewise
433:−
417:piecewise
56:talk page
10724:(1999).
10615:28705788
10568:96229316
10419:EPJ Plus
10081:T-spline
10071:M-spline
10066:I-spline
10045:See also
7298:Because
6463:Gaussian
6135:supports
5995:. Here,
4701:P-spline
3756:At
3633:At
2207:if
1253:if
989:for all
818:if
811:non-zero
322:B-spline
18:B-Spline
10595:Bibcode
10548:Bibcode
10437:Bibcode
10317:7003455
10282:2407104
4360:alone.
3077:FORTRAN
3066:support
1816:(where
334:support
312:In the
248:scholar
148:Please
10763:
10744:
10710:
10684:
10661:
10613:
10566:
10512:
10481:
10455:
10315:
10280:
10191:
10181:
9153:, and
8747:, and
7584:, and
6406:where
6278:
5447:. The
4707:fitted
4009:
3953:
3807:
3772:
3684:
3649:
2727:, and
346:domain
344:, and
338:degree
330:spline
250:
243:
236:
229:
221:
10819:(PDF)
10812:(PDF)
10611:S2CID
10564:S2CID
10453:S2CID
10427:arXiv
10313:S2CID
10278:S2CID
10087:Notes
9802:with
9127:NURBS
6622:, or
5806:with
2949:is a
415:is a
328:is a
255:JSTOR
241:books
10761:ISBN
10742:ISBN
10708:ISBN
10682:ISBN
10659:ISBN
10510:ISBN
10479:ISBN
10189:OCLC
10179:ISBN
9665:and
9657:and
7291:way.
7130:and
6745:and
6490:and
6461:and
5962:and
5654:norm
5505:norm
5376:norm
4693:See
4684:sinc
4298:and
4240:-th
3594:<
3516:<
3423:<
2227:<
1273:<
1138:and
1036:and
838:<
471:are
389:term
367:and
356:and
320:, a
227:news
10603:doi
10556:doi
10500:"8"
10445:doi
10423:133
10398:doi
10358:doi
10305:doi
10270:doi
10216:doi
10212:109
10171:doi
9387:),
9375:),
9145:In
6486:In
6474:or
6274:all
4686:in
3801:0.5
3678:0.5
3083:at
2864:to
2694:to
383:at
363:In
324:or
210:by
152:to
10864::
10795:.
10609:.
10601:.
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10589:.
10585:.
10562:.
10554:.
10544:38
10542:.
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10435:.
10421:.
10394:67
10392:.
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10311:.
10301:36
10299:.
10276:.
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10210:.
10187:.
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6478:.
6251:,
6215:A
6096:A
5182:on
4721:.
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3072:.
2318::=
2190::=
1399::=
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340:,
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10828:.
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10769:.
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10716:.
10690:.
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9433:i
9425:=
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9419:u
9416:(
9413:C
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9301:i
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9260:n
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9216:u
9213:(
9210:C
9190:d
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9104:.
9101:)
9096:3
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9040:=
9031:3
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8986:b
8981:(
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8968:=
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8927:+
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8912:2
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8904:3
8901:1
8896:=
8887:1
8882:P
8873:,
8870:)
8865:2
8860:b
8855:+
8850:1
8845:b
8840:4
8837:+
8832:0
8827:b
8822:(
8817:6
8814:1
8809:=
8800:0
8795:P
8764:3
8758:P
8733:2
8727:P
8702:1
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8671:0
8665:P
8646:.
8632:)
8627:)
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8597:4
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8579:(
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8573:t
8570:)
8565:2
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8500:3
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8464:3
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8430:+
8425:2
8420:b
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8397:3
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8376:(
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8364:6
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8353:)
8350:t
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8317:i
8312:b
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8300:(
8295:i
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8285:3
8280:0
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8266:=
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8260:t
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8219:t
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8198:)
8195:t
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8160:+
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8112:=
8105:)
8102:t
8099:(
8094:2
8090:B
8082:)
8079:4
8076:+
8071:2
8067:t
8063:6
8055:3
8051:t
8047:3
8044:(
8039:6
8036:1
8031:=
8024:)
8021:t
8018:(
8013:1
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8001:)
7998:1
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7992:t
7989:3
7981:2
7977:t
7973:3
7970:+
7965:3
7961:t
7954:(
7949:6
7946:1
7941:=
7934:)
7931:t
7928:(
7923:0
7919:B
7899:.
7885:]
7877:3
7872:b
7861:2
7856:b
7845:1
7840:b
7829:0
7824:b
7816:[
7809:]
7803:0
7798:1
7793:4
7788:1
7781:0
7776:3
7771:0
7766:3
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7746:6
7738:3
7731:1
7726:3
7718:3
7713:1
7704:[
7697:]
7691:1
7686:t
7679:2
7675:t
7667:3
7663:t
7656:[
7648:6
7645:1
7640:=
7637:)
7634:t
7631:(
7627:C
7601:3
7595:b
7570:2
7564:b
7539:1
7533:b
7508:0
7502:b
7473:]
7470:1
7467:,
7464:0
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7455:t
7435:)
7432:t
7429:(
7425:C
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7383:x
7380:(
7375:n
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7344:1
7341:=
7338:)
7335:x
7332:(
7327:n
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7321:i
7317:B
7311:i
7276:i
7272:P
7242:i
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7217:)
7212:i
7208:z
7204:,
7199:i
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7191:,
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7178:(
7175:=
7170:i
7166:P
7143:i
7139:z
7116:i
7112:y
7089:i
7085:x
7057:.
7054:)
7051:t
7048:(
7043:n
7040:,
7037:i
7033:B
7027:i
7023:z
7017:i
7009:=
7002:)
6999:t
6996:(
6993:Z
6986:,
6983:)
6980:t
6977:(
6972:n
6969:,
6966:i
6962:B
6956:i
6952:y
6946:i
6938:=
6931:)
6928:t
6925:(
6922:Y
6915:,
6912:)
6909:t
6906:(
6901:n
6898:,
6895:i
6891:B
6885:i
6881:x
6875:i
6867:=
6860:)
6857:t
6854:(
6851:X
6819:n
6815:t
6811:,
6805:,
6800:2
6796:t
6792:,
6787:1
6783:t
6762:)
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2084:=
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1247:1
1241:{
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997:t
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971:=
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951:i
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929:m
924:0
921:=
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880:.
870:0
863:,
858:1
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852:p
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846:i
842:t
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804:{
799:=
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793:t
790:(
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782:,
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775:B
751:)
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436:1
430:n
403:n
295:)
289:(
277:)
271:(
266:)
262:(
252:·
245:·
238:·
231:·
204:.
175:)
169:(
164:)
160:(
146:.
120:)
114:(
109:)
105:(
101:.
91:.
66:)
62:(
20:)
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