185:") through points laid out on the floor of a large design loft, a technique borrowed from ship-hull design. For years the practice of ship design had employed models to design in the small. The successful design was then plotted on graph paper and the key points of the plot were re-plotted on larger graph paper to full size. The thin wooden strips provided an interpolation of the key points into smooth curves. The strips would be held in place at discrete points (called "ducks" by Forrest; Schoenberg used "dogs" or "rats") and between these points would assume shapes of minimum strain energy. According to Forrest, one possible impetus for a mathematical model for this process was the potential loss of the critical design components for an entire aircraft should the loft be hit by an enemy bomb. This gave rise to "conic lofting", which used conic sections to model the position of the curve between the ducks. Conic lofting was replaced by what we would call splines in the early 1960s based on work by
134:
of roughness (for example integral squared curvature) subject to the interpolation constraints. Smoothing splines may be viewed as generalizations of interpolation splines where the functions are determined to minimize a weighted combination of the average squared approximation error over observed data and the roughness measure. For a number of meaningful definitions of the roughness measure, the spline functions are found to be finite dimensional in nature, which is the primary reason for their utility in computations and representation. For the rest of this section, we focus entirely on one-dimensional, polynomial splines and use the term "spline" in this restricted sense.
4143:
27:
3582:
276:
6715:
1191:
3110:
1750:
5279:
4138:{\displaystyle {\begin{aligned}S_{i}(x_{i})&=y_{i}=S_{i-1}(x_{i}),&&i=1,\ \ldots ,\ n-1,\\S_{0}(x_{0})&=y_{0},\\S_{n-1}(x_{n})&=y_{n},\\S'_{i}(x_{i})&=S'_{i-1}(x_{i}),&&i=1,\ \ldots ,\ n-1,\\S''_{i}(x_{i})&=S''_{i-1}(x_{i}),&&i=1,\ \ldots ,\ n-1,\\S''_{0}(x_{0})&=S''_{n-1}(x_{n})=0.\end{aligned}}}
2812:
4907:
6455:
913:
161:
were used, polynomials were generally preferred because they were easier to work with. Through the advent of computers, splines have gained importance. They were first used as a replacement for polynomials in interpolation, then as a tool to construct smooth and flexible shapes in computer graphics.
133:
The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The data may be either one-dimensional or multi-dimensional. Spline functions for interpolation are normally determined as the minimizers of suitable measures
2821:
1430:
4997:
2542:
802:
6905:); which implies that all interior knots are double. Several methods have been invented to fit such splines to given data points; that is, to make them into interpolating splines, and to do so by estimating plausible tangent values where each two polynomial pieces meet (giving us
6255:
4655:
5512:
7240:
However, the evaluation and summation steps are often combined in clever ways. For example, Bernstein polynomials are a basis for polynomials that can be evaluated in linear combinations efficiently using special recurrence relations. This is the essence of
2354:
169:, which is probably the first place that the word "spline" is used in connection with smooth, piecewise polynomial approximation. However, the ideas have their roots in the aircraft and shipbuilding industries. In the foreword to (Bartels et al., 1987),
4553:
3396:
442:
3587:
6710:{\displaystyle {\begin{aligned}a&=t_{0}<\underbrace {t_{1}=\cdots =t_{1}} _{j_{1}}<\cdots <\underbrace {t_{k-2}=\cdots =t_{k-2}} _{j_{k-2}}<t_{k-1}=b\\j_{i}&\leq n+1,\qquad i=1,\ldots ,k-2.\end{aligned}}}
5864:
1186:{\displaystyle {\begin{aligned}S(t)&=P_{0}(t),&&t_{0}\leq t<t_{1},\\S(t)&=P_{1}(t),&&t_{1}\leq t<t_{2},\\&\vdots \\S(t)&=P_{k-1}(t),&&t_{k-1}\leq t\leq t_{k}.\end{aligned}}}
3121:
is not quadratic, the result is still called a quadratic spline. This demonstrates that the degree of a spline is the maximum degree of its polynomial parts.) The extended knot vector for this type of spline would be
6792:
If a type of spline has additional linear conditions imposed upon it, then the resulting spline will lie in a subspace. The space of all natural cubic splines, for instance, is a subspace of the space of all cubic
1435:
918:
5316:
5002:
2826:
2547:
447:
3105:{\displaystyle {\begin{aligned}S(t)&=P_{0}(t)=-2-2t^{2},&&0\leq t<1\\S(t)&=P_{1}(t)=1-6t+t^{2},&&1\leq t<2\\S(t)&=P_{2}(t)=-1+t-2t^{2},&&2\leq t\leq 3\end{aligned}}}
1745:{\displaystyle {\begin{aligned}P_{i-1}^{(0)}(t_{i})&=P_{i}^{(0)}(t_{i}),\\P_{i-1}^{(1)}(t_{i})&=P_{i}^{(1)}(t_{i}),\\\vdots &\\P_{i-1}^{(r_{i})}(t_{i})&=P_{i}^{(r_{i})}(t_{i}).\end{aligned}}}
5274:{\displaystyle {\begin{aligned}c_{j}&=z_{j}-\mu _{j}c_{j+1},\\b_{j}&={\frac {a_{j+1}-a_{j}}{h_{j}}}-{\frac {h_{j}(c_{j+1}+2c_{j})}{3}},\\d_{j}&={\frac {c_{j+1}-c_{j}}{3h_{j}}}.\end{aligned}}}
2486:
6460:
2167:
6920:
For each of the representations, some means of evaluation must be found so that values of the spline can be produced on demand. For those representations that express each individual polynomial piece
6446:. Briefly this means that adding any two splines of a given type produces spline of that given type, and multiplying a spline of a given type by any constant produces a spline of that given type. The
2807:{\displaystyle {\begin{aligned}S(t)&=P_{0}(t)=-1+4t-t^{2},&&0\leq t<1\\S(t)&=P_{1}(t)=2t,&&1\leq t<2\\S(t)&=P_{2}(t)=2-t+t^{2},&&2\leq t\leq 3\end{aligned}}}
3215:
3154:, i.e. the values and first and second derivatives are continuous. Natural means that the second derivatives of the spline polynomials are zero at the endpoints of the interval of interpolation.
5321:
4660:
1928:
110:. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through
243:(see Birkhoff and de Boor, 1965), all for work occurring in the very early 1960s or late 1950s. At least one of de Casteljau's papers was published, but not widely, in 1959. De Boor's work at
7238:
4902:{\displaystyle {\begin{aligned}l_{i}&=2(x_{i+1}-x_{i-1})-h_{i-1}\mu _{i-1},\\\mu _{i}&={\frac {h_{i}}{l_{i}}},\\z_{i}&={\frac {\alpha _{i}-h_{i-1}z_{i-1}}{l_{i}}}.\end{aligned}}}
6319:
4637:
2128:
7132:
6787:
5706:
5588:
4412:
4975:
891:
7056:
5590:
at the location of this high multiplicity. By convention, any such situation indicates a simple discontinuity between the two adjacent polynomial pieces. This means that if a knot
4386:
254:
Work was also being done at Pratt & Whitney
Aircraft, where two of the authors of (Ahlberg et al., 1967) — the first book-length treatment of splines — were employed, and the
6384:
423:
5809:
1321:
4290:
370:
6423:
3543:
3230:
1810:
1366:
7002:
1878:. Equipped with the operation of adding two functions (pointwise addition) and taking real multiples of functions, this set becomes a real vector space. This
186:
2393:
194:
6888:
is a spline that is expressed using
Hermite polynomials to represent each of the individual polynomial pieces. These are most often used with
797:{\displaystyle {\begin{aligned}&,\quad i=0,\ldots ,k-1\\&=\\&a=t_{0}\leq t_{1}\leq \cdots \leq t_{k-1}\leq t_{k}=b\end{aligned}}}
7256:
For a representation that defines a spline as a linear combination of basis splines, however, something more sophisticated is needed. The
5633:, etc. have the same meaning. It is commonly assumed that any knot vector defining any type of spline has been culled in this fashion.
6250:{\displaystyle S_{i}(x)={\frac {z_{i}(x-t_{i-1})^{3}}{6h_{i}}}+{\frac {z_{i-1}(t_{i}-x)^{3}}{6h_{i}}}+\left(x-t_{i-1})+\left(t_{i}-x)}
2150:. This leads to a more general understanding of a knot vector. The continuity loss at any point can be considered to be the result of
7167:
2021:
5643:
5535:
5818:
Many computer-aided design systems that are designed for high-end graphics and animation use extended knot vectors, for example
211:
The use of splines for modeling automobile bodies seems to have several independent beginnings. Credit is claimed on behalf of
820:
5507:{\displaystyle {\begin{aligned}S_{i,a}=a_{i},\\S_{i,b}=b_{i},\\S_{i,c}=c_{i},\\S_{i,d}=d_{i},\\S_{i,x}=x_{i}.\end{aligned}}}
2512:. Suppose the polynomial pieces are to be of degree 2, and the pieces on and must join in value and first derivative (at
1885:
379:
3159:
7416:
6884:
Often a special name was chosen for a type of spline satisfying two or more of the main items above. For example, the
1401:
share common derivative values from the derivative of order 0 (the function value) up through the derivative of order
6800:
The literature of splines is replete with names for special types of splines. These names have been associated with:
316:
7250:
7136:
Look up the coefficients of the linear combination of those basis polynomials that give the spline on that interval
6823:
5754:
153:
Before computers were used, numerical calculations were done by hand. Although piecewise-defined functions like the
7334:
2349:{\displaystyle (t_{0},t_{1},\cdots ,t_{1},t_{2},\cdots ,t_{2},t_{3},\cdots ,t_{k-2},t_{k-1},\cdots ,t_{k-1},t_{k})}
4574:
7067:
6723:
3403:
7483:
4914:
247:
resulted in a number of papers being published in the early 1960s, including some of the fundamental work on
5732:), natural spline, which is a spline of this classical type with additional conditions imposed at endpoints
7009:
5823:
4331:
198:
6265:
7242:
6330:
262:
is detailed nicely in (Birkhoff, 1990) and (Young, 1997). Davis (1997) summarizes some of this material.
7304:
Birkhoff and de Boor, Piecewise polynomial interpolation and approximation, in: H. L. Garabedian (ed.),
259:
244:
240:
7340:
Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions,
298:
6450:
of the space containing all splines of a certain type can be counted from the extended knot vector:
1286:
4254:
1951:, are taken to approach one another and become coincident has an easy answer. The polynomial piece
350:
294:
255:
72:
7357:
Larry, Schumaker, "Spline
Functions: Computational Methods", SIAM, ISBN 978-1-611973-89-1, (2015).
7290:
Birkhoff, Fluid dynamics, reactor computations, and surface representation, in: Steve Nash (ed.),
4548:{\displaystyle \alpha _{i}={\frac {3}{h_{i}}}(a_{i+1}-a_{i})-{\frac {3}{h_{i-1}}}(a_{i}-a_{i-1}).}
6914:
6390:
1932:
In the mathematical study of polynomial splines the question of what happens when two knots, say
1238:
for the spline. If the knots are equidistantly distributed in the interval we say the spline is
7429:
3137:. A closed linear spline (i.e, the first knot and the last are the same) in the plane is just a
7488:
6808:
6438:
For a given interval and a given extended knot vector on that interval, the splines of degree
1781:
1337:
340:
166:
76:
52:
5743:
Another type of spline that is much used in graphics, for example in drawing programs such as
2500:
are spline functions of the same degree with the same extended knot vectors on that interval.
165:
It is commonly accepted that the first mathematical reference to splines is the 1946 paper by
7408:
91:
80:
39:. Triple knots at both ends of the interval ensure that the curve interpolates the end points
36:
7157:
Add up that linear combination of basis polynomial values to get the value of the spline at
3219:
Thus, the graph of the spline is a straight line outside of the interval, but still smooth.
6949:
6896:
6819:
5830:
205:
68:
8:
5612:
th can be removed without changing the character of the spline, since all multiplicities
286:
143:
142:
According to Gerald Farin, B-splines were explored as early as the nineteenth century by
103:
7323:
7402:
6910:
2154:
located at that point, and a spline type can be completely characterized by its degree
30:
Single knots at 1/3 and 2/3 establish a spline of three cubic polynomials meeting with
7257:
5822:. Computer-aided design systems often use an extended concept of a spline known as a
5744:
212:
122:
95:
6822:
as employed by Pierre Bézier to represent each polynomial piece (giving us the name
5708:
which means that every two adjacent polynomial pieces meet in their value and first
3391:{\displaystyle S_{j}(x)=a_{j}+b_{j}(x-x_{j})+c_{j}(x-x_{j})^{2}+d_{j}(x-x_{j})^{3}.}
2387:
228:
147:
87:
7312:
220:
6906:
6447:
5532:
multiple knots in a knot vector have, since this would lead to continuities like
1331:
236:
7246:
6899:. In this degree they may additionally be chosen to be only tangent-continuous (
7372:
7299:
An
Introduction to Splines for Use in Computer Graphics and Geometric Modeling,
6885:
5715:
derivatives at each knot. The mathematical spline that most closely models the
429:
to be piecewise defined. To accomplish this, let the interval be covered by
7477:
7412:
5819:
5748:
3130:
434:
170:
158:
154:
111:
64:
181:
to construct templates for airplanes by passing thin wooden strips (called "
6443:
373:
178:
7450:
5836:
5716:
182:
118:
44:
26:
20:
6717:
The dimension is equal to the sum of the degree plus the multiplicities
7397:
5812:
5604:
times in an extended knot vector, all instances of it in excess of the
337:
330:
232:
60:
216:
334:
56:
7438:
7261:
6831:
The choices made in forming the extended knot vector, for example:
6812:
5829:
If sampled data from a function or a physical object is available,
248:
3114:
would be a member of that type. (Note: while the polynomial piece
5833:
is an approach to creating a spline that approximates that data.
5815:
as well as in the definition of some computer typographic fonts.
3138:
1408:(in other words, the two adjacent polynomial pieces connect with
224:
174:
3222:
3133:. The next most simple spline has degree 1. It is also called a
190:
7385:
7354:
Chapra, Canale, "Numerical
Methods for Engineers" 5th edition.
6874:
requiring that given data values be on the spline (giving us
107:
7444:
5751:, has pieces that are cubic but has continuity only at most
2481:{\displaystyle G(t)={\bigl (}X(t),Y(t){\bigr )},\quad t\in }
177:", a technique used in the British aircraft industry during
6856:
Any special conditions imposed on the spline, for example:
6804:
The choices made for representing the spline, for example:
7333:
Stoer & Bulirsch, Introduction to
Numerical Analysis.
5859:
with the natural condition can be found using the formula
6840:
continuity and spacing these knots evenly on (giving us
4182:
correspond to coefficients in the form shown earlier and
7386:
XLL Excel Addin
Function Implementation of cubic spline
75:
because it yields similar results, even when using low
7006:
Look up the polynomial basis chosen for that interval
6847:
using knots with no restriction on spacing (giving us
3129:
The simplest spline has degree 0. It is also called a
810:"pieces" of , we want to define a polynomial, call it
16:
Mathematical function defined piecewise by polynomials
7170:
7070:
7012:
6952:
6726:
6458:
6393:
6333:
6268:
5867:
5757:
5646:
5538:
5319:
5000:
4917:
4658:
4577:
4415:
4334:
4257:
3585:
3406:
3233:
3162:
2824:
2545:
2519:) while the pieces on and join simply in value (at
2396:
2170:
2024:
1888:
1784:
1433:
1340:
1289:
916:
823:
445:
382:
353:
7308:
pp. 164–190. Elsevier, New York and
Amsterdam, 1965.
6811:
functions for the entire spline (giving us the name
1859:, one can consider the set of all splines of degree
6938:polynomials, this is conceptually straightforward:
2508:Suppose the interval is and the subintervals are
1923:{\displaystyle S_{n}^{\mathbf {r} }(\mathbf {t} ).}
7232:
7126:
7050:
6996:
6781:
6709:
6417:
6378:
6313:
6249:
5803:
5700:
5582:
5506:
5273:
4969:
4901:
4631:
4547:
4380:
4284:
4137:
3537:
3390:
3209:
3104:
2806:
2480:
2348:
2122:
1922:
1804:
1744:
1360:
1315:
1185:
885:
796:
417:
364:
347:, takes values from an interval and maps them to
102:more frequently refers to a piecewise polynomial (
7347:Young, Garrett Birkhoff and applied mathematics,
7475:
7466:Curves and surfaces for CAGD: a practical guide
7430:Notes, PPT, Mathcad, Maple, Mathematica, Matlab
7405:Interactive simulation of various cubic splines
7280:J. ACM, vol. 11, no. 2, pp. 221-228, Apr. 1964.
7233:{\displaystyle \sum _{j=0}^{k-2}c_{j}P_{j}(t).}
6321:are the values of the second derivative at the
3148:. A cubic spline has degree 3 with continuity
1999:join with the sum of the smoothness losses for
4200:Algorithm for computing Natural Cubic Splines:
7285:The Theory of Splines and Their Applications,
3527:
3415:
3223:Algorithm for computing natural cubic splines
2448:
2414:
6433:
4632:{\displaystyle l_{0}=1,{\mu }_{0}=z_{0}=0\,}
2123:{\displaystyle S(t)\in C^{n-j_{i}-j_{i+1}},}
297:. There might be a discussion about this on
7127:{\displaystyle P_{0}(t),\ldots ,P_{k-2}(t)}
7060:Find the value of each basis polynomial at
6782:{\displaystyle d=n+\sum _{i=1}^{k-2}j_{i}.}
5701:{\displaystyle S(t)\in \mathrm {C} ^{n-1},}
5583:{\displaystyle S(t)\in C^{-m},\quad m>0}
5640:used in numerical analysis has continuity
7398:Online Cubic Spline Interpolation Utility
5528:It might be asked what meaning more than
4966:
4628:
4377:
4281:
4222:Output: set splines which is composed of
3180:
2816:would be a member of that type, and also
876:
408:
355:
317:Learn how and when to remove this message
4970:{\displaystyle l_{n}=1;z_{n}=c_{n}=0.\,}
117:The term spline comes from the flexible
25:
7306:Proc. General Motors Symposium of 1964,
886:{\displaystyle P_{i}:\to \mathbb {R} .}
329:We begin by limiting our discussion to
7476:
7373:An Interactive Introduction to Splines
7260:is an efficient method for evaluating
6934:in terms of some basis for the degree
6425:are the values of the function at the
5855:interpolating cubic spline at a point
5726:), twice continuously differentiable (
2526:). This would define a type of spline
7351:vol. 44, no. 11, pp. 1446–1449, 1997.
7051:{\displaystyle P_{0},\ldots ,P_{k-2}}
6946:, find the interval in which it lies
6859:enforcing zero second derivatives at
4381:{\displaystyle h_{i}=x_{i+1}-x_{i}\,}
258:, by Feodor Theilheimer. The work at
7445:Sisl: Opensource C-library for NURBS
7434:Holistic Numerical Methods Institute
7344:vol. 4, pp. 45–99 and 112–141, 1946.
6314:{\displaystyle z_{i}=S_{i}''(t_{i})}
5636:The classical spline type of degree
1778:such that the spline has smoothness
269:
204:The word "spline" was originally an
7292:A History of Scientific Computation
7278:Multi-variable curve interpolation,
6379:{\displaystyle h_{i}=t_{i}-t_{i-1}}
1330:, then the spline is said to be of
1260:, then the spline is said to be of
418:{\displaystyle S:\to \mathbb {R} .}
13:
7417:The Wolfram Demonstrations Project
6942:For a given value of the argument
5664:
3210:{\displaystyle S''(a)\,=S''(b)=0.}
14:
7500:
7361:
6917:, depending on the method used).
5811:This spline type is also used in
121:devices used by shipbuilders and
1910:
1900:
274:
6675:
5845:The general expression for the
5570:
5517:
5288:
5284:
4147:Let us define one cubic spline
2456:
491:
333:. In this case, a spline is a
128:
7458:
7313:B-splines and Geometric design
7224:
7218:
7121:
7115:
7087:
7081:
6991:
6959:
6412:
6397:
6308:
6295:
6244:
6225:
6166:
6147:
6130:
6105:
6052:
6039:
5998:
5978:
5929:
5903:
5884:
5878:
5804:{\displaystyle S(t)\in C^{1}.}
5795:
5783:
5767:
5761:
5692:
5680:
5656:
5650:
5548:
5542:
5181:
5146:
4721:
4683:
4539:
4507:
4478:
4446:
4122:
4109:
4080:
4067:
4005:
3992:
3963:
3950:
3888:
3875:
3846:
3833:
3793:
3780:
3737:
3724:
3665:
3652:
3613:
3600:
3522:
3496:
3478:
3452:
3446:
3420:
3376:
3356:
3334:
3314:
3298:
3279:
3250:
3244:
3227:Cubic splines are of the form
3198:
3192:
3177:
3171:
3043:
3037:
3017:
3011:
2952:
2946:
2926:
2920:
2864:
2858:
2838:
2832:
2751:
2745:
2725:
2719:
2679:
2673:
2653:
2647:
2585:
2579:
2559:
2553:
2475:
2463:
2443:
2437:
2428:
2422:
2406:
2400:
2343:
2171:
2114:
2082:
2034:
2028:
1914:
1906:
1732:
1719:
1714:
1701:
1681:
1668:
1663:
1650:
1616:
1603:
1598:
1592:
1572:
1559:
1554:
1548:
1522:
1509:
1504:
1498:
1478:
1465:
1460:
1454:
1316:{\displaystyle S\in C^{r_{i}}}
1132:
1126:
1100:
1094:
1036:
1030:
1010:
1004:
956:
950:
930:
924:
872:
869:
837:
709:
696:
690:
658:
652:
614:
602:
576:
570:
544:
538:
526:
485:
453:
404:
401:
389:
114:and interactive curve design.
1:
7297:Bartels, Beatty, and Barsky,
7283:Ahlberg, Nielson, and Walsh,
7270:
4285:{\displaystyle a_{i}=y_{i}\,}
365:{\displaystyle \mathbb {R} ,}
265:
7337:. p. 93-106. ISBN 0387904204
5824:Nonuniform rational B-spline
199:British Aircraft Corporation
79:polynomials, while avoiding
7:
6418:{\displaystyle f(t_{i}^{})}
3538:{\displaystyle C={\bigl },}
2503:
1965:disappears, and the pieces
1847:, and a smoothness vector
331:polynomials in one variable
19:For the drafting tool, see
10:
7505:
7468:. Morgan Kaufmann. p. 119.
5841:interpolating cubic spline
4203:Input: set of coordinates
3400:Given set of coordinates
1382:the two polynomial pieces
137:
18:
7409:Symmetrical Spline Curves
6434:Representations and names
5837:General expression for a
1805:{\displaystyle C^{r_{i}}}
1361:{\displaystyle C^{r_{i}}}
1256:each have degree at most
1249:If the polynomial pieces
1242:, otherwise we say it is
343:. This function, call it
7453:, vbnumericalmethods.com
7451:VBA Spline Interpolation
7243:De Casteljau's algorithm
6915:Kochanek-Bartels splines
5523:
256:David Taylor Model Basin
193:and (somewhat later) by
73:polynomial interpolation
7403:Learning by Simulations
6834:using single knots for
3545:we wish to find set of
3144:A common spline is the
1882:is commonly denoted by
1872:and smoothness vector
125:to draw smooth shapes.
7330:vol. 98, no. 26, 1998.
7234:
7197:
7128:
7052:
6998:
6783:
6765:
6711:
6419:
6380:
6315:
6251:
5805:
5702:
5584:
5508:
5275:
4971:
4903:
4633:
4549:
4382:
4286:
4139:
3539:
3392:
3211:
3106:
2808:
2482:
2350:
2124:
1924:
1806:
1746:
1362:
1317:
1187:
887:
798:
419:
366:
71:is often preferred to
40:
7484:Splines (mathematics)
7464:Farin, G. E. (2002).
7319:vol. 29, no. 5, 1996.
7235:
7171:
7129:
7053:
6999:
6997:{\displaystyle t\in }
6897:Cubic Hermite splines
6876:interpolating splines
6820:Bernstein polynomials
6784:
6739:
6712:
6420:
6381:
6316:
6252:
5806:
5703:
5585:
5509:
5276:
4972:
4904:
4634:
4550:
4383:
4287:
4140:
3540:
3393:
3212:
3107:
2809:
2483:
2351:
2125:
1925:
1807:
1747:
1363:
1323:in a neighborhood of
1318:
1188:
888:
799:
420:
367:
92:computer-aided design
83:for higher degrees.
37:parametric continuity
29:
7245:, which features in
7168:
7068:
7010:
6950:
6724:
6456:
6391:
6331:
6266:
5865:
5831:spline interpolation
5755:
5644:
5536:
5317:
5291:. Populate it with
4998:
4994:, set the following:
4915:
4656:
4575:
4413:
4332:
4255:
3583:
3577:These must satisfy:
3404:
3231:
3160:
3146:natural cubic spline
2822:
2543:
2394:
2168:
2022:
1886:
1837:Given a knot vector
1782:
1431:
1338:
1287:
914:
897:th subinterval of ,
821:
443:
380:
351:
287:confusing or unclear
69:spline interpolation
7349:Notices of the AMS,
7342:Quart. Appl. Math.,
7276:Ferguson, James C,
6911:Catmull-Rom splines
6411:
6294:
5313:set the following:
4108:
4066:
3991:
3949:
3874:
3832:
1905:
1866:having knot vector
1718:
1667:
1602:
1558:
1508:
1464:
295:clarify the article
144:Nikolai Lobachevsky
7411:, an animation by
7324:History of Splines
7230:
7124:
7048:
6994:
6849:nonuniform splines
6779:
6707:
6705:
6616:
6596:
6537:
6523:
6415:
6400:
6376:
6311:
6282:
6247:
5801:
5698:
5597:appears more than
5580:
5504:
5502:
5271:
5269:
4967:
4899:
4897:
4652:set the following:
4629:
4557:Create new arrays
4545:
4378:
4294:Create new arrays
4282:
4135:
4133:
4090:
4054:
3973:
3937:
3856:
3820:
3535:
3388:
3207:
3102:
3100:
2804:
2802:
2478:
2346:
2120:
1920:
1889:
1802:
1742:
1740:
1691:
1634:
1582:
1532:
1488:
1438:
1410:loss of smoothness
1358:
1313:
1183:
1181:
883:
794:
792:
415:
362:
81:Runge's phenomenon
41:
7258:de Boor algorithm
6549:
6547:
6488:
6486:
6218:
6180:
6098:
6066:
6023:
5954:
5745:Adobe Illustrator
5262:
5188:
5128:
4890:
4805:
4505:
4444:
4390:Create new array
4309:Create new array
4230:Create new array
4037:
4028:
3920:
3911:
3697:
3688:
2390:on the interval
1832:smoothness vector
806:On each of these
327:
326:
319:
96:computer graphics
7496:
7469:
7462:
7439:various routines
7392:Online utilities
7239:
7237:
7236:
7231:
7217:
7216:
7207:
7206:
7196:
7185:
7160:
7154:
7133:
7131:
7130:
7125:
7114:
7113:
7080:
7079:
7063:
7057:
7055:
7054:
7049:
7047:
7046:
7022:
7021:
7003:
7001:
7000:
6995:
6990:
6989:
6971:
6970:
6945:
6937:
6933:
6907:Cardinal splines
6904:
6894:
6866:
6862:
6839:
6788:
6786:
6785:
6780:
6775:
6774:
6764:
6753:
6716:
6714:
6713:
6708:
6706:
6655:
6654:
6635:
6634:
6615:
6614:
6613:
6597:
6592:
6591:
6590:
6566:
6565:
6536:
6535:
6534:
6524:
6519:
6518:
6517:
6499:
6498:
6482:
6481:
6441:
6428:
6424:
6422:
6421:
6416:
6410:
6408:
6385:
6383:
6382:
6377:
6375:
6374:
6356:
6355:
6343:
6342:
6324:
6320:
6318:
6317:
6312:
6307:
6306:
6290:
6278:
6277:
6256:
6254:
6253:
6248:
6237:
6236:
6224:
6220:
6219:
6214:
6213:
6212:
6203:
6202:
6186:
6181:
6179:
6178:
6169:
6165:
6164:
6142:
6129:
6128:
6104:
6100:
6099:
6094:
6093:
6092:
6083:
6082:
6072:
6067:
6065:
6064:
6055:
6051:
6050:
6034:
6024:
6022:
6021:
6020:
6007:
6006:
6005:
5990:
5989:
5977:
5976:
5960:
5955:
5953:
5952:
5951:
5938:
5937:
5936:
5927:
5926:
5902:
5901:
5891:
5877:
5876:
5854:
5848:
5810:
5808:
5807:
5802:
5782:
5781:
5739:
5735:
5731:
5725:
5714:
5707:
5705:
5704:
5699:
5679:
5678:
5667:
5639:
5632:
5625:
5618:
5611:
5603:
5596:
5589:
5587:
5586:
5581:
5566:
5565:
5531:
5519:
5513:
5511:
5510:
5505:
5503:
5496:
5495:
5483:
5482:
5460:
5459:
5447:
5446:
5424:
5423:
5411:
5410:
5388:
5387:
5375:
5374:
5352:
5351:
5339:
5338:
5312:
5298:
5294:
5290:
5286:
5280:
5278:
5277:
5272:
5270:
5263:
5261:
5260:
5259:
5246:
5245:
5244:
5232:
5231:
5215:
5206:
5205:
5189:
5184:
5180:
5179:
5164:
5163:
5145:
5144:
5134:
5129:
5127:
5126:
5117:
5116:
5115:
5103:
5102:
5086:
5077:
5076:
5060:
5059:
5044:
5043:
5031:
5030:
5014:
5013:
4993:
4976:
4974:
4973:
4968:
4959:
4958:
4946:
4945:
4927:
4926:
4908:
4906:
4905:
4900:
4898:
4891:
4889:
4888:
4879:
4878:
4877:
4862:
4861:
4843:
4842:
4832:
4823:
4822:
4806:
4804:
4803:
4794:
4793:
4784:
4775:
4774:
4758:
4757:
4742:
4741:
4720:
4719:
4701:
4700:
4672:
4671:
4651:
4638:
4636:
4635:
4630:
4621:
4620:
4608:
4607:
4602:
4587:
4586:
4567:
4560:
4554:
4552:
4551:
4546:
4538:
4537:
4519:
4518:
4506:
4504:
4503:
4485:
4477:
4476:
4464:
4463:
4445:
4443:
4442:
4430:
4425:
4424:
4408:
4397:
4393:
4387:
4385:
4384:
4379:
4376:
4375:
4363:
4362:
4344:
4343:
4327:
4316:
4312:
4305:
4301:
4297:
4291:
4289:
4288:
4283:
4280:
4279:
4267:
4266:
4250:
4240:
4233:
4225:
4220:
4214:
4206:
4195:
4188:
4181:
4177:
4150:
4144:
4142:
4141:
4136:
4134:
4121:
4120:
4104:
4079:
4078:
4062:
4035:
4026:
4012:
4004:
4003:
3987:
3962:
3961:
3945:
3918:
3909:
3895:
3887:
3886:
3870:
3845:
3844:
3828:
3812:
3811:
3792:
3791:
3779:
3778:
3756:
3755:
3736:
3735:
3723:
3722:
3695:
3686:
3672:
3664:
3663:
3651:
3650:
3632:
3631:
3612:
3611:
3599:
3598:
3573:
3562:
3548:
3544:
3542:
3541:
3536:
3531:
3530:
3521:
3520:
3508:
3507:
3477:
3476:
3464:
3463:
3445:
3444:
3432:
3431:
3419:
3418:
3397:
3395:
3394:
3389:
3384:
3383:
3374:
3373:
3355:
3354:
3342:
3341:
3332:
3331:
3313:
3312:
3297:
3296:
3278:
3277:
3265:
3264:
3243:
3242:
3216:
3214:
3213:
3208:
3191:
3170:
3153:
3125:
3120:
3111:
3109:
3108:
3103:
3101:
3081:
3076:
3075:
3036:
3035:
2987:
2982:
2981:
2945:
2944:
2896:
2891:
2890:
2857:
2856:
2813:
2811:
2810:
2805:
2803:
2783:
2778:
2777:
2744:
2743:
2695:
2672:
2671:
2623:
2618:
2617:
2578:
2577:
2536:
2525:
2518:
2511:
2499:
2495:
2487:
2485:
2484:
2479:
2452:
2451:
2418:
2417:
2388:parametric curve
2382:
2371:
2364:
2355:
2353:
2352:
2347:
2342:
2341:
2329:
2328:
2304:
2303:
2285:
2284:
2260:
2259:
2247:
2246:
2228:
2227:
2215:
2214:
2196:
2195:
2183:
2182:
2157:
2149:
2129:
2127:
2126:
2121:
2113:
2112:
2094:
2093:
2081:
2080:
2079:
2078:
2060:
2059:
2017:
2005:
1998:
1981:
1964:
1950:
1938:
1929:
1927:
1926:
1921:
1913:
1904:
1903:
1897:
1877:
1871:
1865:
1858:
1852:
1846:
1842:
1834:for the spline.
1829:
1818:
1811:
1809:
1808:
1803:
1801:
1800:
1799:
1798:
1777:
1751:
1749:
1748:
1743:
1741:
1731:
1730:
1717:
1713:
1712:
1699:
1680:
1679:
1666:
1662:
1661:
1648:
1630:
1615:
1614:
1601:
1590:
1571:
1570:
1557:
1546:
1521:
1520:
1507:
1496:
1477:
1476:
1463:
1452:
1424:
1407:
1400:
1393:
1381:
1374:
1367:
1365:
1364:
1359:
1357:
1356:
1355:
1354:
1329:
1322:
1320:
1319:
1314:
1312:
1311:
1310:
1309:
1279:
1269:
1259:
1255:
1233:
1208:
1201:
1192:
1190:
1189:
1184:
1182:
1175:
1174:
1156:
1155:
1139:
1125:
1124:
1080:
1073:
1072:
1054:
1053:
1043:
1029:
1028:
993:
992:
974:
973:
963:
949:
948:
907:
900:
896:
892:
890:
889:
884:
879:
868:
867:
849:
848:
833:
832:
816:
809:
803:
801:
800:
795:
793:
783:
782:
770:
769:
745:
744:
732:
731:
715:
708:
707:
689:
688:
676:
675:
651:
650:
632:
631:
601:
600:
588:
587:
569:
568:
556:
555:
522:
484:
483:
465:
464:
449:
432:
428:
424:
422:
421:
416:
411:
371:
369:
368:
363:
358:
346:
322:
315:
311:
308:
302:
278:
277:
270:
148:Kazan University
88:computer science
35:
7504:
7503:
7499:
7498:
7497:
7495:
7494:
7493:
7474:
7473:
7472:
7463:
7459:
7364:
7335:Springer-Verlag
7273:
7267:
7212:
7208:
7202:
7198:
7186:
7175:
7169:
7166:
7165:
7158:
7153:
7143:
7137:
7103:
7099:
7075:
7071:
7069:
7066:
7065:
7061:
7036:
7032:
7017:
7013:
7011:
7008:
7007:
6979:
6975:
6966:
6962:
6951:
6948:
6947:
6943:
6935:
6926:
6921:
6900:
6889:
6869:natural splines
6864:
6860:
6842:uniform splines
6835:
6770:
6766:
6754:
6743:
6725:
6722:
6721:
6704:
6703:
6656:
6650:
6646:
6643:
6642:
6624:
6620:
6603:
6599:
6598:
6580:
6576:
6555:
6551:
6550:
6548:
6530:
6526:
6525:
6513:
6509:
6494:
6490:
6489:
6487:
6477:
6473:
6466:
6459:
6457:
6454:
6453:
6439:
6436:
6426:
6409:
6404:
6392:
6389:
6388:
6364:
6360:
6351:
6347:
6338:
6334:
6332:
6329:
6328:
6322:
6302:
6298:
6286:
6273:
6269:
6267:
6264:
6263:
6232:
6228:
6208:
6204:
6192:
6188:
6187:
6185:
6174:
6170:
6154:
6150:
6143:
6141:
6140:
6136:
6118:
6114:
6088:
6084:
6078:
6074:
6073:
6071:
6060:
6056:
6046:
6042:
6035:
6033:
6032:
6028:
6016:
6012:
6008:
6001:
5997:
5985:
5981:
5966:
5962:
5961:
5959:
5947:
5943:
5939:
5932:
5928:
5916:
5912:
5897:
5893:
5892:
5890:
5872:
5868:
5866:
5863:
5862:
5850:
5846:
5843:
5777:
5773:
5756:
5753:
5752:
5737:
5733:
5727:
5720:
5709:
5668:
5663:
5662:
5645:
5642:
5641:
5637:
5627:
5620:
5613:
5605:
5598:
5595:
5591:
5558:
5554:
5537:
5534:
5533:
5529:
5526:
5501:
5500:
5491:
5487:
5472:
5468:
5465:
5464:
5455:
5451:
5436:
5432:
5429:
5428:
5419:
5415:
5400:
5396:
5393:
5392:
5383:
5379:
5364:
5360:
5357:
5356:
5347:
5343:
5328:
5324:
5320:
5318:
5315:
5314:
5303:
5296:
5292:
5283:Create new set
5268:
5267:
5255:
5251:
5247:
5240:
5236:
5221:
5217:
5216:
5214:
5207:
5201:
5197:
5194:
5193:
5175:
5171:
5153:
5149:
5140:
5136:
5135:
5133:
5122:
5118:
5111:
5107:
5092:
5088:
5087:
5085:
5078:
5072:
5068:
5065:
5064:
5049:
5045:
5039:
5035:
5026:
5022:
5015:
5009:
5005:
5001:
4999:
4996:
4995:
4980:
4954:
4950:
4941:
4937:
4922:
4918:
4916:
4913:
4912:
4896:
4895:
4884:
4880:
4867:
4863:
4851:
4847:
4838:
4834:
4833:
4831:
4824:
4818:
4814:
4811:
4810:
4799:
4795:
4789:
4785:
4783:
4776:
4770:
4766:
4763:
4762:
4747:
4743:
4731:
4727:
4709:
4705:
4690:
4686:
4673:
4667:
4663:
4659:
4657:
4654:
4653:
4642:
4616:
4612:
4603:
4598:
4597:
4582:
4578:
4576:
4573:
4572:
4562:
4561:, each of size
4558:
4527:
4523:
4514:
4510:
4493:
4489:
4484:
4472:
4468:
4453:
4449:
4438:
4434:
4429:
4420:
4416:
4414:
4411:
4410:
4399:
4395:
4391:
4371:
4367:
4352:
4348:
4339:
4335:
4333:
4330:
4329:
4318:
4314:
4310:
4303:
4302:, each of size
4299:
4295:
4275:
4271:
4262:
4258:
4256:
4253:
4252:
4242:
4235:
4231:
4223:
4221:
4210:
4208:
4204:
4202:
4194:
4190:
4187:
4183:
4179:
4174:
4152:
4148:
4132:
4131:
4116:
4112:
4094:
4083:
4074:
4070:
4058:
4051:
4050:
4011:
3999:
3995:
3977:
3966:
3957:
3953:
3941:
3934:
3933:
3894:
3882:
3878:
3860:
3849:
3840:
3836:
3824:
3817:
3816:
3807:
3803:
3796:
3787:
3783:
3768:
3764:
3761:
3760:
3751:
3747:
3740:
3731:
3727:
3718:
3714:
3711:
3710:
3671:
3659:
3655:
3640:
3636:
3627:
3623:
3616:
3607:
3603:
3594:
3590:
3586:
3584:
3581:
3580:
3564:
3555:
3550:
3546:
3526:
3525:
3516:
3512:
3503:
3499:
3472:
3468:
3459:
3455:
3440:
3436:
3427:
3423:
3414:
3413:
3405:
3402:
3401:
3379:
3375:
3369:
3365:
3350:
3346:
3337:
3333:
3327:
3323:
3308:
3304:
3292:
3288:
3273:
3269:
3260:
3256:
3238:
3234:
3232:
3229:
3228:
3225:
3184:
3163:
3161:
3158:
3157:
3149:
3124:(0, 1, 2, 2, 3)
3123:
3115:
3099:
3098:
3080:
3071:
3067:
3031:
3027:
3020:
3005:
3004:
2986:
2977:
2973:
2940:
2936:
2929:
2914:
2913:
2895:
2886:
2882:
2852:
2848:
2841:
2825:
2823:
2820:
2819:
2801:
2800:
2782:
2773:
2769:
2739:
2735:
2728:
2713:
2712:
2694:
2667:
2663:
2656:
2641:
2640:
2622:
2613:
2609:
2573:
2569:
2562:
2546:
2544:
2541:
2540:
2527:
2520:
2513:
2509:
2506:
2497:
2493:
2447:
2446:
2413:
2412:
2395:
2392:
2391:
2373:
2370:
2366:
2363:
2359:
2337:
2333:
2318:
2314:
2293:
2289:
2274:
2270:
2255:
2251:
2242:
2238:
2223:
2219:
2210:
2206:
2191:
2187:
2178:
2174:
2169:
2166:
2165:
2155:
2147:
2136:
2131:
2102:
2098:
2089:
2085:
2068:
2064:
2055:
2051:
2044:
2040:
2023:
2020:
2019:
2016:
2007:
2004:
2000:
1992:
1983:
1975:
1966:
1957:
1952:
1949:
1940:
1937:
1933:
1909:
1899:
1898:
1893:
1887:
1884:
1883:
1873:
1867:
1860:
1854:
1848:
1844:
1838:
1820:
1817:
1813:
1794:
1790:
1789:
1785:
1783:
1780:
1779:
1775:
1765:
1755:
1739:
1738:
1726:
1722:
1708:
1704:
1700:
1695:
1684:
1675:
1671:
1657:
1653:
1649:
1638:
1631:
1629:
1623:
1622:
1610:
1606:
1591:
1586:
1575:
1566:
1562:
1547:
1536:
1529:
1528:
1516:
1512:
1497:
1492:
1481:
1472:
1468:
1453:
1442:
1434:
1432:
1429:
1428:
1422:
1413:
1406:
1402:
1399:
1395:
1392:
1383:
1380:
1376:
1373:
1369:
1350:
1346:
1345:
1341:
1339:
1336:
1335:
1328:
1324:
1305:
1301:
1300:
1296:
1288:
1285:
1284:
1274:
1264:
1257:
1254:
1250:
1230:
1224:
1214:
1207:
1203:
1196:
1180:
1179:
1170:
1166:
1145:
1141:
1138:
1114:
1110:
1103:
1088:
1087:
1078:
1077:
1068:
1064:
1049:
1045:
1042:
1024:
1020:
1013:
998:
997:
988:
984:
969:
965:
962:
944:
940:
933:
917:
915:
912:
911:
906:
902:
898:
894:
875:
857:
853:
844:
840:
828:
824:
822:
819:
818:
815:
811:
807:
791:
790:
778:
774:
759:
755:
740:
736:
727:
723:
713:
712:
703:
699:
684:
680:
665:
661:
640:
636:
621:
617:
596:
592:
583:
579:
564:
560:
551:
547:
520:
519:
473:
469:
460:
456:
446:
444:
441:
440:
430:
426:
407:
381:
378:
377:
354:
352:
349:
348:
344:
323:
312:
306:
303:
292:
279:
275:
268:
140:
131:
31:
24:
17:
12:
11:
5:
7502:
7492:
7491:
7486:
7471:
7470:
7456:
7455:
7454:
7448:
7442:
7436:
7421:
7420:
7406:
7400:
7389:
7388:
7380:Excel Function
7377:
7376:
7363:
7362:External links
7360:
7359:
7358:
7355:
7352:
7345:
7338:
7331:
7320:
7309:
7302:
7295:
7288:
7281:
7272:
7269:
7251:Bézier splines
7229:
7226:
7223:
7220:
7215:
7211:
7205:
7201:
7195:
7192:
7189:
7184:
7181:
7178:
7174:
7163:
7162:
7155:
7148:
7141:
7134:
7123:
7120:
7117:
7112:
7109:
7106:
7102:
7098:
7095:
7092:
7089:
7086:
7083:
7078:
7074:
7058:
7045:
7042:
7039:
7035:
7031:
7028:
7025:
7020:
7016:
7004:
6993:
6988:
6985:
6982:
6978:
6974:
6969:
6965:
6961:
6958:
6955:
6924:
6895:; that is, as
6886:Hermite spline
6882:
6881:
6880:
6879:
6872:
6854:
6853:
6852:
6845:
6829:
6828:
6827:
6824:Bézier splines
6816:
6790:
6789:
6778:
6773:
6769:
6763:
6760:
6757:
6752:
6749:
6746:
6742:
6738:
6735:
6732:
6729:
6702:
6699:
6696:
6693:
6690:
6687:
6684:
6681:
6678:
6674:
6671:
6668:
6665:
6662:
6659:
6657:
6653:
6649:
6645:
6644:
6641:
6638:
6633:
6630:
6627:
6623:
6619:
6612:
6609:
6606:
6602:
6595:
6589:
6586:
6583:
6579:
6575:
6572:
6569:
6564:
6561:
6558:
6554:
6546:
6543:
6540:
6533:
6529:
6522:
6516:
6512:
6508:
6505:
6502:
6497:
6493:
6485:
6480:
6476:
6472:
6469:
6467:
6465:
6462:
6461:
6435:
6432:
6431:
6430:
6414:
6407:
6403:
6399:
6396:
6386:
6373:
6370:
6367:
6363:
6359:
6354:
6350:
6346:
6341:
6337:
6326:
6310:
6305:
6301:
6297:
6293:
6289:
6285:
6281:
6276:
6272:
6246:
6243:
6240:
6235:
6231:
6227:
6223:
6217:
6211:
6207:
6201:
6198:
6195:
6191:
6184:
6177:
6173:
6168:
6163:
6160:
6157:
6153:
6149:
6146:
6139:
6135:
6132:
6127:
6124:
6121:
6117:
6113:
6110:
6107:
6103:
6097:
6091:
6087:
6081:
6077:
6070:
6063:
6059:
6054:
6049:
6045:
6041:
6038:
6031:
6027:
6019:
6015:
6011:
6004:
6000:
5996:
5993:
5988:
5984:
5980:
5975:
5972:
5969:
5965:
5958:
5950:
5946:
5942:
5935:
5931:
5925:
5922:
5919:
5915:
5911:
5908:
5905:
5900:
5896:
5889:
5886:
5883:
5880:
5875:
5871:
5842:
5835:
5800:
5797:
5794:
5791:
5788:
5785:
5780:
5776:
5772:
5769:
5766:
5763:
5760:
5697:
5694:
5691:
5688:
5685:
5682:
5677:
5674:
5671:
5666:
5661:
5658:
5655:
5652:
5649:
5593:
5579:
5576:
5573:
5569:
5564:
5561:
5557:
5553:
5550:
5547:
5544:
5541:
5525:
5522:
5521:
5520:
5514:
5499:
5494:
5490:
5486:
5481:
5478:
5475:
5471:
5467:
5466:
5463:
5458:
5454:
5450:
5445:
5442:
5439:
5435:
5431:
5430:
5427:
5422:
5418:
5414:
5409:
5406:
5403:
5399:
5395:
5394:
5391:
5386:
5382:
5378:
5373:
5370:
5367:
5363:
5359:
5358:
5355:
5350:
5346:
5342:
5337:
5334:
5331:
5327:
5323:
5322:
5300:
5281:
5266:
5258:
5254:
5250:
5243:
5239:
5235:
5230:
5227:
5224:
5220:
5213:
5210:
5208:
5204:
5200:
5196:
5195:
5192:
5187:
5183:
5178:
5174:
5170:
5167:
5162:
5159:
5156:
5152:
5148:
5143:
5139:
5132:
5125:
5121:
5114:
5110:
5106:
5101:
5098:
5095:
5091:
5084:
5081:
5079:
5075:
5071:
5067:
5066:
5063:
5058:
5055:
5052:
5048:
5042:
5038:
5034:
5029:
5025:
5021:
5018:
5016:
5012:
5008:
5004:
5003:
4977:
4965:
4962:
4957:
4953:
4949:
4944:
4940:
4936:
4933:
4930:
4925:
4921:
4909:
4894:
4887:
4883:
4876:
4873:
4870:
4866:
4860:
4857:
4854:
4850:
4846:
4841:
4837:
4830:
4827:
4825:
4821:
4817:
4813:
4812:
4809:
4802:
4798:
4792:
4788:
4782:
4779:
4777:
4773:
4769:
4765:
4764:
4761:
4756:
4753:
4750:
4746:
4740:
4737:
4734:
4730:
4726:
4723:
4718:
4715:
4712:
4708:
4704:
4699:
4696:
4693:
4689:
4685:
4682:
4679:
4676:
4674:
4670:
4666:
4662:
4661:
4639:
4627:
4624:
4619:
4615:
4611:
4606:
4601:
4596:
4593:
4590:
4585:
4581:
4569:
4555:
4544:
4541:
4536:
4533:
4530:
4526:
4522:
4517:
4513:
4509:
4502:
4499:
4496:
4492:
4488:
4483:
4480:
4475:
4471:
4467:
4462:
4459:
4456:
4452:
4448:
4441:
4437:
4433:
4428:
4423:
4419:
4388:
4374:
4370:
4366:
4361:
4358:
4355:
4351:
4347:
4342:
4338:
4307:
4292:
4278:
4274:
4270:
4265:
4261:
4192:
4185:
4172:
4130:
4127:
4124:
4119:
4115:
4111:
4107:
4103:
4100:
4097:
4093:
4089:
4086:
4084:
4082:
4077:
4073:
4069:
4065:
4061:
4057:
4053:
4052:
4049:
4046:
4043:
4040:
4034:
4031:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4002:
3998:
3994:
3990:
3986:
3983:
3980:
3976:
3972:
3969:
3967:
3965:
3960:
3956:
3952:
3948:
3944:
3940:
3936:
3935:
3932:
3929:
3926:
3923:
3917:
3914:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3885:
3881:
3877:
3873:
3869:
3866:
3863:
3859:
3855:
3852:
3850:
3848:
3843:
3839:
3835:
3831:
3827:
3823:
3819:
3818:
3815:
3810:
3806:
3802:
3799:
3797:
3795:
3790:
3786:
3782:
3777:
3774:
3771:
3767:
3763:
3762:
3759:
3754:
3750:
3746:
3743:
3741:
3739:
3734:
3730:
3726:
3721:
3717:
3713:
3712:
3709:
3706:
3703:
3700:
3694:
3691:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3662:
3658:
3654:
3649:
3646:
3643:
3639:
3635:
3630:
3626:
3622:
3619:
3617:
3615:
3610:
3606:
3602:
3597:
3593:
3589:
3588:
3553:
3534:
3529:
3524:
3519:
3515:
3511:
3506:
3502:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3475:
3471:
3467:
3462:
3458:
3454:
3451:
3448:
3443:
3439:
3435:
3430:
3426:
3422:
3417:
3412:
3409:
3387:
3382:
3378:
3372:
3368:
3364:
3361:
3358:
3353:
3349:
3345:
3340:
3336:
3330:
3326:
3322:
3319:
3316:
3311:
3307:
3303:
3300:
3295:
3291:
3287:
3284:
3281:
3276:
3272:
3268:
3263:
3259:
3255:
3252:
3249:
3246:
3241:
3237:
3224:
3221:
3206:
3203:
3200:
3197:
3194:
3190:
3187:
3183:
3179:
3176:
3173:
3169:
3166:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3074:
3070:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3034:
3030:
3026:
3023:
3021:
3019:
3016:
3013:
3010:
3007:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2980:
2976:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2943:
2939:
2935:
2932:
2930:
2928:
2925:
2922:
2919:
2916:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2889:
2885:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2855:
2851:
2847:
2844:
2842:
2840:
2837:
2834:
2831:
2828:
2827:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2776:
2772:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2742:
2738:
2734:
2731:
2729:
2727:
2724:
2721:
2718:
2715:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2670:
2666:
2662:
2659:
2657:
2655:
2652:
2649:
2646:
2643:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2616:
2612:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2576:
2572:
2568:
2565:
2563:
2561:
2558:
2555:
2552:
2549:
2548:
2505:
2502:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2455:
2450:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2416:
2411:
2408:
2405:
2402:
2399:
2368:
2361:
2345:
2340:
2336:
2332:
2327:
2324:
2321:
2317:
2313:
2310:
2307:
2302:
2299:
2296:
2292:
2288:
2283:
2280:
2277:
2273:
2269:
2266:
2263:
2258:
2254:
2250:
2245:
2241:
2237:
2234:
2231:
2226:
2222:
2218:
2213:
2209:
2205:
2202:
2199:
2194:
2190:
2186:
2181:
2177:
2173:
2152:multiple knots
2145:
2134:
2119:
2116:
2111:
2108:
2105:
2101:
2097:
2092:
2088:
2084:
2077:
2074:
2071:
2067:
2063:
2058:
2054:
2050:
2047:
2043:
2039:
2036:
2033:
2030:
2027:
2011:
2002:
1987:
1970:
1955:
1944:
1935:
1919:
1916:
1912:
1908:
1902:
1896:
1892:
1815:
1797:
1793:
1788:
1770:
1763:
1737:
1734:
1729:
1725:
1721:
1716:
1711:
1707:
1703:
1698:
1694:
1690:
1687:
1685:
1683:
1678:
1674:
1670:
1665:
1660:
1656:
1652:
1647:
1644:
1641:
1637:
1633:
1632:
1628:
1625:
1624:
1621:
1618:
1613:
1609:
1605:
1600:
1597:
1594:
1589:
1585:
1581:
1578:
1576:
1574:
1569:
1565:
1561:
1556:
1553:
1550:
1545:
1542:
1539:
1535:
1531:
1530:
1527:
1524:
1519:
1515:
1511:
1506:
1503:
1500:
1495:
1491:
1487:
1484:
1482:
1480:
1475:
1471:
1467:
1462:
1459:
1456:
1451:
1448:
1445:
1441:
1437:
1436:
1420:
1404:
1397:
1387:
1378:
1375:. That is, at
1371:
1353:
1349:
1344:
1326:
1308:
1304:
1299:
1295:
1292:
1252:
1228:
1222:
1205:
1178:
1173:
1169:
1165:
1162:
1159:
1154:
1151:
1148:
1144:
1140:
1137:
1134:
1131:
1128:
1123:
1120:
1117:
1113:
1109:
1106:
1104:
1102:
1099:
1096:
1093:
1090:
1089:
1086:
1083:
1081:
1079:
1076:
1071:
1067:
1063:
1060:
1057:
1052:
1048:
1044:
1041:
1038:
1035:
1032:
1027:
1023:
1019:
1016:
1014:
1012:
1009:
1006:
1003:
1000:
999:
996:
991:
987:
983:
980:
977:
972:
968:
964:
961:
958:
955:
952:
947:
943:
939:
936:
934:
932:
929:
926:
923:
920:
919:
904:
901:is defined by
882:
878:
874:
871:
866:
863:
860:
856:
852:
847:
843:
839:
836:
831:
827:
813:
789:
786:
781:
777:
773:
768:
765:
762:
758:
754:
751:
748:
743:
739:
735:
730:
726:
722:
719:
716:
714:
711:
706:
702:
698:
695:
692:
687:
683:
679:
674:
671:
668:
664:
660:
657:
654:
649:
646:
643:
639:
635:
630:
627:
624:
620:
616:
613:
610:
607:
604:
599:
595:
591:
586:
582:
578:
575:
572:
567:
563:
559:
554:
550:
546:
543:
540:
537:
534:
531:
528:
525:
523:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
490:
487:
482:
479:
476:
472:
468:
463:
459:
455:
452:
450:
448:
437:subintervals,
414:
410:
406:
403:
400:
397:
394:
391:
388:
385:
361:
357:
325:
324:
282:
280:
273:
267:
264:
260:General Motors
245:General Motors
241:General Motors
208:dialect word.
187:J. C. Ferguson
139:
136:
130:
127:
15:
9:
6:
4:
3:
2:
7501:
7490:
7489:Interpolation
7487:
7485:
7482:
7481:
7479:
7467:
7461:
7457:
7452:
7449:
7446:
7443:
7440:
7437:
7435:
7431:
7428:
7427:
7426:
7425:
7424:Computer Code
7418:
7414:
7413:Theodore Gray
7410:
7407:
7404:
7401:
7399:
7396:
7395:
7394:
7393:
7387:
7384:
7383:
7382:
7381:
7375:, ibiblio.org
7374:
7371:
7370:
7369:
7368:
7356:
7353:
7350:
7346:
7343:
7339:
7336:
7332:
7329:
7325:
7321:
7318:
7314:
7310:
7307:
7303:
7300:
7296:
7293:
7289:
7286:
7282:
7279:
7275:
7274:
7268:
7265:
7263:
7259:
7254:
7252:
7248:
7247:Bézier curves
7244:
7227:
7221:
7213:
7209:
7203:
7199:
7193:
7190:
7187:
7182:
7179:
7176:
7172:
7156:
7151:
7147:
7140:
7135:
7118:
7110:
7107:
7104:
7100:
7096:
7093:
7090:
7084:
7076:
7072:
7059:
7043:
7040:
7037:
7033:
7029:
7026:
7023:
7018:
7014:
7005:
6986:
6983:
6980:
6976:
6972:
6967:
6963:
6956:
6953:
6941:
6940:
6939:
6931:
6927:
6918:
6916:
6912:
6908:
6903:
6898:
6892:
6887:
6877:
6873:
6870:
6858:
6857:
6855:
6850:
6846:
6843:
6838:
6833:
6832:
6830:
6825:
6821:
6817:
6814:
6810:
6806:
6805:
6803:
6802:
6801:
6798:
6796:
6776:
6771:
6767:
6761:
6758:
6755:
6750:
6747:
6744:
6740:
6736:
6733:
6730:
6727:
6720:
6719:
6718:
6700:
6697:
6694:
6691:
6688:
6685:
6682:
6679:
6676:
6672:
6669:
6666:
6663:
6660:
6658:
6651:
6647:
6639:
6636:
6631:
6628:
6625:
6621:
6617:
6610:
6607:
6604:
6600:
6593:
6587:
6584:
6581:
6577:
6573:
6570:
6567:
6562:
6559:
6556:
6552:
6544:
6541:
6538:
6531:
6527:
6520:
6514:
6510:
6506:
6503:
6500:
6495:
6491:
6483:
6478:
6474:
6470:
6468:
6463:
6451:
6449:
6445:
6405:
6401:
6394:
6387:
6371:
6368:
6365:
6361:
6357:
6352:
6348:
6344:
6339:
6335:
6327:
6303:
6299:
6291:
6287:
6283:
6279:
6274:
6270:
6262:
6261:
6260:
6257:
6241:
6238:
6233:
6229:
6221:
6215:
6209:
6205:
6199:
6196:
6193:
6189:
6182:
6175:
6171:
6161:
6158:
6155:
6151:
6144:
6137:
6133:
6125:
6122:
6119:
6115:
6111:
6108:
6101:
6095:
6089:
6085:
6079:
6075:
6068:
6061:
6057:
6047:
6043:
6036:
6029:
6025:
6017:
6013:
6009:
6002:
5994:
5991:
5986:
5982:
5973:
5970:
5967:
5963:
5956:
5948:
5944:
5940:
5933:
5923:
5920:
5917:
5913:
5909:
5906:
5898:
5894:
5887:
5881:
5873:
5869:
5860:
5858:
5853:
5840:
5834:
5832:
5827:
5825:
5821:
5820:Autodesk Maya
5816:
5814:
5798:
5792:
5789:
5786:
5778:
5774:
5770:
5764:
5758:
5750:
5749:Adobe Systems
5746:
5741:
5730:
5723:
5718:
5712:
5695:
5689:
5686:
5683:
5675:
5672:
5669:
5659:
5653:
5647:
5634:
5630:
5623:
5616:
5609:
5601:
5577:
5574:
5571:
5567:
5562:
5559:
5555:
5551:
5545:
5539:
5515:
5497:
5492:
5488:
5484:
5479:
5476:
5473:
5469:
5461:
5456:
5452:
5448:
5443:
5440:
5437:
5433:
5425:
5420:
5416:
5412:
5407:
5404:
5401:
5397:
5389:
5384:
5380:
5376:
5371:
5368:
5365:
5361:
5353:
5348:
5344:
5340:
5335:
5332:
5329:
5325:
5310:
5306:
5301:
5282:
5264:
5256:
5252:
5248:
5241:
5237:
5233:
5228:
5225:
5222:
5218:
5211:
5209:
5202:
5198:
5190:
5185:
5176:
5172:
5168:
5165:
5160:
5157:
5154:
5150:
5141:
5137:
5130:
5123:
5119:
5112:
5108:
5104:
5099:
5096:
5093:
5089:
5082:
5080:
5073:
5069:
5061:
5056:
5053:
5050:
5046:
5040:
5036:
5032:
5027:
5023:
5019:
5017:
5010:
5006:
4991:
4987:
4983:
4978:
4963:
4960:
4955:
4951:
4947:
4942:
4938:
4934:
4931:
4928:
4923:
4919:
4910:
4892:
4885:
4881:
4874:
4871:
4868:
4864:
4858:
4855:
4852:
4848:
4844:
4839:
4835:
4828:
4826:
4819:
4815:
4807:
4800:
4796:
4790:
4786:
4780:
4778:
4771:
4767:
4759:
4754:
4751:
4748:
4744:
4738:
4735:
4732:
4728:
4724:
4716:
4713:
4710:
4706:
4702:
4697:
4694:
4691:
4687:
4680:
4677:
4675:
4668:
4664:
4649:
4645:
4640:
4625:
4622:
4617:
4613:
4609:
4604:
4599:
4594:
4591:
4588:
4583:
4579:
4570:
4565:
4556:
4542:
4534:
4531:
4528:
4524:
4520:
4515:
4511:
4500:
4497:
4494:
4490:
4486:
4481:
4473:
4469:
4465:
4460:
4457:
4454:
4450:
4439:
4435:
4431:
4426:
4421:
4417:
4406:
4402:
4389:
4372:
4368:
4364:
4359:
4356:
4353:
4349:
4345:
4340:
4336:
4325:
4321:
4308:
4293:
4276:
4272:
4268:
4263:
4259:
4249:
4245:
4238:
4229:
4228:
4227:
4218:
4213:
4201:
4197:
4175:
4168:
4164:
4160:
4156:
4151:as a 5-tuple
4145:
4128:
4125:
4117:
4113:
4105:
4101:
4098:
4095:
4091:
4087:
4085:
4075:
4071:
4063:
4059:
4055:
4047:
4044:
4041:
4038:
4032:
4029:
4023:
4020:
4017:
4014:
4008:
4000:
3996:
3988:
3984:
3981:
3978:
3974:
3970:
3968:
3958:
3954:
3946:
3942:
3938:
3930:
3927:
3924:
3921:
3915:
3912:
3906:
3903:
3900:
3897:
3891:
3883:
3879:
3871:
3867:
3864:
3861:
3857:
3853:
3851:
3841:
3837:
3829:
3825:
3821:
3813:
3808:
3804:
3800:
3798:
3788:
3784:
3775:
3772:
3769:
3765:
3757:
3752:
3748:
3744:
3742:
3732:
3728:
3719:
3715:
3707:
3704:
3701:
3698:
3692:
3689:
3683:
3680:
3677:
3674:
3668:
3660:
3656:
3647:
3644:
3641:
3637:
3633:
3628:
3624:
3620:
3618:
3608:
3604:
3595:
3591:
3578:
3575:
3571:
3567:
3560:
3556:
3532:
3517:
3513:
3509:
3504:
3500:
3493:
3490:
3487:
3484:
3481:
3473:
3469:
3465:
3460:
3456:
3449:
3441:
3437:
3433:
3428:
3424:
3410:
3407:
3398:
3385:
3380:
3370:
3366:
3362:
3359:
3351:
3347:
3343:
3338:
3328:
3324:
3320:
3317:
3309:
3305:
3301:
3293:
3289:
3285:
3282:
3274:
3270:
3266:
3261:
3257:
3253:
3247:
3239:
3235:
3220:
3217:
3204:
3201:
3195:
3188:
3185:
3181:
3174:
3167:
3164:
3155:
3152:
3147:
3142:
3140:
3136:
3135:linear spline
3132:
3131:step function
3127:
3119:
3112:
3095:
3092:
3089:
3086:
3083:
3077:
3072:
3068:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3040:
3032:
3028:
3024:
3022:
3014:
3008:
3001:
2998:
2995:
2992:
2989:
2983:
2978:
2974:
2970:
2967:
2964:
2961:
2958:
2955:
2949:
2941:
2937:
2933:
2931:
2923:
2917:
2910:
2907:
2904:
2901:
2898:
2892:
2887:
2883:
2879:
2876:
2873:
2870:
2867:
2861:
2853:
2849:
2845:
2843:
2835:
2829:
2817:
2814:
2797:
2794:
2791:
2788:
2785:
2779:
2774:
2770:
2766:
2763:
2760:
2757:
2754:
2748:
2740:
2736:
2732:
2730:
2722:
2716:
2709:
2706:
2703:
2700:
2697:
2691:
2688:
2685:
2682:
2676:
2668:
2664:
2660:
2658:
2650:
2644:
2637:
2634:
2631:
2628:
2625:
2619:
2614:
2610:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2582:
2574:
2570:
2566:
2564:
2556:
2550:
2538:
2534:
2530:
2523:
2516:
2501:
2491:
2472:
2469:
2466:
2460:
2457:
2453:
2440:
2434:
2431:
2425:
2419:
2409:
2403:
2397:
2389:
2384:
2380:
2376:
2356:
2338:
2334:
2330:
2325:
2322:
2319:
2315:
2311:
2308:
2305:
2300:
2297:
2294:
2290:
2286:
2281:
2278:
2275:
2271:
2267:
2264:
2261:
2256:
2252:
2248:
2243:
2239:
2235:
2232:
2229:
2224:
2220:
2216:
2211:
2207:
2203:
2200:
2197:
2192:
2188:
2184:
2179:
2175:
2163:
2161:
2153:
2148:
2141:
2137:
2117:
2109:
2106:
2103:
2099:
2095:
2090:
2086:
2075:
2072:
2069:
2065:
2061:
2056:
2052:
2048:
2045:
2041:
2037:
2031:
2025:
2014:
2010:
1996:
1990:
1986:
1979:
1973:
1969:
1962:
1958:
1947:
1943:
1930:
1917:
1894:
1890:
1881:
1876:
1870:
1864:
1857:
1851:
1841:
1835:
1833:
1827:
1823:
1795:
1791:
1786:
1773:
1769:
1762:
1758:
1752:
1735:
1727:
1723:
1709:
1705:
1696:
1692:
1688:
1686:
1676:
1672:
1658:
1654:
1645:
1642:
1639:
1635:
1626:
1619:
1611:
1607:
1595:
1587:
1583:
1579:
1577:
1567:
1563:
1551:
1543:
1540:
1537:
1533:
1525:
1517:
1513:
1501:
1493:
1489:
1485:
1483:
1473:
1469:
1457:
1449:
1446:
1443:
1439:
1426:
1423:
1416:
1411:
1390:
1386:
1351:
1347:
1342:
1333:
1306:
1302:
1297:
1293:
1290:
1281:
1277:
1273:
1268:
1263:
1247:
1245:
1241:
1237:
1231:
1221:
1217:
1213:. The vector
1212:
1199:
1193:
1176:
1171:
1167:
1163:
1160:
1157:
1152:
1149:
1146:
1142:
1135:
1129:
1121:
1118:
1115:
1111:
1107:
1105:
1097:
1091:
1084:
1082:
1074:
1069:
1065:
1061:
1058:
1055:
1050:
1046:
1039:
1033:
1025:
1021:
1017:
1015:
1007:
1001:
994:
989:
985:
981:
978:
975:
970:
966:
959:
953:
945:
941:
937:
935:
927:
921:
909:
880:
864:
861:
858:
854:
850:
845:
841:
834:
829:
825:
804:
787:
784:
779:
775:
771:
766:
763:
760:
756:
752:
749:
746:
741:
737:
733:
728:
724:
720:
717:
704:
700:
693:
685:
681:
677:
672:
669:
666:
662:
655:
647:
644:
641:
637:
633:
628:
625:
622:
618:
611:
608:
605:
597:
593:
589:
584:
580:
573:
565:
561:
557:
552:
548:
541:
535:
532:
529:
524:
516:
513:
510:
507:
504:
501:
498:
495:
492:
488:
480:
477:
474:
470:
466:
461:
457:
451:
438:
436:
412:
398:
395:
392:
386:
383:
375:
359:
342:
339:
336:
332:
321:
318:
310:
307:February 2009
300:
299:the talk page
296:
290:
288:
283:This article
281:
272:
271:
263:
261:
257:
252:
250:
246:
242:
238:
234:
230:
226:
222:
221:Pierre Bézier
218:
214:
209:
207:
202:
200:
196:
192:
188:
184:
180:
176:
172:
171:Robin Forrest
168:
163:
160:
159:step function
156:
155:sign function
151:
149:
145:
135:
126:
124:
120:
115:
113:
112:curve fitting
109:
105:
101:
97:
93:
90:subfields of
89:
84:
82:
78:
74:
70:
66:
65:interpolating
62:
58:
54:
50:
46:
38:
34:
28:
22:
7465:
7460:
7433:
7423:
7422:
7391:
7390:
7379:
7378:
7366:
7365:
7348:
7341:
7327:
7316:
7305:
7298:
7291:
7284:
7277:
7266:
7255:
7164:
7149:
7145:
7138:
6929:
6922:
6919:
6901:
6890:
6883:
6875:
6868:
6848:
6841:
6836:
6799:
6794:
6791:
6452:
6444:vector space
6437:
6258:
5861:
5856:
5851:
5844:
5838:
5828:
5817:
5742:
5728:
5721:
5719:is a cubic (
5710:
5635:
5628:
5621:
5614:
5607:
5599:
5527:
5308:
5304:
5287:and call it
4989:
4985:
4981:
4647:
4643:
4563:
4404:
4400:
4323:
4319:
4247:
4243:
4236:
4216:
4211:
4199:
4198:
4189:is equal to
4170:
4166:
4162:
4158:
4154:
4146:
3579:
3576:
3569:
3565:
3558:
3551:
3399:
3226:
3218:
3156:
3150:
3145:
3143:
3134:
3128:
3117:
3113:
2818:
2815:
2539:
2532:
2528:
2521:
2514:
2507:
2490:spline curve
2489:
2385:
2378:
2374:
2365:is repeated
2357:
2164:
2162:knot vector
2159:
2151:
2143:
2139:
2132:
2012:
2008:
1994:
1988:
1984:
1977:
1971:
1967:
1960:
1953:
1945:
1941:
1931:
1880:spline space
1879:
1874:
1868:
1862:
1855:
1849:
1839:
1836:
1831:
1830:is called a
1825:
1821:
1771:
1767:
1760:
1756:
1753:
1427:
1418:
1414:
1409:
1388:
1384:
1282:
1275:
1271:
1266:
1261:
1248:
1243:
1239:
1235:
1234:is called a
1226:
1219:
1215:
1210:
1197:
1194:
910:
805:
439:
374:real numbers
328:
313:
304:
293:Please help
284:
253:
213:de Casteljau
210:
206:East Anglian
203:
179:World War II
164:
152:
141:
132:
129:Introduction
116:
99:
85:
48:
42:
32:
6867:(giving us
5717:flat spline
2018:. That is,
1843:, a degree
1412:of at most
1334:(at least)
1244:non-uniform
1236:knot vector
1209:are called
372:the set of
173:describes "
150:in Russia.
98:, the term
61:polynomials
45:mathematics
21:Flat spline
7478:Categories
7328:NA Digest,
7322:Epperson,
7317:SIAM News,
7271:References
5813:PostScript
5518:output_set
5289:output_set
4559:c, l, μ, z
4226:5-tuples.
4180:a, b, c, d
2537:for which
2372:times for
1332:smoothness
1195:The given
338:polynomial
289:to readers
266:Definition
233:Garabedian
195:M.A. Sabin
167:Schoenberg
104:parametric
67:problems,
7262:B-splines
7191:−
7173:∑
7108:−
7094:…
7041:−
7027:…
6957:∈
6813:B-splines
6797:splines.
6759:−
6741:∑
6698:−
6689:…
6661:≤
6629:−
6608:−
6594:⏟
6585:−
6571:⋯
6560:−
6542:⋯
6521:⏟
6504:⋯
6448:dimension
6369:−
6358:−
6239:−
6197:−
6183:−
6159:−
6123:−
6112:−
6069:−
5992:−
5971:−
5921:−
5910:−
5826:(NURBS).
5771:∈
5713:− 1
5673:−
5660:∈
5560:−
5552:∈
5234:−
5131:−
5105:−
5037:μ
5033:−
4992:– 2, …, 0
4872:−
4856:−
4845:−
4836:α
4768:μ
4752:−
4745:μ
4736:−
4725:−
4714:−
4703:−
4600:μ
4532:−
4521:−
4498:−
4482:−
4466:−
4418:α
4365:−
4215:| =
4099:−
4042:−
4030:…
3982:−
3925:−
3913:…
3865:−
3773:−
3702:−
3690:…
3645:−
3363:−
3321:−
3286:−
3093:≤
3087:≤
3062:−
3050:−
2993:≤
2962:−
2902:≤
2877:−
2871:−
2795:≤
2789:≤
2761:−
2701:≤
2629:≤
2607:−
2592:−
2461:∈
2323:−
2309:⋯
2298:−
2279:−
2265:⋯
2233:⋯
2201:⋯
2062:−
2049:−
2038:∈
1754:A vector
1643:−
1627:⋮
1541:−
1447:−
1294:∈
1164:≤
1158:≤
1150:−
1119:−
1085:⋮
1056:≤
976:≤
873:→
772:≤
764:−
753:≤
750:⋯
747:≤
734:≤
694:∪
670:−
656:∪
645:−
626:−
612:∪
609:⋯
606:∪
574:∪
514:−
505:…
433:ordered,
405:→
335:piecewise
249:B-splines
123:draftsmen
57:piecewise
7447:, SINTEF
6429:th knot.
6325:th knot.
6292:″
5307:= 0, …,
5295:splines
4646:= 1, …,
4403:= 1, …,
4398:and for
4394:of size
4322:= 0, …,
4317:and for
4313:of size
4246:= 0, …,
4241:and for
4234:of size
4106:″
4064:″
3989:″
3947:″
3872:′
3830:′
3568:= 0, …,
3549:splines
3189:″
3168:″
2504:Examples
2492:if both
2377:= 1, …,
2160:extended
2158:and its
1861:≤
1824:= 1, …,
1265:≤
435:disjoint
425:We want
341:function
229:Birkhoff
55:defined
53:function
7419:, 2007.
7311:Davis,
7294:, 1990.
7144:, ...,
6442:form a
5516:Output
5285:Splines
4207:, with
3139:polygon
1270:(or of
1240:uniform
1202:points
893:On the
285:may be
237:de Boor
225:Renault
217:Citroën
183:splines
175:lofting
138:History
86:In the
7441:, NTCC
7367:Theory
6913:, and
6818:using
6807:using
6259:where
4209:|
4178:where
4036:
4027:
3919:
3910:
3696:
3687:
2358:where
2130:where
1262:degree
235:, and
227:, and
191:Boeing
119:spline
100:spline
77:degree
49:spline
7301:1987.
7287:1967.
7249:and
6809:basis
5747:from
5524:Notes
4988:– 1,
2488:is a
1853:for
1766:, …,
1272:order
1225:, …,
1211:knots
108:curve
63:. In
51:is a
6863:and
6618:<
6545:<
6539:<
6484:<
5736:and
5610:+ 1)
5575:>
5302:For
4979:For
4911:Set
4641:For
4571:Set
4409:set
4328:set
4298:and
4251:set
3563:for
2999:<
2908:<
2707:<
2635:<
2510:, ,
2496:and
2006:and
1982:and
1939:and
1819:for
1394:and
1062:<
982:<
94:and
47:, a
7253:).
6893:= 3
5849:th
5724:= 3
5631:+ 3
5624:+ 2
5617:+ 1
5602:+ 1
5311:– 1
4650:– 1
4566:+ 1
4407:– 1
4326:– 1
4239:+ 1
4219:+ 1
3572:– 1
2524:= 2
2517:= 1
2381:– 1
1828:– 1
1812:at
1759:= (
1368:at
1283:If
1280:).
1278:+ 1
1218:= (
1200:+ 1
239:at
223:at
215:at
197:at
189:at
157:or
146:at
59:by
43:In
7480::
7432:,
7415:,
7326:,
7315:,
7264:.
7152:–2
7064::
6909:,
6701:2.
5740:.
5626:,
5619:,
4984:=
4964:0.
4196:.
4169:,
4165:,
4161:,
4157:,
4129:0.
3574:.
3205:0.
3141:.
3126:.
2386:A
2383:.
2142:–
2138:=
2015:+1
1991:+1
1974:−1
1948:+1
1774:–1
1425:)
1417:–
1391:–1
1246:.
908:,
817:.
376:,
251:.
231:,
219:,
201:.
106:)
7228:.
7225:)
7222:t
7219:(
7214:j
7210:P
7204:j
7200:c
7194:2
7188:k
7183:0
7180:=
7177:j
7161::
7159:t
7150:k
7146:c
7142:0
7139:c
7122:)
7119:t
7116:(
7111:2
7105:k
7101:P
7097:,
7091:,
7088:)
7085:t
7082:(
7077:0
7073:P
7062:t
7044:2
7038:k
7034:P
7030:,
7024:,
7019:0
7015:P
6992:]
6987:1
6984:+
6981:i
6977:t
6973:,
6968:i
6964:t
6960:[
6954:t
6944:t
6936:n
6932:)
6930:t
6928:(
6925:i
6923:P
6902:C
6891:n
6878:)
6871:)
6865:b
6861:a
6851:)
6844:)
6837:C
6826:)
6815:)
6795:C
6777:.
6772:i
6768:j
6762:2
6756:k
6751:1
6748:=
6745:i
6737:+
6734:n
6731:=
6728:d
6695:k
6692:,
6686:,
6683:1
6680:=
6677:i
6673:,
6670:1
6667:+
6664:n
6652:i
6648:j
6640:b
6637:=
6632:1
6626:k
6622:t
6611:2
6605:k
6601:j
6588:2
6582:k
6578:t
6574:=
6568:=
6563:2
6557:k
6553:t
6532:1
6528:j
6515:1
6511:t
6507:=
6501:=
6496:1
6492:t
6479:0
6475:t
6471:=
6464:a
6440:n
6427:i
6413:)
6406:i
6402:t
6398:(
6395:f
6372:1
6366:i
6362:t
6353:i
6349:t
6345:=
6340:i
6336:h
6323:i
6309:)
6304:i
6300:t
6296:(
6288:i
6284:S
6280:=
6275:i
6271:z
6245:)
6242:x
6234:i
6230:t
6226:(
6222:]
6216:6
6210:i
6206:h
6200:1
6194:i
6190:z
6176:i
6172:h
6167:)
6162:1
6156:i
6152:t
6148:(
6145:f
6138:[
6134:+
6131:)
6126:1
6120:i
6116:t
6109:x
6106:(
6102:]
6096:6
6090:i
6086:h
6080:i
6076:z
6062:i
6058:h
6053:)
6048:i
6044:t
6040:(
6037:f
6030:[
6026:+
6018:i
6014:h
6010:6
6003:3
5999:)
5995:x
5987:i
5983:t
5979:(
5974:1
5968:i
5964:z
5957:+
5949:i
5945:h
5941:6
5934:3
5930:)
5924:1
5918:i
5914:t
5907:x
5904:(
5899:i
5895:z
5888:=
5885:)
5882:x
5879:(
5874:i
5870:S
5857:x
5852:C
5847:i
5839:C
5799:.
5796:]
5793:b
5790:,
5787:a
5784:[
5779:1
5775:C
5768:)
5765:t
5762:(
5759:S
5738:b
5734:a
5729:C
5722:n
5711:n
5696:,
5693:]
5690:b
5687:,
5684:a
5681:[
5676:1
5670:n
5665:C
5657:)
5654:t
5651:(
5648:S
5638:n
5629:n
5622:n
5615:n
5608:n
5606:(
5600:n
5594:i
5592:t
5578:0
5572:m
5568:,
5563:m
5556:C
5549:)
5546:t
5543:(
5540:S
5530:n
5498:.
5493:i
5489:x
5485:=
5480:x
5477:,
5474:i
5470:S
5462:,
5457:i
5453:d
5449:=
5444:d
5441:,
5438:i
5434:S
5426:,
5421:i
5417:c
5413:=
5408:c
5405:,
5402:i
5398:S
5390:,
5385:i
5381:b
5377:=
5372:b
5369:,
5366:i
5362:S
5354:,
5349:i
5345:a
5341:=
5336:a
5333:,
5330:i
5326:S
5309:n
5305:i
5299:.
5297:S
5293:n
5265:.
5257:j
5253:h
5249:3
5242:j
5238:c
5229:1
5226:+
5223:j
5219:c
5212:=
5203:j
5199:d
5191:,
5186:3
5182:)
5177:j
5173:c
5169:2
5166:+
5161:1
5158:+
5155:j
5151:c
5147:(
5142:j
5138:h
5124:j
5120:h
5113:j
5109:a
5100:1
5097:+
5094:j
5090:a
5083:=
5074:j
5070:b
5062:,
5057:1
5054:+
5051:j
5047:c
5041:j
5028:j
5024:z
5020:=
5011:j
5007:c
4990:n
4986:n
4982:j
4961:=
4956:n
4952:c
4948:=
4943:n
4939:z
4935:;
4932:1
4929:=
4924:n
4920:l
4893:.
4886:i
4882:l
4875:1
4869:i
4865:z
4859:1
4853:i
4849:h
4840:i
4829:=
4820:i
4816:z
4808:,
4801:i
4797:l
4791:i
4787:h
4781:=
4772:i
4760:,
4755:1
4749:i
4739:1
4733:i
4729:h
4722:)
4717:1
4711:i
4707:x
4698:1
4695:+
4692:i
4688:x
4684:(
4681:2
4678:=
4669:i
4665:l
4648:n
4644:i
4626:0
4623:=
4618:0
4614:z
4610:=
4605:0
4595:,
4592:1
4589:=
4584:0
4580:l
4568:.
4564:n
4543:.
4540:)
4535:1
4529:i
4525:a
4516:i
4512:a
4508:(
4501:1
4495:i
4491:h
4487:3
4479:)
4474:i
4470:a
4461:1
4458:+
4455:i
4451:a
4447:(
4440:i
4436:h
4432:3
4427:=
4422:i
4405:n
4401:i
4396:n
4392:α
4373:i
4369:x
4360:1
4357:+
4354:i
4350:x
4346:=
4341:i
4337:h
4324:n
4320:i
4315:n
4311:h
4306:.
4304:n
4300:d
4296:b
4277:i
4273:y
4269:=
4264:i
4260:a
4248:n
4244:i
4237:n
4232:a
4224:n
4217:n
4212:C
4205:C
4193:j
4191:x
4186:t
4184:x
4176:)
4173:t
4171:x
4167:d
4163:c
4159:b
4155:a
4153:(
4149:S
4126:=
4123:)
4118:n
4114:x
4110:(
4102:1
4096:n
4092:S
4088:=
4081:)
4076:0
4072:x
4068:(
4060:0
4056:S
4048:,
4045:1
4039:n
4033:,
4024:,
4021:1
4018:=
4015:i
4009:,
4006:)
4001:i
3997:x
3993:(
3985:1
3979:i
3975:S
3971:=
3964:)
3959:i
3955:x
3951:(
3943:i
3939:S
3931:,
3928:1
3922:n
3916:,
3907:,
3904:1
3901:=
3898:i
3892:,
3889:)
3884:i
3880:x
3876:(
3868:1
3862:i
3858:S
3854:=
3847:)
3842:i
3838:x
3834:(
3826:i
3822:S
3814:,
3809:n
3805:y
3801:=
3794:)
3789:n
3785:x
3781:(
3776:1
3770:n
3766:S
3758:,
3753:0
3749:y
3745:=
3738:)
3733:0
3729:x
3725:(
3720:0
3716:S
3708:,
3705:1
3699:n
3693:,
3684:,
3681:1
3678:=
3675:i
3669:,
3666:)
3661:i
3657:x
3653:(
3648:1
3642:i
3638:S
3634:=
3629:i
3625:y
3621:=
3614:)
3609:i
3605:x
3601:(
3596:i
3592:S
3570:n
3566:i
3561:)
3559:x
3557:(
3554:i
3552:S
3547:n
3533:,
3528:]
3523:)
3518:n
3514:y
3510:,
3505:n
3501:x
3497:(
3494:,
3491:.
3488:.
3485:.
3482:,
3479:)
3474:1
3470:y
3466:,
3461:1
3457:x
3453:(
3450:,
3447:)
3442:0
3438:y
3434:,
3429:0
3425:x
3421:(
3416:[
3411:=
3408:C
3386:.
3381:3
3377:)
3371:j
3367:x
3360:x
3357:(
3352:j
3348:d
3344:+
3339:2
3335:)
3329:j
3325:x
3318:x
3315:(
3310:j
3306:c
3302:+
3299:)
3294:j
3290:x
3283:x
3280:(
3275:j
3271:b
3267:+
3262:j
3258:a
3254:=
3251:)
3248:x
3245:(
3240:j
3236:S
3202:=
3199:)
3196:b
3193:(
3186:S
3182:=
3178:)
3175:a
3172:(
3165:S
3151:C
3118:t
3116:2
3096:3
3090:t
3084:2
3078:,
3073:2
3069:t
3065:2
3059:t
3056:+
3053:1
3047:=
3044:)
3041:t
3038:(
3033:2
3029:P
3025:=
3018:)
3015:t
3012:(
3009:S
3002:2
2996:t
2990:1
2984:,
2979:2
2975:t
2971:+
2968:t
2965:6
2959:1
2956:=
2953:)
2950:t
2947:(
2942:1
2938:P
2934:=
2927:)
2924:t
2921:(
2918:S
2911:1
2905:t
2899:0
2893:,
2888:2
2884:t
2880:2
2874:2
2868:=
2865:)
2862:t
2859:(
2854:0
2850:P
2846:=
2839:)
2836:t
2833:(
2830:S
2798:3
2792:t
2786:2
2780:,
2775:2
2771:t
2767:+
2764:t
2758:2
2755:=
2752:)
2749:t
2746:(
2741:2
2737:P
2733:=
2726:)
2723:t
2720:(
2717:S
2710:2
2704:t
2698:1
2692:,
2689:t
2686:2
2683:=
2680:)
2677:t
2674:(
2669:1
2665:P
2661:=
2654:)
2651:t
2648:(
2645:S
2638:1
2632:t
2626:0
2620:,
2615:2
2611:t
2604:t
2601:4
2598:+
2595:1
2589:=
2586:)
2583:t
2580:(
2575:0
2571:P
2567:=
2560:)
2557:t
2554:(
2551:S
2535:)
2533:t
2531:(
2529:S
2522:t
2515:t
2498:Y
2494:X
2476:]
2473:b
2470:,
2467:a
2464:[
2458:t
2454:,
2449:)
2444:)
2441:t
2438:(
2435:Y
2432:,
2429:)
2426:t
2423:(
2420:X
2415:(
2410:=
2407:)
2404:t
2401:(
2398:G
2379:k
2375:i
2369:i
2367:j
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2339:k
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2331:,
2326:1
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2316:t
2312:,
2306:,
2301:1
2295:k
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2287:,
2282:2
2276:k
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2268:,
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2257:3
2253:t
2249:,
2244:2
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2221:t
2217:,
2212:1
2208:t
2204:,
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2185:,
2180:0
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2172:(
2156:n
2146:i
2144:r
2140:n
2135:i
2133:j
2118:,
2115:]
2110:1
2107:+
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2100:t
2096:=
2091:i
2087:t
2083:[
2076:1
2073:+
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2066:j
2057:i
2053:j
2046:n
2042:C
2035:)
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2029:(
2026:S
2013:i
2009:t
2003:i
2001:t
1997:)
1995:t
1993:(
1989:i
1985:P
1980:)
1978:t
1976:(
1972:i
1968:P
1963:)
1961:t
1959:(
1956:i
1954:P
1946:i
1942:t
1936:i
1934:t
1918:.
1915:)
1911:t
1907:(
1901:r
1895:n
1891:S
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1869:t
1863:n
1856:t
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1733:)
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1720:(
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1697:i
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1689:=
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1612:i
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1580:=
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1505:)
1502:0
1499:(
1494:i
1490:P
1486:=
1479:)
1474:i
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1466:(
1461:)
1458:0
1455:(
1450:1
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1421:i
1419:r
1415:n
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1389:i
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1276:n
1267:n
1258:n
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1232:)
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1227:t
1223:0
1220:t
1216:t
1206:i
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1198:k
1177:.
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1153:1
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1143:t
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1130:t
1127:(
1122:1
1116:k
1112:P
1108:=
1101:)
1098:t
1095:(
1092:S
1075:,
1070:2
1066:t
1059:t
1051:1
1047:t
1040:,
1037:)
1034:t
1031:(
1026:1
1022:P
1018:=
1011:)
1008:t
1005:(
1002:S
995:,
990:1
986:t
979:t
971:0
967:t
960:,
957:)
954:t
951:(
946:0
942:P
938:=
931:)
928:t
925:(
922:S
905:i
903:P
899:S
895:i
881:.
877:R
870:]
865:1
862:+
859:i
855:t
851:,
846:i
842:t
838:[
835::
830:i
826:P
814:i
812:P
808:k
788:b
785:=
780:k
776:t
767:1
761:k
757:t
742:1
738:t
729:0
725:t
721:=
718:a
710:]
705:k
701:t
697:[
691:)
686:k
682:t
678:,
673:1
667:k
663:t
659:[
653:)
648:1
642:k
638:t
634:,
629:2
623:k
619:t
615:[
603:)
598:2
594:t
590:,
585:1
581:t
577:[
571:)
566:1
562:t
558:,
553:0
549:t
545:[
542:=
539:]
536:b
533:,
530:a
527:[
517:1
511:k
508:,
502:,
499:0
496:=
493:i
489:,
486:]
481:1
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475:i
471:t
467:,
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458:t
454:[
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413:.
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390:[
387::
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360:,
356:R
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320:)
314:(
309:)
305:(
301:.
291:.
33:C
23:.
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