196:") 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
145:
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.
4154:
38:
3593:
287:
6726:
1202:
3121:
1761:
5290:
4149:{\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}}}
2823:
4918:
6466:
924:
172:
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.
144:
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
2832:
1441:
5008:
2553:
813:
6916:); 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
6266:
4666:
5523:
7251:
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
2365:
180:, 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),
4564:
3407:
453:
3598:
6721:{\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}}}
5875:
1197:{\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}}}
3132:
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
6803:
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
1446:
929:
5327:
5013:
2837:
2558:
458:
3116:{\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}}}
1756:{\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}}}
5285:{\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}}}
2497:
6471:
2178:
6931:
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
6457:. 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
2818:{\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}}}
3226:
3165:, 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.
5332:
4671:
1939:
121:. 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
254:(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
7249:
4913:{\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}}}
6330:
4648:
2139:
7143:
6798:
5717:
5599:
4423:
4986:
902:
7067:
5601:
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
4397:
265:
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
6395:
434:
5820:
1332:
4301:
381:
6434:
3554:
3241:
1821:
1377:
7013:
1889:. Equipped with the operation of adding two functions (pointwise addition) and taking real multiples of functions, this set becomes a real vector space. This
197:
2404:
205:
6899:
is a spline that is expressed using
Hermite polynomials to represent each of the individual polynomial pieces. These are most often used with
808:{\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}}}
7267:
For a representation that defines a spline as a linear combination of basis splines, however, something more sophisticated is needed. The
5644:, etc. have the same meaning. It is commonly assumed that any knot vector defining any type of spline has been culled in this fashion.
6261:{\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)}
2161:. 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
7178:
2032:
5654:
5546:
5829:
Many computer-aided design systems that are designed for high-end graphics and animation use extended knot vectors, for example
222:
The use of splines for modeling automobile bodies seems to have several independent beginnings. Credit is claimed on behalf of
831:
5518:{\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}}}
2523:. Suppose the polynomial pieces are to be of degree 2, and the pieces on and must join in value and first derivative (at
1896:
390:
3170:
7427:
6895:
Often a special name was chosen for a type of spline satisfying two or more of the main items above. For example, the
1412:
share common derivative values from the derivative of order 0 (the function value) up through the derivative of order
6811:
The literature of splines is replete with names for special types of splines. These names have been associated with:
327:
7261:
7147:
Look up the coefficients of the linear combination of those basis polynomials that give the spline on that interval
6834:
5765:
164:
Before computers were used, numerical calculations were done by hand. Although piecewise-defined functions like the
7345:
2360:{\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})}
4585:
7078:
6734:
3414:
7494:
4925:
258:
resulted in a number of papers being published in the early 1960s, including some of the fundamental work on
5743:), natural spline, which is a spline of this classical type with additional conditions imposed at endpoints
7020:
5834:
4342:
209:
17:
6276:
7253:
6341:
273:
is detailed nicely in (Birkhoff, 1990) and (Young, 1997). Davis (1997) summarizes some of this material.
7315:
Birkhoff and de Boor, Piecewise polynomial interpolation and approximation, in: H. L. Garabedian (ed.),
270:
255:
251:
7351:
Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions,
309:
6461:
of the space containing all splines of a certain type can be counted from the extended knot vector:
1297:
4265:
1962:, are taken to approach one another and become coincident has an easy answer. The polynomial piece
361:
305:
266:
83:
7368:
Larry, Schumaker, "Spline
Functions: Computational Methods", SIAM, ISBN 978-1-611973-89-1, (2015).
7301:
Birkhoff, Fluid dynamics, reactor computations, and surface representation, in: Steve Nash (ed.),
4559:{\displaystyle \alpha _{i}={\frac {3}{h_{i}}}(a_{i+1}-a_{i})-{\frac {3}{h_{i-1}}}(a_{i}-a_{i-1}).}
6925:
6401:
1943:
In the mathematical study of polynomial splines the question of what happens when two knots, say
1249:
for the spline. If the knots are equidistantly distributed in the interval we say the spline is
7440:
3148:. A closed linear spline (i.e, the first knot and the last are the same) in the plane is just a
7499:
6819:
6449:
For a given interval and a given extended knot vector on that interval, the splines of degree
1792:
1348:
351:
177:
87:
63:
5754:
Another type of spline that is much used in graphics, for example in drawing programs such as
2511:
are spline functions of the same degree with the same extended knot vectors on that interval.
176:
It is commonly accepted that the first mathematical reference to splines is the 1946 paper by
7419:
102:
91:
50:. Triple knots at both ends of the interval ensure that the curve interpolates the end points
47:
7168:
Add up that linear combination of basis polynomial values to get the value of the spline at
3230:
Thus, the graph of the spline is a straight line outside of the interval, but still smooth.
6960:
6907:
6830:
5841:
216:
79:
8:
5623:
th can be removed without changing the character of the spline, since all multiplicities
297:
154:
153:
According to Gerald Farin, B-splines were explored as early as the nineteenth century by
114:
7334:
7413:
6921:
2165:
located at that point, and a spline type can be completely characterized by its degree
41:
Single knots at 1/3 and 2/3 establish a spline of three cubic polynomials meeting with
7268:
5833:. Computer-aided design systems often use an extended concept of a spline known as a
5755:
223:
133:
106:
6833:
as employed by Pierre Bézier to represent each polynomial piece (giving us the name
5719:
which means that every two adjacent polynomial pieces meet in their value and first
3402:{\displaystyle S_{j}(x)=a_{j}+b_{j}(x-x_{j})+c_{j}(x-x_{j})^{2}+d_{j}(x-x_{j})^{3}.}
2398:
239:
158:
98:
7323:
231:
6917:
6458:
5543:
multiple knots in a knot vector have, since this would lead to continuities like
1342:
247:
7257:
6910:. In this degree they may additionally be chosen to be only tangent-continuous (
7383:
7310:
An
Introduction to Splines for Use in Computer Graphics and Geometric Modeling,
6896:
5726:
derivatives at each knot. The mathematical spline that most closely models the
440:
to be piecewise defined. To accomplish this, let the interval be covered by
7488:
7423:
5830:
5759:
3141:
445:
181:
169:
165:
122:
75:
192:
to construct templates for airplanes by passing thin wooden strips (called "
6454:
384:
189:
7461:
5847:
5727:
193:
129:
55:
37:
31:
6728:
The dimension is equal to the sum of the degree plus the multiplicities
7408:
5823:
5615:
times in an extended knot vector, all instances of it in excess of the
348:
341:
243:
71:
227:
345:
67:
7449:
7272:
6842:
The choices made in forming the extended knot vector, for example:
6823:
5840:
If sampled data from a function or a physical object is available,
259:
3125:
would be a member of that type. (Note: while the polynomial piece
5844:
is an approach to creating a spline that approximates that data.
5826:
as well as in the definition of some computer typographic fonts.
3149:
1419:(in other words, the two adjacent polynomial pieces connect with
235:
185:
3233:
3144:. The next most simple spline has degree 1. It is also called a
201:
7396:
7365:
Chapra, Canale, "Numerical
Methods for Engineers" 5th edition.
6885:
requiring that given data values be on the spline (giving us
118:
7455:
5762:, has pieces that are cubic but has continuity only at most
2492:{\displaystyle G(t)={\bigl (}X(t),Y(t){\bigr )},\quad t\in }
188:", a technique used in the British aircraft industry during
6867:
Any special conditions imposed on the spline, for example:
6815:
The choices made for representing the spline, for example:
7344:
Stoer & Bulirsch, Introduction to
Numerical Analysis.
5870:
with the natural condition can be found using the formula
6851:
continuity and spacing these knots evenly on (giving us
4193:
correspond to coefficients in the form shown earlier and
7397:
XLL Excel Addin
Function Implementation of cubic spline
86:
because it yields similar results, even when using low
7017:
Look up the polynomial basis chosen for that interval
6858:
using knots with no restriction on spacing (giving us
3140:
The simplest spline has degree 0. It is also called a
821:"pieces" of , we want to define a polynomial, call it
27:
Mathematical function defined piecewise by polynomials
7181:
7081:
7023:
6963:
6737:
6469:
6404:
6344:
6279:
5878:
5768:
5657:
5549:
5330:
5011:
4928:
4669:
4588:
4426:
4345:
4268:
3596:
3417:
3244:
3173:
2835:
2556:
2530:) while the pieces on and join simply in value (at
2407:
2181:
2035:
1899:
1795:
1444:
1351:
1300:
927:
834:
456:
393:
364:
7319:
pp. 164–190. Elsevier, New York and
Amsterdam, 1965.
6822:
functions for the entire spline (giving us the name
1870:, one can consider the set of all splines of degree
6949:polynomials, this is conceptually straightforward:
2519:Suppose the interval is and the subintervals are
1934:{\displaystyle S_{n}^{\mathbf {r} }(\mathbf {t} ).}
7243:
7137:
7061:
7007:
6792:
6720:
6428:
6389:
6324:
6260:
5814:
5711:
5593:
5517:
5284:
4980:
4912:
4642:
4558:
4391:
4295:
4148:
3548:
3401:
3220:
3115:
2817:
2491:
2359:
2133:
1933:
1815:
1755:
1371:
1326:
1196:
896:
807:
428:
375:
358:, takes values from an interval and maps them to
113:more frequently refers to a piecewise polynomial (
7358:Young, Garrett Birkhoff and applied mathematics,
7486:
7477:Curves and surfaces for CAGD: a practical guide
7441:Notes, PPT, Mathcad, Maple, Mathematica, Matlab
7416:Interactive simulation of various cubic splines
7291:J. ACM, vol. 11, no. 2, pp. 221-228, Apr. 1964.
7244:{\displaystyle \sum _{j=0}^{k-2}c_{j}P_{j}(t).}
6332:are the values of the second derivative at the
3159:. A cubic spline has degree 3 with continuity
2010:join with the sum of the smoothness losses for
4211:Algorithm for computing Natural Cubic Splines:
7296:The Theory of Splines and Their Applications,
3538:
3426:
3234:Algorithm for computing natural cubic splines
2459:
2425:
6444:
4643:{\displaystyle l_{0}=1,{\mu }_{0}=z_{0}=0\,}
2134:{\displaystyle S(t)\in C^{n-j_{i}-j_{i+1}},}
308:. There might be a discussion about this on
7138:{\displaystyle P_{0}(t),\ldots ,P_{k-2}(t)}
7071:Find the value of each basis polynomial at
6793:{\displaystyle d=n+\sum _{i=1}^{k-2}j_{i}.}
5712:{\displaystyle S(t)\in \mathrm {C} ^{n-1},}
5594:{\displaystyle S(t)\in C^{-m},\quad m>0}
5651:used in numerical analysis has continuity
7409:Online Cubic Spline Interpolation Utility
5539:It might be asked what meaning more than
4977:
4639:
4388:
4292:
4233:Output: set splines which is composed of
3191:
2827:would be a member of that type, and also
887:
419:
366:
328:Learn how and when to remove this message
4981:{\displaystyle l_{n}=1;z_{n}=c_{n}=0.\,}
128:The term spline comes from the flexible
36:
7317:Proc. General Motors Symposium of 1964,
897:{\displaystyle P_{i}:\to \mathbb {R} .}
340:We begin by limiting our discussion to
14:
7487:
7384:An Interactive Introduction to Splines
7271:is an efficient method for evaluating
6945:in terms of some basis for the degree
6436:are the values of the function at the
5866:interpolating cubic spline at a point
5737:), twice continuously differentiable (
2537:). This would define a type of spline
7362:vol. 44, no. 11, pp. 1446–1449, 1997.
7062:{\displaystyle P_{0},\ldots ,P_{k-2}}
6957:, find the interval in which it lies
6870:enforcing zero second derivatives at
4392:{\displaystyle h_{i}=x_{i+1}-x_{i}\,}
269:, by Feodor Theilheimer. The work at
7456:Sisl: Opensource C-library for NURBS
7445:Holistic Numerical Methods Institute
7355:vol. 4, pp. 45–99 and 112–141, 1946.
6325:{\displaystyle z_{i}=S_{i}''(t_{i})}
5647:The classical spline type of degree
1789:such that the spline has smoothness
280:
215:The word "spline" was originally an
7303:A History of Scientific Computation
7289:Multi-variable curve interpolation,
6390:{\displaystyle h_{i}=t_{i}-t_{i-1}}
1341:, then the spline is said to be of
1271:, then the spline is said to be of
429:{\displaystyle S:\to \mathbb {R} .}
24:
7428:The Wolfram Demonstrations Project
6953:For a given value of the argument
5675:
3221:{\displaystyle S''(a)\,=S''(b)=0.}
25:
7511:
7372:
6928:, depending on the method used).
5822:This spline type is also used in
132:devices used by shipbuilders and
1921:
1911:
285:
6686:
5856:The general expression for the
5581:
5528:
5299:
5295:
4158:Let us define one cubic spline
2467:
502:
344:. In this case, a spline is a
139:
7469:
7324:B-splines and Geometric design
7235:
7229:
7132:
7126:
7098:
7092:
7002:
6970:
6423:
6408:
6319:
6306:
6255:
6236:
6177:
6158:
6141:
6116:
6063:
6050:
6009:
5989:
5940:
5914:
5895:
5889:
5815:{\displaystyle S(t)\in C^{1}.}
5806:
5794:
5778:
5772:
5703:
5691:
5667:
5661:
5559:
5553:
5192:
5157:
4732:
4694:
4550:
4518:
4489:
4457:
4133:
4120:
4091:
4078:
4016:
4003:
3974:
3961:
3899:
3886:
3857:
3844:
3804:
3791:
3748:
3735:
3676:
3663:
3624:
3611:
3533:
3507:
3489:
3463:
3457:
3431:
3387:
3367:
3345:
3325:
3309:
3290:
3261:
3255:
3238:Cubic splines are of the form
3209:
3203:
3188:
3182:
3054:
3048:
3028:
3022:
2963:
2957:
2937:
2931:
2875:
2869:
2849:
2843:
2762:
2756:
2736:
2730:
2690:
2684:
2664:
2658:
2596:
2590:
2570:
2564:
2486:
2474:
2454:
2448:
2439:
2433:
2417:
2411:
2354:
2182:
2125:
2093:
2045:
2039:
1925:
1917:
1743:
1730:
1725:
1712:
1692:
1679:
1674:
1661:
1627:
1614:
1609:
1603:
1583:
1570:
1565:
1559:
1533:
1520:
1515:
1509:
1489:
1476:
1471:
1465:
1327:{\displaystyle S\in C^{r_{i}}}
1143:
1137:
1111:
1105:
1047:
1041:
1021:
1015:
967:
961:
941:
935:
883:
880:
848:
720:
707:
701:
669:
663:
625:
613:
587:
581:
555:
549:
537:
496:
464:
415:
412:
400:
125:and interactive curve design.
13:
1:
7308:Bartels, Beatty, and Barsky,
7294:Ahlberg, Nielson, and Walsh,
7281:
4296:{\displaystyle a_{i}=y_{i}\,}
376:{\displaystyle \mathbb {R} ,}
276:
7348:. p. 93-106. ISBN 0387904204
5835:Nonuniform rational B-spline
210:British Aircraft Corporation
90:polynomials, while avoiding
7:
6429:{\displaystyle f(t_{i}^{})}
3549:{\displaystyle C={\bigl },}
2514:
1976:disappears, and the pieces
1858:, and a smoothness vector
342:polynomials in one variable
30:For the drafting tool, see
10:
7516:
7479:. Morgan Kaufmann. p. 119.
5852:interpolating cubic spline
4214:Input: set of coordinates
3411:Given set of coordinates
1393:the two polynomial pieces
148:
29:
7420:Symmetrical Spline Curves
6445:Representations and names
5848:General expression for a
1816:{\displaystyle C^{r_{i}}}
1372:{\displaystyle C^{r_{i}}}
1267:each have degree at most
1260:If the polynomial pieces
1253:, otherwise we say it is
354:. This function, call it
7464:, vbnumericalmethods.com
7462:VBA Spline Interpolation
7254:De Casteljau's algorithm
6926:Kochanek-Bartels splines
5534:
267:David Taylor Model Basin
204:and (somewhat later) by
84:polynomial interpolation
7414:Learning by Simulations
6845:using single knots for
3556:we wish to find set of
3155:A common spline is the
1893:is commonly denoted by
1883:and smoothness vector
136:to draw smooth shapes.
7341:vol. 98, no. 26, 1998.
7245:
7208:
7139:
7063:
7009:
6794:
6776:
6722:
6430:
6391:
6326:
6262:
5816:
5713:
5595:
5519:
5286:
4982:
4914:
4644:
4560:
4393:
4297:
4150:
3550:
3403:
3222:
3117:
2819:
2493:
2361:
2135:
1935:
1817:
1757:
1373:
1328:
1198:
898:
809:
430:
377:
82:is often preferred to
51:
7495:Splines (mathematics)
7475:Farin, G. E. (2002).
7330:vol. 29, no. 5, 1996.
7246:
7182:
7140:
7064:
7010:
7008:{\displaystyle t\in }
6908:Cubic Hermite splines
6887:interpolating splines
6831:Bernstein polynomials
6795:
6750:
6723:
6431:
6392:
6327:
6263:
5817:
5714:
5596:
5520:
5287:
4983:
4915:
4645:
4561:
4394:
4298:
4151:
3551:
3404:
3223:
3118:
2820:
2494:
2362:
2136:
1936:
1818:
1758:
1374:
1334:in a neighborhood of
1329:
1199:
899:
810:
431:
378:
103:computer-aided design
94:for higher degrees.
48:parametric continuity
40:
7256:, which features in
7179:
7079:
7021:
6961:
6735:
6467:
6402:
6342:
6277:
5876:
5842:spline interpolation
5766:
5655:
5547:
5328:
5302:. Populate it with
5009:
5005:, set the following:
4926:
4667:
4586:
4424:
4343:
4266:
3594:
3588:These must satisfy:
3415:
3242:
3171:
3157:natural cubic spline
2833:
2554:
2405:
2179:
2033:
1897:
1848:Given a knot vector
1793:
1442:
1349:
1298:
925:
908:th subinterval of ,
832:
454:
391:
362:
298:confusing or unclear
80:spline interpolation
7360:Notices of the AMS,
7353:Quart. Appl. Math.,
7287:Ferguson, James C,
6922:Catmull-Rom splines
6422:
6305:
5324:set the following:
4119:
4077:
4002:
3960:
3885:
3843:
1916:
1877:having knot vector
1729:
1678:
1613:
1569:
1519:
1475:
306:clarify the article
155:Nikolai Lobachevsky
7422:, an animation by
7335:History of Splines
7241:
7135:
7059:
7005:
6860:nonuniform splines
6790:
6718:
6716:
6627:
6607:
6548:
6534:
6426:
6411:
6387:
6322:
6293:
6258:
5812:
5709:
5608:appears more than
5591:
5515:
5513:
5282:
5280:
4978:
4910:
4908:
4663:set the following:
4640:
4568:Create new arrays
4556:
4389:
4305:Create new arrays
4293:
4146:
4144:
4101:
4065:
3984:
3948:
3867:
3831:
3546:
3399:
3218:
3113:
3111:
2815:
2813:
2489:
2357:
2131:
1931:
1900:
1813:
1753:
1751:
1702:
1645:
1593:
1543:
1499:
1449:
1421:loss of smoothness
1369:
1324:
1194:
1192:
894:
805:
803:
426:
373:
92:Runge's phenomenon
52:
7269:de Boor algorithm
6560:
6558:
6499:
6497:
6229:
6191:
6109:
6077:
6034:
5965:
5756:Adobe Illustrator
5273:
5199:
5139:
4901:
4816:
4516:
4455:
4401:Create new array
4320:Create new array
4241:Create new array
4048:
4039:
3931:
3922:
3708:
3699:
2401:on the interval
1843:smoothness vector
817:On each of these
338:
337:
330:
107:computer graphics
16:(Redirected from
7507:
7480:
7473:
7450:various routines
7403:Online utilities
7250:
7248:
7247:
7242:
7228:
7227:
7218:
7217:
7207:
7196:
7171:
7165:
7144:
7142:
7141:
7136:
7125:
7124:
7091:
7090:
7074:
7068:
7066:
7065:
7060:
7058:
7057:
7033:
7032:
7014:
7012:
7011:
7006:
7001:
7000:
6982:
6981:
6956:
6948:
6944:
6918:Cardinal splines
6915:
6905:
6877:
6873:
6850:
6799:
6797:
6796:
6791:
6786:
6785:
6775:
6764:
6727:
6725:
6724:
6719:
6717:
6666:
6665:
6646:
6645:
6626:
6625:
6624:
6608:
6603:
6602:
6601:
6577:
6576:
6547:
6546:
6545:
6535:
6530:
6529:
6528:
6510:
6509:
6493:
6492:
6452:
6439:
6435:
6433:
6432:
6427:
6421:
6419:
6396:
6394:
6393:
6388:
6386:
6385:
6367:
6366:
6354:
6353:
6335:
6331:
6329:
6328:
6323:
6318:
6317:
6301:
6289:
6288:
6267:
6265:
6264:
6259:
6248:
6247:
6235:
6231:
6230:
6225:
6224:
6223:
6214:
6213:
6197:
6192:
6190:
6189:
6180:
6176:
6175:
6153:
6140:
6139:
6115:
6111:
6110:
6105:
6104:
6103:
6094:
6093:
6083:
6078:
6076:
6075:
6066:
6062:
6061:
6045:
6035:
6033:
6032:
6031:
6018:
6017:
6016:
6001:
6000:
5988:
5987:
5971:
5966:
5964:
5963:
5962:
5949:
5948:
5947:
5938:
5937:
5913:
5912:
5902:
5888:
5887:
5865:
5859:
5821:
5819:
5818:
5813:
5793:
5792:
5750:
5746:
5742:
5736:
5725:
5718:
5716:
5715:
5710:
5690:
5689:
5678:
5650:
5643:
5636:
5629:
5622:
5614:
5607:
5600:
5598:
5597:
5592:
5577:
5576:
5542:
5530:
5524:
5522:
5521:
5516:
5514:
5507:
5506:
5494:
5493:
5471:
5470:
5458:
5457:
5435:
5434:
5422:
5421:
5399:
5398:
5386:
5385:
5363:
5362:
5350:
5349:
5323:
5309:
5305:
5301:
5297:
5291:
5289:
5288:
5283:
5281:
5274:
5272:
5271:
5270:
5257:
5256:
5255:
5243:
5242:
5226:
5217:
5216:
5200:
5195:
5191:
5190:
5175:
5174:
5156:
5155:
5145:
5140:
5138:
5137:
5128:
5127:
5126:
5114:
5113:
5097:
5088:
5087:
5071:
5070:
5055:
5054:
5042:
5041:
5025:
5024:
5004:
4987:
4985:
4984:
4979:
4970:
4969:
4957:
4956:
4938:
4937:
4919:
4917:
4916:
4911:
4909:
4902:
4900:
4899:
4890:
4889:
4888:
4873:
4872:
4854:
4853:
4843:
4834:
4833:
4817:
4815:
4814:
4805:
4804:
4795:
4786:
4785:
4769:
4768:
4753:
4752:
4731:
4730:
4712:
4711:
4683:
4682:
4662:
4649:
4647:
4646:
4641:
4632:
4631:
4619:
4618:
4613:
4598:
4597:
4578:
4571:
4565:
4563:
4562:
4557:
4549:
4548:
4530:
4529:
4517:
4515:
4514:
4496:
4488:
4487:
4475:
4474:
4456:
4454:
4453:
4441:
4436:
4435:
4419:
4408:
4404:
4398:
4396:
4395:
4390:
4387:
4386:
4374:
4373:
4355:
4354:
4338:
4327:
4323:
4316:
4312:
4308:
4302:
4300:
4299:
4294:
4291:
4290:
4278:
4277:
4261:
4251:
4244:
4236:
4231:
4225:
4217:
4206:
4199:
4192:
4188:
4161:
4155:
4153:
4152:
4147:
4145:
4132:
4131:
4115:
4090:
4089:
4073:
4046:
4037:
4023:
4015:
4014:
3998:
3973:
3972:
3956:
3929:
3920:
3906:
3898:
3897:
3881:
3856:
3855:
3839:
3823:
3822:
3803:
3802:
3790:
3789:
3767:
3766:
3747:
3746:
3734:
3733:
3706:
3697:
3683:
3675:
3674:
3662:
3661:
3643:
3642:
3623:
3622:
3610:
3609:
3584:
3573:
3559:
3555:
3553:
3552:
3547:
3542:
3541:
3532:
3531:
3519:
3518:
3488:
3487:
3475:
3474:
3456:
3455:
3443:
3442:
3430:
3429:
3408:
3406:
3405:
3400:
3395:
3394:
3385:
3384:
3366:
3365:
3353:
3352:
3343:
3342:
3324:
3323:
3308:
3307:
3289:
3288:
3276:
3275:
3254:
3253:
3227:
3225:
3224:
3219:
3202:
3181:
3164:
3136:
3131:
3122:
3120:
3119:
3114:
3112:
3092:
3087:
3086:
3047:
3046:
2998:
2993:
2992:
2956:
2955:
2907:
2902:
2901:
2868:
2867:
2824:
2822:
2821:
2816:
2814:
2794:
2789:
2788:
2755:
2754:
2706:
2683:
2682:
2634:
2629:
2628:
2589:
2588:
2547:
2536:
2529:
2522:
2510:
2506:
2498:
2496:
2495:
2490:
2463:
2462:
2429:
2428:
2399:parametric curve
2393:
2382:
2375:
2366:
2364:
2363:
2358:
2353:
2352:
2340:
2339:
2315:
2314:
2296:
2295:
2271:
2270:
2258:
2257:
2239:
2238:
2226:
2225:
2207:
2206:
2194:
2193:
2168:
2160:
2140:
2138:
2137:
2132:
2124:
2123:
2105:
2104:
2092:
2091:
2090:
2089:
2071:
2070:
2028:
2016:
2009:
1992:
1975:
1961:
1949:
1940:
1938:
1937:
1932:
1924:
1915:
1914:
1908:
1888:
1882:
1876:
1869:
1863:
1857:
1853:
1845:for the spline.
1840:
1829:
1822:
1820:
1819:
1814:
1812:
1811:
1810:
1809:
1788:
1762:
1760:
1759:
1754:
1752:
1742:
1741:
1728:
1724:
1723:
1710:
1691:
1690:
1677:
1673:
1672:
1659:
1641:
1626:
1625:
1612:
1601:
1582:
1581:
1568:
1557:
1532:
1531:
1518:
1507:
1488:
1487:
1474:
1463:
1435:
1418:
1411:
1404:
1392:
1385:
1378:
1376:
1375:
1370:
1368:
1367:
1366:
1365:
1340:
1333:
1331:
1330:
1325:
1323:
1322:
1321:
1320:
1290:
1280:
1270:
1266:
1244:
1219:
1212:
1203:
1201:
1200:
1195:
1193:
1186:
1185:
1167:
1166:
1150:
1136:
1135:
1091:
1084:
1083:
1065:
1064:
1054:
1040:
1039:
1004:
1003:
985:
984:
974:
960:
959:
918:
911:
907:
903:
901:
900:
895:
890:
879:
878:
860:
859:
844:
843:
827:
820:
814:
812:
811:
806:
804:
794:
793:
781:
780:
756:
755:
743:
742:
726:
719:
718:
700:
699:
687:
686:
662:
661:
643:
642:
612:
611:
599:
598:
580:
579:
567:
566:
533:
495:
494:
476:
475:
460:
443:
439:
435:
433:
432:
427:
422:
382:
380:
379:
374:
369:
357:
333:
326:
322:
319:
313:
289:
288:
281:
159:Kazan University
99:computer science
46:
21:
7515:
7514:
7510:
7509:
7508:
7506:
7505:
7504:
7485:
7484:
7483:
7474:
7470:
7375:
7346:Springer-Verlag
7284:
7278:
7223:
7219:
7213:
7209:
7197:
7186:
7180:
7177:
7176:
7169:
7164:
7154:
7148:
7114:
7110:
7086:
7082:
7080:
7077:
7076:
7072:
7047:
7043:
7028:
7024:
7022:
7019:
7018:
6990:
6986:
6977:
6973:
6962:
6959:
6958:
6954:
6946:
6937:
6932:
6911:
6900:
6880:natural splines
6875:
6871:
6853:uniform splines
6846:
6781:
6777:
6765:
6754:
6736:
6733:
6732:
6715:
6714:
6667:
6661:
6657:
6654:
6653:
6635:
6631:
6614:
6610:
6609:
6591:
6587:
6566:
6562:
6561:
6559:
6541:
6537:
6536:
6524:
6520:
6505:
6501:
6500:
6498:
6488:
6484:
6477:
6470:
6468:
6465:
6464:
6450:
6447:
6437:
6420:
6415:
6403:
6400:
6399:
6375:
6371:
6362:
6358:
6349:
6345:
6343:
6340:
6339:
6333:
6313:
6309:
6297:
6284:
6280:
6278:
6275:
6274:
6243:
6239:
6219:
6215:
6203:
6199:
6198:
6196:
6185:
6181:
6165:
6161:
6154:
6152:
6151:
6147:
6129:
6125:
6099:
6095:
6089:
6085:
6084:
6082:
6071:
6067:
6057:
6053:
6046:
6044:
6043:
6039:
6027:
6023:
6019:
6012:
6008:
5996:
5992:
5977:
5973:
5972:
5970:
5958:
5954:
5950:
5943:
5939:
5927:
5923:
5908:
5904:
5903:
5901:
5883:
5879:
5877:
5874:
5873:
5861:
5857:
5854:
5788:
5784:
5767:
5764:
5763:
5748:
5744:
5738:
5731:
5720:
5679:
5674:
5673:
5656:
5653:
5652:
5648:
5638:
5631:
5624:
5616:
5609:
5606:
5602:
5569:
5565:
5548:
5545:
5544:
5540:
5537:
5512:
5511:
5502:
5498:
5483:
5479:
5476:
5475:
5466:
5462:
5447:
5443:
5440:
5439:
5430:
5426:
5411:
5407:
5404:
5403:
5394:
5390:
5375:
5371:
5368:
5367:
5358:
5354:
5339:
5335:
5331:
5329:
5326:
5325:
5314:
5307:
5303:
5294:Create new set
5279:
5278:
5266:
5262:
5258:
5251:
5247:
5232:
5228:
5227:
5225:
5218:
5212:
5208:
5205:
5204:
5186:
5182:
5164:
5160:
5151:
5147:
5146:
5144:
5133:
5129:
5122:
5118:
5103:
5099:
5098:
5096:
5089:
5083:
5079:
5076:
5075:
5060:
5056:
5050:
5046:
5037:
5033:
5026:
5020:
5016:
5012:
5010:
5007:
5006:
4991:
4965:
4961:
4952:
4948:
4933:
4929:
4927:
4924:
4923:
4907:
4906:
4895:
4891:
4878:
4874:
4862:
4858:
4849:
4845:
4844:
4842:
4835:
4829:
4825:
4822:
4821:
4810:
4806:
4800:
4796:
4794:
4787:
4781:
4777:
4774:
4773:
4758:
4754:
4742:
4738:
4720:
4716:
4701:
4697:
4684:
4678:
4674:
4670:
4668:
4665:
4664:
4653:
4627:
4623:
4614:
4609:
4608:
4593:
4589:
4587:
4584:
4583:
4573:
4572:, each of size
4569:
4538:
4534:
4525:
4521:
4504:
4500:
4495:
4483:
4479:
4464:
4460:
4449:
4445:
4440:
4431:
4427:
4425:
4422:
4421:
4410:
4406:
4402:
4382:
4378:
4363:
4359:
4350:
4346:
4344:
4341:
4340:
4329:
4325:
4321:
4314:
4313:, each of size
4310:
4306:
4286:
4282:
4273:
4269:
4267:
4264:
4263:
4253:
4246:
4242:
4234:
4232:
4221:
4219:
4215:
4213:
4205:
4201:
4198:
4194:
4190:
4185:
4163:
4159:
4143:
4142:
4127:
4123:
4105:
4094:
4085:
4081:
4069:
4062:
4061:
4022:
4010:
4006:
3988:
3977:
3968:
3964:
3952:
3945:
3944:
3905:
3893:
3889:
3871:
3860:
3851:
3847:
3835:
3828:
3827:
3818:
3814:
3807:
3798:
3794:
3779:
3775:
3772:
3771:
3762:
3758:
3751:
3742:
3738:
3729:
3725:
3722:
3721:
3682:
3670:
3666:
3651:
3647:
3638:
3634:
3627:
3618:
3614:
3605:
3601:
3597:
3595:
3592:
3591:
3575:
3566:
3561:
3557:
3537:
3536:
3527:
3523:
3514:
3510:
3483:
3479:
3470:
3466:
3451:
3447:
3438:
3434:
3425:
3424:
3416:
3413:
3412:
3390:
3386:
3380:
3376:
3361:
3357:
3348:
3344:
3338:
3334:
3319:
3315:
3303:
3299:
3284:
3280:
3271:
3267:
3249:
3245:
3243:
3240:
3239:
3236:
3195:
3174:
3172:
3169:
3168:
3160:
3135:(0, 1, 2, 2, 3)
3134:
3126:
3110:
3109:
3091:
3082:
3078:
3042:
3038:
3031:
3016:
3015:
2997:
2988:
2984:
2951:
2947:
2940:
2925:
2924:
2906:
2897:
2893:
2863:
2859:
2852:
2836:
2834:
2831:
2830:
2812:
2811:
2793:
2784:
2780:
2750:
2746:
2739:
2724:
2723:
2705:
2678:
2674:
2667:
2652:
2651:
2633:
2624:
2620:
2584:
2580:
2573:
2557:
2555:
2552:
2551:
2538:
2531:
2524:
2520:
2517:
2508:
2504:
2458:
2457:
2424:
2423:
2406:
2403:
2402:
2384:
2381:
2377:
2374:
2370:
2348:
2344:
2329:
2325:
2304:
2300:
2285:
2281:
2266:
2262:
2253:
2249:
2234:
2230:
2221:
2217:
2202:
2198:
2189:
2185:
2180:
2177:
2176:
2166:
2158:
2147:
2142:
2113:
2109:
2100:
2096:
2079:
2075:
2066:
2062:
2055:
2051:
2034:
2031:
2030:
2027:
2018:
2015:
2011:
2003:
1994:
1986:
1977:
1968:
1963:
1960:
1951:
1948:
1944:
1920:
1910:
1909:
1904:
1898:
1895:
1894:
1884:
1878:
1871:
1865:
1859:
1855:
1849:
1831:
1828:
1824:
1805:
1801:
1800:
1796:
1794:
1791:
1790:
1786:
1776:
1766:
1750:
1749:
1737:
1733:
1719:
1715:
1711:
1706:
1695:
1686:
1682:
1668:
1664:
1660:
1649:
1642:
1640:
1634:
1633:
1621:
1617:
1602:
1597:
1586:
1577:
1573:
1558:
1547:
1540:
1539:
1527:
1523:
1508:
1503:
1492:
1483:
1479:
1464:
1453:
1445:
1443:
1440:
1439:
1433:
1424:
1417:
1413:
1410:
1406:
1403:
1394:
1391:
1387:
1384:
1380:
1361:
1357:
1356:
1352:
1350:
1347:
1346:
1339:
1335:
1316:
1312:
1311:
1307:
1299:
1296:
1295:
1285:
1275:
1268:
1265:
1261:
1241:
1235:
1225:
1218:
1214:
1207:
1191:
1190:
1181:
1177:
1156:
1152:
1149:
1125:
1121:
1114:
1099:
1098:
1089:
1088:
1079:
1075:
1060:
1056:
1053:
1035:
1031:
1024:
1009:
1008:
999:
995:
980:
976:
973:
955:
951:
944:
928:
926:
923:
922:
917:
913:
909:
905:
886:
868:
864:
855:
851:
839:
835:
833:
830:
829:
826:
822:
818:
802:
801:
789:
785:
770:
766:
751:
747:
738:
734:
724:
723:
714:
710:
695:
691:
676:
672:
651:
647:
632:
628:
607:
603:
594:
590:
575:
571:
562:
558:
531:
530:
484:
480:
471:
467:
457:
455:
452:
451:
441:
437:
418:
392:
389:
388:
365:
363:
360:
359:
355:
334:
323:
317:
314:
303:
290:
286:
279:
151:
142:
42:
35:
28:
23:
22:
15:
12:
11:
5:
7513:
7503:
7502:
7497:
7482:
7481:
7467:
7466:
7465:
7459:
7453:
7447:
7432:
7431:
7417:
7411:
7400:
7399:
7391:Excel Function
7388:
7387:
7374:
7373:External links
7371:
7370:
7369:
7366:
7363:
7356:
7349:
7342:
7331:
7320:
7313:
7306:
7299:
7292:
7283:
7280:
7262:Bézier splines
7240:
7237:
7234:
7231:
7226:
7222:
7216:
7212:
7206:
7203:
7200:
7195:
7192:
7189:
7185:
7174:
7173:
7166:
7159:
7152:
7145:
7134:
7131:
7128:
7123:
7120:
7117:
7113:
7109:
7106:
7103:
7100:
7097:
7094:
7089:
7085:
7069:
7056:
7053:
7050:
7046:
7042:
7039:
7036:
7031:
7027:
7015:
7004:
6999:
6996:
6993:
6989:
6985:
6980:
6976:
6972:
6969:
6966:
6935:
6906:; that is, as
6897:Hermite spline
6893:
6892:
6891:
6890:
6883:
6865:
6864:
6863:
6856:
6840:
6839:
6838:
6835:Bézier splines
6827:
6801:
6800:
6789:
6784:
6780:
6774:
6771:
6768:
6763:
6760:
6757:
6753:
6749:
6746:
6743:
6740:
6713:
6710:
6707:
6704:
6701:
6698:
6695:
6692:
6689:
6685:
6682:
6679:
6676:
6673:
6670:
6668:
6664:
6660:
6656:
6655:
6652:
6649:
6644:
6641:
6638:
6634:
6630:
6623:
6620:
6617:
6613:
6606:
6600:
6597:
6594:
6590:
6586:
6583:
6580:
6575:
6572:
6569:
6565:
6557:
6554:
6551:
6544:
6540:
6533:
6527:
6523:
6519:
6516:
6513:
6508:
6504:
6496:
6491:
6487:
6483:
6480:
6478:
6476:
6473:
6472:
6446:
6443:
6442:
6441:
6425:
6418:
6414:
6410:
6407:
6397:
6384:
6381:
6378:
6374:
6370:
6365:
6361:
6357:
6352:
6348:
6337:
6321:
6316:
6312:
6308:
6304:
6300:
6296:
6292:
6287:
6283:
6257:
6254:
6251:
6246:
6242:
6238:
6234:
6228:
6222:
6218:
6212:
6209:
6206:
6202:
6195:
6188:
6184:
6179:
6174:
6171:
6168:
6164:
6160:
6157:
6150:
6146:
6143:
6138:
6135:
6132:
6128:
6124:
6121:
6118:
6114:
6108:
6102:
6098:
6092:
6088:
6081:
6074:
6070:
6065:
6060:
6056:
6052:
6049:
6042:
6038:
6030:
6026:
6022:
6015:
6011:
6007:
6004:
5999:
5995:
5991:
5986:
5983:
5980:
5976:
5969:
5961:
5957:
5953:
5946:
5942:
5936:
5933:
5930:
5926:
5922:
5919:
5916:
5911:
5907:
5900:
5897:
5894:
5891:
5886:
5882:
5853:
5846:
5811:
5808:
5805:
5802:
5799:
5796:
5791:
5787:
5783:
5780:
5777:
5774:
5771:
5708:
5705:
5702:
5699:
5696:
5693:
5688:
5685:
5682:
5677:
5672:
5669:
5666:
5663:
5660:
5604:
5590:
5587:
5584:
5580:
5575:
5572:
5568:
5564:
5561:
5558:
5555:
5552:
5536:
5533:
5532:
5531:
5525:
5510:
5505:
5501:
5497:
5492:
5489:
5486:
5482:
5478:
5477:
5474:
5469:
5465:
5461:
5456:
5453:
5450:
5446:
5442:
5441:
5438:
5433:
5429:
5425:
5420:
5417:
5414:
5410:
5406:
5405:
5402:
5397:
5393:
5389:
5384:
5381:
5378:
5374:
5370:
5369:
5366:
5361:
5357:
5353:
5348:
5345:
5342:
5338:
5334:
5333:
5311:
5292:
5277:
5269:
5265:
5261:
5254:
5250:
5246:
5241:
5238:
5235:
5231:
5224:
5221:
5219:
5215:
5211:
5207:
5206:
5203:
5198:
5194:
5189:
5185:
5181:
5178:
5173:
5170:
5167:
5163:
5159:
5154:
5150:
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5136:
5132:
5125:
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5117:
5112:
5109:
5106:
5102:
5095:
5092:
5090:
5086:
5082:
5078:
5077:
5074:
5069:
5066:
5063:
5059:
5053:
5049:
5045:
5040:
5036:
5032:
5029:
5027:
5023:
5019:
5015:
5014:
4988:
4976:
4973:
4968:
4964:
4960:
4955:
4951:
4947:
4944:
4941:
4936:
4932:
4920:
4905:
4898:
4894:
4887:
4884:
4881:
4877:
4871:
4868:
4865:
4861:
4857:
4852:
4848:
4841:
4838:
4836:
4832:
4828:
4824:
4823:
4820:
4813:
4809:
4803:
4799:
4793:
4790:
4788:
4784:
4780:
4776:
4775:
4772:
4767:
4764:
4761:
4757:
4751:
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4741:
4737:
4734:
4729:
4726:
4723:
4719:
4715:
4710:
4707:
4704:
4700:
4696:
4693:
4690:
4687:
4685:
4681:
4677:
4673:
4672:
4650:
4638:
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4626:
4622:
4617:
4612:
4607:
4604:
4601:
4596:
4592:
4580:
4566:
4555:
4552:
4547:
4544:
4541:
4537:
4533:
4528:
4524:
4520:
4513:
4510:
4507:
4503:
4499:
4494:
4491:
4486:
4482:
4478:
4473:
4470:
4467:
4463:
4459:
4452:
4448:
4444:
4439:
4434:
4430:
4399:
4385:
4381:
4377:
4372:
4369:
4366:
4362:
4358:
4353:
4349:
4318:
4303:
4289:
4285:
4281:
4276:
4272:
4203:
4196:
4183:
4141:
4138:
4135:
4130:
4126:
4122:
4118:
4114:
4111:
4108:
4104:
4100:
4097:
4095:
4093:
4088:
4084:
4080:
4076:
4072:
4068:
4064:
4063:
4060:
4057:
4054:
4051:
4045:
4042:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4013:
4009:
4005:
4001:
3997:
3994:
3991:
3987:
3983:
3980:
3978:
3976:
3971:
3967:
3963:
3959:
3955:
3951:
3947:
3946:
3943:
3940:
3937:
3934:
3928:
3925:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3896:
3892:
3888:
3884:
3880:
3877:
3874:
3870:
3866:
3863:
3861:
3859:
3854:
3850:
3846:
3842:
3838:
3834:
3830:
3829:
3826:
3821:
3817:
3813:
3810:
3808:
3806:
3801:
3797:
3793:
3788:
3785:
3782:
3778:
3774:
3773:
3770:
3765:
3761:
3757:
3754:
3752:
3750:
3745:
3741:
3737:
3732:
3728:
3724:
3723:
3720:
3717:
3714:
3711:
3705:
3702:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3673:
3669:
3665:
3660:
3657:
3654:
3650:
3646:
3641:
3637:
3633:
3630:
3628:
3626:
3621:
3617:
3613:
3608:
3604:
3600:
3599:
3564:
3545:
3540:
3535:
3530:
3526:
3522:
3517:
3513:
3509:
3506:
3503:
3500:
3497:
3494:
3491:
3486:
3482:
3478:
3473:
3469:
3465:
3462:
3459:
3454:
3450:
3446:
3441:
3437:
3433:
3428:
3423:
3420:
3398:
3393:
3389:
3383:
3379:
3375:
3372:
3369:
3364:
3360:
3356:
3351:
3347:
3341:
3337:
3333:
3330:
3327:
3322:
3318:
3314:
3311:
3306:
3302:
3298:
3295:
3292:
3287:
3283:
3279:
3274:
3270:
3266:
3263:
3260:
3257:
3252:
3248:
3235:
3232:
3217:
3214:
3211:
3208:
3205:
3201:
3198:
3194:
3190:
3187:
3184:
3180:
3177:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3085:
3081:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3045:
3041:
3037:
3034:
3032:
3030:
3027:
3024:
3021:
3018:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2991:
2987:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2954:
2950:
2946:
2943:
2941:
2939:
2936:
2933:
2930:
2927:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2900:
2896:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2866:
2862:
2858:
2855:
2853:
2851:
2848:
2845:
2842:
2839:
2838:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2787:
2783:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2753:
2749:
2745:
2742:
2740:
2738:
2735:
2732:
2729:
2726:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2681:
2677:
2673:
2670:
2668:
2666:
2663:
2660:
2657:
2654:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2627:
2623:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2587:
2583:
2579:
2576:
2574:
2572:
2569:
2566:
2563:
2560:
2559:
2516:
2513:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2466:
2461:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2427:
2422:
2419:
2416:
2413:
2410:
2379:
2372:
2356:
2351:
2347:
2343:
2338:
2335:
2332:
2328:
2324:
2321:
2318:
2313:
2310:
2307:
2303:
2299:
2294:
2291:
2288:
2284:
2280:
2277:
2274:
2269:
2265:
2261:
2256:
2252:
2248:
2245:
2242:
2237:
2233:
2229:
2224:
2220:
2216:
2213:
2210:
2205:
2201:
2197:
2192:
2188:
2184:
2163:multiple knots
2156:
2145:
2130:
2127:
2122:
2119:
2116:
2112:
2108:
2103:
2099:
2095:
2088:
2085:
2082:
2078:
2074:
2069:
2065:
2061:
2058:
2054:
2050:
2047:
2044:
2041:
2038:
2022:
2013:
1998:
1981:
1966:
1955:
1946:
1930:
1927:
1923:
1919:
1913:
1907:
1903:
1826:
1808:
1804:
1799:
1781:
1774:
1748:
1745:
1740:
1736:
1732:
1727:
1722:
1718:
1714:
1709:
1705:
1701:
1698:
1696:
1694:
1689:
1685:
1681:
1676:
1671:
1667:
1663:
1658:
1655:
1652:
1648:
1644:
1643:
1639:
1636:
1635:
1632:
1629:
1624:
1620:
1616:
1611:
1608:
1605:
1600:
1596:
1592:
1589:
1587:
1585:
1580:
1576:
1572:
1567:
1564:
1561:
1556:
1553:
1550:
1546:
1542:
1541:
1538:
1535:
1530:
1526:
1522:
1517:
1514:
1511:
1506:
1502:
1498:
1495:
1493:
1491:
1486:
1482:
1478:
1473:
1470:
1467:
1462:
1459:
1456:
1452:
1448:
1447:
1431:
1415:
1408:
1398:
1389:
1386:. That is, at
1382:
1364:
1360:
1355:
1337:
1319:
1315:
1310:
1306:
1303:
1263:
1239:
1233:
1216:
1189:
1184:
1180:
1176:
1173:
1170:
1165:
1162:
1159:
1155:
1151:
1148:
1145:
1142:
1139:
1134:
1131:
1128:
1124:
1120:
1117:
1115:
1113:
1110:
1107:
1104:
1101:
1100:
1097:
1094:
1092:
1090:
1087:
1082:
1078:
1074:
1071:
1068:
1063:
1059:
1055:
1052:
1049:
1046:
1043:
1038:
1034:
1030:
1027:
1025:
1023:
1020:
1017:
1014:
1011:
1010:
1007:
1002:
998:
994:
991:
988:
983:
979:
975:
972:
969:
966:
963:
958:
954:
950:
947:
945:
943:
940:
937:
934:
931:
930:
915:
912:is defined by
893:
889:
885:
882:
877:
874:
871:
867:
863:
858:
854:
850:
847:
842:
838:
824:
800:
797:
792:
788:
784:
779:
776:
773:
769:
765:
762:
759:
754:
750:
746:
741:
737:
733:
730:
727:
725:
722:
717:
713:
709:
706:
703:
698:
694:
690:
685:
682:
679:
675:
671:
668:
665:
660:
657:
654:
650:
646:
641:
638:
635:
631:
627:
624:
621:
618:
615:
610:
606:
602:
597:
593:
589:
586:
583:
578:
574:
570:
565:
561:
557:
554:
551:
548:
545:
542:
539:
536:
534:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
501:
498:
493:
490:
487:
483:
479:
474:
470:
466:
463:
461:
459:
448:subintervals,
425:
421:
417:
414:
411:
408:
405:
402:
399:
396:
372:
368:
336:
335:
293:
291:
284:
278:
275:
271:General Motors
256:General Motors
252:General Motors
219:dialect word.
198:J. C. Ferguson
150:
147:
141:
138:
26:
9:
6:
4:
3:
2:
7512:
7501:
7500:Interpolation
7498:
7496:
7493:
7492:
7490:
7478:
7472:
7468:
7463:
7460:
7457:
7454:
7451:
7448:
7446:
7442:
7439:
7438:
7437:
7436:
7435:Computer Code
7429:
7425:
7424:Theodore Gray
7421:
7418:
7415:
7412:
7410:
7407:
7406:
7405:
7404:
7398:
7395:
7394:
7393:
7392:
7386:, ibiblio.org
7385:
7382:
7381:
7380:
7379:
7367:
7364:
7361:
7357:
7354:
7350:
7347:
7343:
7340:
7336:
7332:
7329:
7325:
7321:
7318:
7314:
7311:
7307:
7304:
7300:
7297:
7293:
7290:
7286:
7285:
7279:
7276:
7274:
7270:
7265:
7263:
7259:
7258:Bézier curves
7255:
7238:
7232:
7224:
7220:
7214:
7210:
7204:
7201:
7198:
7193:
7190:
7187:
7183:
7167:
7162:
7158:
7151:
7146:
7129:
7121:
7118:
7115:
7111:
7107:
7104:
7101:
7095:
7087:
7083:
7070:
7054:
7051:
7048:
7044:
7040:
7037:
7034:
7029:
7025:
7016:
6997:
6994:
6991:
6987:
6983:
6978:
6974:
6967:
6964:
6952:
6951:
6950:
6942:
6938:
6929:
6927:
6923:
6919:
6914:
6909:
6903:
6898:
6888:
6884:
6881:
6869:
6868:
6866:
6861:
6857:
6854:
6849:
6844:
6843:
6841:
6836:
6832:
6828:
6825:
6821:
6817:
6816:
6814:
6813:
6812:
6809:
6807:
6787:
6782:
6778:
6772:
6769:
6766:
6761:
6758:
6755:
6751:
6747:
6744:
6741:
6738:
6731:
6730:
6729:
6711:
6708:
6705:
6702:
6699:
6696:
6693:
6690:
6687:
6683:
6680:
6677:
6674:
6671:
6669:
6662:
6658:
6650:
6647:
6642:
6639:
6636:
6632:
6628:
6621:
6618:
6615:
6611:
6604:
6598:
6595:
6592:
6588:
6584:
6581:
6578:
6573:
6570:
6567:
6563:
6555:
6552:
6549:
6542:
6538:
6531:
6525:
6521:
6517:
6514:
6511:
6506:
6502:
6494:
6489:
6485:
6481:
6479:
6474:
6462:
6460:
6456:
6416:
6412:
6405:
6398:
6382:
6379:
6376:
6372:
6368:
6363:
6359:
6355:
6350:
6346:
6338:
6314:
6310:
6302:
6298:
6294:
6290:
6285:
6281:
6273:
6272:
6271:
6268:
6252:
6249:
6244:
6240:
6232:
6226:
6220:
6216:
6210:
6207:
6204:
6200:
6193:
6186:
6182:
6172:
6169:
6166:
6162:
6155:
6148:
6144:
6136:
6133:
6130:
6126:
6122:
6119:
6112:
6106:
6100:
6096:
6090:
6086:
6079:
6072:
6068:
6058:
6054:
6047:
6040:
6036:
6028:
6024:
6020:
6013:
6005:
6002:
5997:
5993:
5984:
5981:
5978:
5974:
5967:
5959:
5955:
5951:
5944:
5934:
5931:
5928:
5924:
5920:
5917:
5909:
5905:
5898:
5892:
5884:
5880:
5871:
5869:
5864:
5851:
5845:
5843:
5838:
5836:
5832:
5831:Autodesk Maya
5827:
5825:
5809:
5803:
5800:
5797:
5789:
5785:
5781:
5775:
5769:
5761:
5760:Adobe Systems
5757:
5752:
5741:
5734:
5729:
5723:
5706:
5700:
5697:
5694:
5686:
5683:
5680:
5670:
5664:
5658:
5645:
5641:
5634:
5627:
5620:
5612:
5588:
5585:
5582:
5578:
5573:
5570:
5566:
5562:
5556:
5550:
5526:
5508:
5503:
5499:
5495:
5490:
5487:
5484:
5480:
5472:
5467:
5463:
5459:
5454:
5451:
5448:
5444:
5436:
5431:
5427:
5423:
5418:
5415:
5412:
5408:
5400:
5395:
5391:
5387:
5382:
5379:
5376:
5372:
5364:
5359:
5355:
5351:
5346:
5343:
5340:
5336:
5321:
5317:
5312:
5293:
5275:
5267:
5263:
5259:
5252:
5248:
5244:
5239:
5236:
5233:
5229:
5222:
5220:
5213:
5209:
5201:
5196:
5187:
5183:
5179:
5176:
5171:
5168:
5165:
5161:
5152:
5148:
5141:
5134:
5130:
5123:
5119:
5115:
5110:
5107:
5104:
5100:
5093:
5091:
5084:
5080:
5072:
5067:
5064:
5061:
5057:
5051:
5047:
5043:
5038:
5034:
5030:
5028:
5021:
5017:
5002:
4998:
4994:
4989:
4974:
4971:
4966:
4962:
4958:
4953:
4949:
4945:
4942:
4939:
4934:
4930:
4921:
4903:
4896:
4892:
4885:
4882:
4879:
4875:
4869:
4866:
4863:
4859:
4855:
4850:
4846:
4839:
4837:
4830:
4826:
4818:
4811:
4807:
4801:
4797:
4791:
4789:
4782:
4778:
4770:
4765:
4762:
4759:
4755:
4749:
4746:
4743:
4739:
4735:
4727:
4724:
4721:
4717:
4713:
4708:
4705:
4702:
4698:
4691:
4688:
4686:
4679:
4675:
4660:
4656:
4651:
4636:
4633:
4628:
4624:
4620:
4615:
4610:
4605:
4602:
4599:
4594:
4590:
4581:
4576:
4567:
4553:
4545:
4542:
4539:
4535:
4531:
4526:
4522:
4511:
4508:
4505:
4501:
4497:
4492:
4484:
4480:
4476:
4471:
4468:
4465:
4461:
4450:
4446:
4442:
4437:
4432:
4428:
4417:
4413:
4400:
4383:
4379:
4375:
4370:
4367:
4364:
4360:
4356:
4351:
4347:
4336:
4332:
4319:
4304:
4287:
4283:
4279:
4274:
4270:
4260:
4256:
4249:
4240:
4239:
4238:
4229:
4224:
4212:
4208:
4186:
4179:
4175:
4171:
4167:
4162:as a 5-tuple
4156:
4139:
4136:
4128:
4124:
4116:
4112:
4109:
4106:
4102:
4098:
4096:
4086:
4082:
4074:
4070:
4066:
4058:
4055:
4052:
4049:
4043:
4040:
4034:
4031:
4028:
4025:
4019:
4011:
4007:
3999:
3995:
3992:
3989:
3985:
3981:
3979:
3969:
3965:
3957:
3953:
3949:
3941:
3938:
3935:
3932:
3926:
3923:
3917:
3914:
3911:
3908:
3902:
3894:
3890:
3882:
3878:
3875:
3872:
3868:
3864:
3862:
3852:
3848:
3840:
3836:
3832:
3824:
3819:
3815:
3811:
3809:
3799:
3795:
3786:
3783:
3780:
3776:
3768:
3763:
3759:
3755:
3753:
3743:
3739:
3730:
3726:
3718:
3715:
3712:
3709:
3703:
3700:
3694:
3691:
3688:
3685:
3679:
3671:
3667:
3658:
3655:
3652:
3648:
3644:
3639:
3635:
3631:
3629:
3619:
3615:
3606:
3602:
3589:
3586:
3582:
3578:
3571:
3567:
3543:
3528:
3524:
3520:
3515:
3511:
3504:
3501:
3498:
3495:
3492:
3484:
3480:
3476:
3471:
3467:
3460:
3452:
3448:
3444:
3439:
3435:
3421:
3418:
3409:
3396:
3391:
3381:
3377:
3373:
3370:
3362:
3358:
3354:
3349:
3339:
3335:
3331:
3328:
3320:
3316:
3312:
3304:
3300:
3296:
3293:
3285:
3281:
3277:
3272:
3268:
3264:
3258:
3250:
3246:
3231:
3228:
3215:
3212:
3206:
3199:
3196:
3192:
3185:
3178:
3175:
3166:
3163:
3158:
3153:
3151:
3147:
3146:linear spline
3143:
3142:step function
3138:
3130:
3123:
3106:
3103:
3100:
3097:
3094:
3088:
3083:
3079:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3051:
3043:
3039:
3035:
3033:
3025:
3019:
3012:
3009:
3006:
3003:
3000:
2994:
2989:
2985:
2981:
2978:
2975:
2972:
2969:
2966:
2960:
2952:
2948:
2944:
2942:
2934:
2928:
2921:
2918:
2915:
2912:
2909:
2903:
2898:
2894:
2890:
2887:
2884:
2881:
2878:
2872:
2864:
2860:
2856:
2854:
2846:
2840:
2828:
2825:
2808:
2805:
2802:
2799:
2796:
2790:
2785:
2781:
2777:
2774:
2771:
2768:
2765:
2759:
2751:
2747:
2743:
2741:
2733:
2727:
2720:
2717:
2714:
2711:
2708:
2702:
2699:
2696:
2693:
2687:
2679:
2675:
2671:
2669:
2661:
2655:
2648:
2645:
2642:
2639:
2636:
2630:
2625:
2621:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2593:
2585:
2581:
2577:
2575:
2567:
2561:
2549:
2545:
2541:
2534:
2527:
2512:
2502:
2483:
2480:
2477:
2471:
2468:
2464:
2451:
2445:
2442:
2436:
2430:
2420:
2414:
2408:
2400:
2395:
2391:
2387:
2367:
2349:
2345:
2341:
2336:
2333:
2330:
2326:
2322:
2319:
2316:
2311:
2308:
2305:
2301:
2297:
2292:
2289:
2286:
2282:
2278:
2275:
2272:
2267:
2263:
2259:
2254:
2250:
2246:
2243:
2240:
2235:
2231:
2227:
2222:
2218:
2214:
2211:
2208:
2203:
2199:
2195:
2190:
2186:
2174:
2172:
2164:
2159:
2152:
2148:
2128:
2120:
2117:
2114:
2110:
2106:
2101:
2097:
2086:
2083:
2080:
2076:
2072:
2067:
2063:
2059:
2056:
2052:
2048:
2042:
2036:
2025:
2021:
2007:
2001:
1997:
1990:
1984:
1980:
1973:
1969:
1958:
1954:
1941:
1928:
1905:
1901:
1892:
1887:
1881:
1875:
1868:
1862:
1852:
1846:
1844:
1838:
1834:
1806:
1802:
1797:
1784:
1780:
1773:
1769:
1763:
1746:
1738:
1734:
1720:
1716:
1707:
1703:
1699:
1697:
1687:
1683:
1669:
1665:
1656:
1653:
1650:
1646:
1637:
1630:
1622:
1618:
1606:
1598:
1594:
1590:
1588:
1578:
1574:
1562:
1554:
1551:
1548:
1544:
1536:
1528:
1524:
1512:
1504:
1500:
1496:
1494:
1484:
1480:
1468:
1460:
1457:
1454:
1450:
1437:
1434:
1427:
1422:
1401:
1397:
1362:
1358:
1353:
1344:
1317:
1313:
1308:
1304:
1301:
1292:
1288:
1284:
1279:
1274:
1258:
1256:
1252:
1248:
1242:
1232:
1228:
1224:. The vector
1223:
1210:
1204:
1187:
1182:
1178:
1174:
1171:
1168:
1163:
1160:
1157:
1153:
1146:
1140:
1132:
1129:
1126:
1122:
1118:
1116:
1108:
1102:
1095:
1093:
1085:
1080:
1076:
1072:
1069:
1066:
1061:
1057:
1050:
1044:
1036:
1032:
1028:
1026:
1018:
1012:
1005:
1000:
996:
992:
989:
986:
981:
977:
970:
964:
956:
952:
948:
946:
938:
932:
920:
891:
875:
872:
869:
865:
861:
856:
852:
845:
840:
836:
815:
798:
795:
790:
786:
782:
777:
774:
771:
767:
763:
760:
757:
752:
748:
744:
739:
735:
731:
728:
715:
711:
704:
696:
692:
688:
683:
680:
677:
673:
666:
658:
655:
652:
648:
644:
639:
636:
633:
629:
622:
619:
616:
608:
604:
600:
595:
591:
584:
576:
572:
568:
563:
559:
552:
546:
543:
540:
535:
527:
524:
521:
518:
515:
512:
509:
506:
503:
499:
491:
488:
485:
481:
477:
472:
468:
462:
449:
447:
423:
409:
406:
403:
397:
394:
386:
370:
353:
350:
347:
343:
332:
329:
321:
318:February 2009
311:
310:the talk page
307:
301:
299:
294:This article
292:
283:
282:
274:
272:
268:
263:
261:
257:
253:
249:
245:
241:
237:
233:
232:Pierre Bézier
229:
225:
220:
218:
213:
211:
207:
203:
199:
195:
191:
187:
183:
182:Robin Forrest
179:
174:
171:
170:step function
167:
166:sign function
162:
160:
156:
146:
137:
135:
131:
126:
124:
123:curve fitting
120:
116:
112:
108:
104:
101:subfields of
100:
95:
93:
89:
85:
81:
77:
76:interpolating
73:
69:
65:
61:
57:
49:
45:
39:
33:
19:
7476:
7471:
7444:
7434:
7433:
7402:
7401:
7390:
7389:
7377:
7376:
7359:
7352:
7338:
7327:
7316:
7309:
7302:
7295:
7288:
7277:
7266:
7175:
7160:
7156:
7149:
6940:
6933:
6930:
6912:
6901:
6894:
6886:
6879:
6859:
6852:
6847:
6810:
6805:
6802:
6463:
6455:vector space
6448:
6269:
5872:
5867:
5862:
5855:
5849:
5839:
5828:
5753:
5739:
5732:
5730:is a cubic (
5721:
5646:
5639:
5632:
5625:
5618:
5610:
5538:
5319:
5315:
5298:and call it
5000:
4996:
4992:
4658:
4654:
4574:
4415:
4411:
4334:
4330:
4258:
4254:
4247:
4227:
4222:
4210:
4209:
4200:is equal to
4181:
4177:
4173:
4169:
4165:
4157:
3590:
3587:
3580:
3576:
3569:
3562:
3410:
3237:
3229:
3167:
3161:
3156:
3154:
3145:
3139:
3128:
3124:
2829:
2826:
2550:
2543:
2539:
2532:
2525:
2518:
2501:spline curve
2500:
2396:
2389:
2385:
2376:is repeated
2368:
2175:
2173:knot vector
2170:
2162:
2154:
2150:
2143:
2023:
2019:
2005:
1999:
1995:
1988:
1982:
1978:
1971:
1964:
1956:
1952:
1942:
1891:spline space
1890:
1885:
1879:
1873:
1866:
1860:
1850:
1847:
1842:
1841:is called a
1836:
1832:
1782:
1778:
1771:
1767:
1764:
1438:
1429:
1425:
1420:
1399:
1395:
1293:
1286:
1282:
1277:
1272:
1259:
1254:
1250:
1246:
1245:is called a
1237:
1230:
1226:
1221:
1208:
1205:
921:
816:
450:
385:real numbers
339:
324:
315:
304:Please help
295:
264:
224:de Casteljau
221:
217:East Anglian
214:
190:World War II
175:
163:
152:
143:
140:Introduction
127:
110:
96:
59:
53:
43:
18:Spline curve
6878:(giving us
5728:flat spline
2029:. That is,
1854:, a degree
1423:of at most
1345:(at least)
1255:non-uniform
1247:knot vector
1220:are called
383:the set of
184:describes "
161:in Russia.
109:, the term
72:polynomials
56:mathematics
32:Flat spline
7489:Categories
7339:NA Digest,
7333:Epperson,
7328:SIAM News,
7282:References
5824:PostScript
5529:output_set
5300:output_set
4570:c, l, μ, z
4237:5-tuples.
4191:a, b, c, d
2548:for which
2383:times for
1343:smoothness
1206:The given
349:polynomial
300:to readers
277:Definition
244:Garabedian
206:M.A. Sabin
178:Schoenberg
115:parametric
78:problems,
7273:B-splines
7202:−
7184:∑
7119:−
7105:…
7052:−
7038:…
6968:∈
6824:B-splines
6808:splines.
6770:−
6752:∑
6709:−
6700:…
6672:≤
6640:−
6619:−
6605:⏟
6596:−
6582:⋯
6571:−
6553:⋯
6532:⏟
6515:⋯
6459:dimension
6380:−
6369:−
6250:−
6208:−
6194:−
6170:−
6134:−
6123:−
6080:−
6003:−
5982:−
5932:−
5921:−
5837:(NURBS).
5782:∈
5724:− 1
5684:−
5671:∈
5571:−
5563:∈
5245:−
5142:−
5116:−
5048:μ
5044:−
5003:– 2, …, 0
4883:−
4867:−
4856:−
4847:α
4779:μ
4763:−
4756:μ
4747:−
4736:−
4725:−
4714:−
4611:μ
4543:−
4532:−
4509:−
4493:−
4477:−
4429:α
4376:−
4226:| =
4110:−
4053:−
4041:…
3993:−
3936:−
3924:…
3876:−
3784:−
3713:−
3701:…
3656:−
3374:−
3332:−
3297:−
3104:≤
3098:≤
3073:−
3061:−
3004:≤
2973:−
2913:≤
2888:−
2882:−
2806:≤
2800:≤
2772:−
2712:≤
2640:≤
2618:−
2603:−
2472:∈
2334:−
2320:⋯
2309:−
2290:−
2276:⋯
2244:⋯
2212:⋯
2073:−
2060:−
2049:∈
1765:A vector
1654:−
1638:⋮
1552:−
1458:−
1305:∈
1175:≤
1169:≤
1161:−
1130:−
1096:⋮
1067:≤
987:≤
884:→
783:≤
775:−
764:≤
761:⋯
758:≤
745:≤
705:∪
681:−
667:∪
656:−
637:−
623:∪
620:⋯
617:∪
585:∪
525:−
516:…
444:ordered,
416:→
346:piecewise
260:B-splines
134:draftsmen
68:piecewise
7458:, SINTEF
6440:th knot.
6336:th knot.
6303:″
5318:= 0, …,
5306:splines
4657:= 1, …,
4414:= 1, …,
4409:and for
4405:of size
4333:= 0, …,
4328:and for
4324:of size
4257:= 0, …,
4252:and for
4245:of size
4117:″
4075:″
4000:″
3958:″
3883:′
3841:′
3579:= 0, …,
3560:splines
3200:″
3179:″
2515:Examples
2503:if both
2388:= 1, …,
2171:extended
2169:and its
1872:≤
1835:= 1, …,
1276:≤
446:disjoint
436:We want
352:function
240:Birkhoff
66:defined
64:function
7430:, 2007.
7322:Davis,
7305:, 1990.
7155:, ...,
6453:form a
5527:Output
5296:Splines
4218:, with
3150:polygon
1281:(or of
1251:uniform
1213:points
904:On the
296:may be
248:de Boor
236:Renault
228:Citroën
194:splines
186:lofting
149:History
97:In the
7452:, NTCC
7378:Theory
6924:, and
6829:using
6818:using
6270:where
4220:|
4189:where
4047:
4038:
3930:
3921:
3707:
3698:
2369:where
2141:where
1273:degree
246:, and
238:, and
202:Boeing
130:spline
111:spline
88:degree
60:spline
7312:1987.
7298:1967.
7260:and
6820:basis
5758:from
5535:Notes
4999:– 1,
2499:is a
1864:for
1777:, …,
1283:order
1236:, …,
1222:knots
119:curve
74:. In
62:is a
6874:and
6629:<
6556:<
6550:<
6495:<
5747:and
5621:+ 1)
5586:>
5313:For
4990:For
4922:Set
4652:For
4582:Set
4420:set
4339:set
4309:and
4262:set
3574:for
3010:<
2919:<
2718:<
2646:<
2521:, ,
2507:and
2017:and
1993:and
1950:and
1830:for
1405:and
1073:<
993:<
105:and
58:, a
7264:).
6904:= 3
5860:th
5735:= 3
5642:+ 3
5635:+ 2
5628:+ 1
5613:+ 1
5322:– 1
4661:– 1
4577:+ 1
4418:– 1
4337:– 1
4250:+ 1
4230:+ 1
3583:– 1
2535:= 2
2528:= 1
2392:– 1
1839:– 1
1823:at
1770:= (
1379:at
1294:If
1291:).
1289:+ 1
1229:= (
1211:+ 1
250:at
234:at
226:at
208:at
200:at
168:or
157:at
70:by
54:In
7491::
7443:,
7426:,
7337:,
7326:,
7275:.
7163:–2
7075::
6920:,
6712:2.
5751:.
5637:,
5630:,
4995:=
4975:0.
4207:.
4180:,
4176:,
4172:,
4168:,
4140:0.
3585:.
3216:0.
3152:.
3137:.
2397:A
2394:.
2153:–
2149:=
2026:+1
2002:+1
1985:−1
1959:+1
1785:–1
1436:)
1428:–
1402:–1
1257:.
919:,
828:.
387:,
262:.
242:,
230:,
212:.
117:)
7239:.
7236:)
7233:t
7230:(
7225:j
7221:P
7215:j
7211:c
7205:2
7199:k
7194:0
7191:=
7188:j
7172::
7170:t
7161:k
7157:c
7153:0
7150:c
7133:)
7130:t
7127:(
7122:2
7116:k
7112:P
7108:,
7102:,
7099:)
7096:t
7093:(
7088:0
7084:P
7073:t
7055:2
7049:k
7045:P
7041:,
7035:,
7030:0
7026:P
7003:]
6998:1
6995:+
6992:i
6988:t
6984:,
6979:i
6975:t
6971:[
6965:t
6955:t
6947:n
6943:)
6941:t
6939:(
6936:i
6934:P
6913:C
6902:n
6889:)
6882:)
6876:b
6872:a
6862:)
6855:)
6848:C
6837:)
6826:)
6806:C
6788:.
6783:i
6779:j
6773:2
6767:k
6762:1
6759:=
6756:i
6748:+
6745:n
6742:=
6739:d
6706:k
6703:,
6697:,
6694:1
6691:=
6688:i
6684:,
6681:1
6678:+
6675:n
6663:i
6659:j
6651:b
6648:=
6643:1
6637:k
6633:t
6622:2
6616:k
6612:j
6599:2
6593:k
6589:t
6585:=
6579:=
6574:2
6568:k
6564:t
6543:1
6539:j
6526:1
6522:t
6518:=
6512:=
6507:1
6503:t
6490:0
6486:t
6482:=
6475:a
6451:n
6438:i
6424:)
6417:i
6413:t
6409:(
6406:f
6383:1
6377:i
6373:t
6364:i
6360:t
6356:=
6351:i
6347:h
6334:i
6320:)
6315:i
6311:t
6307:(
6299:i
6295:S
6291:=
6286:i
6282:z
6256:)
6253:x
6245:i
6241:t
6237:(
6233:]
6227:6
6221:i
6217:h
6211:1
6205:i
6201:z
6187:i
6183:h
6178:)
6173:1
6167:i
6163:t
6159:(
6156:f
6149:[
6145:+
6142:)
6137:1
6131:i
6127:t
6120:x
6117:(
6113:]
6107:6
6101:i
6097:h
6091:i
6087:z
6073:i
6069:h
6064:)
6059:i
6055:t
6051:(
6048:f
6041:[
6037:+
6029:i
6025:h
6021:6
6014:3
6010:)
6006:x
5998:i
5994:t
5990:(
5985:1
5979:i
5975:z
5968:+
5960:i
5956:h
5952:6
5945:3
5941:)
5935:1
5929:i
5925:t
5918:x
5915:(
5910:i
5906:z
5899:=
5896:)
5893:x
5890:(
5885:i
5881:S
5868:x
5863:C
5858:i
5850:C
5810:.
5807:]
5804:b
5801:,
5798:a
5795:[
5790:1
5786:C
5779:)
5776:t
5773:(
5770:S
5749:b
5745:a
5740:C
5733:n
5722:n
5707:,
5704:]
5701:b
5698:,
5695:a
5692:[
5687:1
5681:n
5676:C
5668:)
5665:t
5662:(
5659:S
5649:n
5640:n
5633:n
5626:n
5619:n
5617:(
5611:n
5605:i
5603:t
5589:0
5583:m
5579:,
5574:m
5567:C
5560:)
5557:t
5554:(
5551:S
5541:n
5509:.
5504:i
5500:x
5496:=
5491:x
5488:,
5485:i
5481:S
5473:,
5468:i
5464:d
5460:=
5455:d
5452:,
5449:i
5445:S
5437:,
5432:i
5428:c
5424:=
5419:c
5416:,
5413:i
5409:S
5401:,
5396:i
5392:b
5388:=
5383:b
5380:,
5377:i
5373:S
5365:,
5360:i
5356:a
5352:=
5347:a
5344:,
5341:i
5337:S
5320:n
5316:i
5310:.
5308:S
5304:n
5276:.
5268:j
5264:h
5260:3
5253:j
5249:c
5240:1
5237:+
5234:j
5230:c
5223:=
5214:j
5210:d
5202:,
5197:3
5193:)
5188:j
5184:c
5180:2
5177:+
5172:1
5169:+
5166:j
5162:c
5158:(
5153:j
5149:h
5135:j
5131:h
5124:j
5120:a
5111:1
5108:+
5105:j
5101:a
5094:=
5085:j
5081:b
5073:,
5068:1
5065:+
5062:j
5058:c
5052:j
5039:j
5035:z
5031:=
5022:j
5018:c
5001:n
4997:n
4993:j
4972:=
4967:n
4963:c
4959:=
4954:n
4950:z
4946:;
4943:1
4940:=
4935:n
4931:l
4904:.
4897:i
4893:l
4886:1
4880:i
4876:z
4870:1
4864:i
4860:h
4851:i
4840:=
4831:i
4827:z
4819:,
4812:i
4808:l
4802:i
4798:h
4792:=
4783:i
4771:,
4766:1
4760:i
4750:1
4744:i
4740:h
4733:)
4728:1
4722:i
4718:x
4709:1
4706:+
4703:i
4699:x
4695:(
4692:2
4689:=
4680:i
4676:l
4659:n
4655:i
4637:0
4634:=
4629:0
4625:z
4621:=
4616:0
4606:,
4603:1
4600:=
4595:0
4591:l
4579:.
4575:n
4554:.
4551:)
4546:1
4540:i
4536:a
4527:i
4523:a
4519:(
4512:1
4506:i
4502:h
4498:3
4490:)
4485:i
4481:a
4472:1
4469:+
4466:i
4462:a
4458:(
4451:i
4447:h
4443:3
4438:=
4433:i
4416:n
4412:i
4407:n
4403:α
4384:i
4380:x
4371:1
4368:+
4365:i
4361:x
4357:=
4352:i
4348:h
4335:n
4331:i
4326:n
4322:h
4317:.
4315:n
4311:d
4307:b
4288:i
4284:y
4280:=
4275:i
4271:a
4259:n
4255:i
4248:n
4243:a
4235:n
4228:n
4223:C
4216:C
4204:j
4202:x
4197:t
4195:x
4187:)
4184:t
4182:x
4178:d
4174:c
4170:b
4166:a
4164:(
4160:S
4137:=
4134:)
4129:n
4125:x
4121:(
4113:1
4107:n
4103:S
4099:=
4092:)
4087:0
4083:x
4079:(
4071:0
4067:S
4059:,
4056:1
4050:n
4044:,
4035:,
4032:1
4029:=
4026:i
4020:,
4017:)
4012:i
4008:x
4004:(
3996:1
3990:i
3986:S
3982:=
3975:)
3970:i
3966:x
3962:(
3954:i
3950:S
3942:,
3939:1
3933:n
3927:,
3918:,
3915:1
3912:=
3909:i
3903:,
3900:)
3895:i
3891:x
3887:(
3879:1
3873:i
3869:S
3865:=
3858:)
3853:i
3849:x
3845:(
3837:i
3833:S
3825:,
3820:n
3816:y
3812:=
3805:)
3800:n
3796:x
3792:(
3787:1
3781:n
3777:S
3769:,
3764:0
3760:y
3756:=
3749:)
3744:0
3740:x
3736:(
3731:0
3727:S
3719:,
3716:1
3710:n
3704:,
3695:,
3692:1
3689:=
3686:i
3680:,
3677:)
3672:i
3668:x
3664:(
3659:1
3653:i
3649:S
3645:=
3640:i
3636:y
3632:=
3625:)
3620:i
3616:x
3612:(
3607:i
3603:S
3581:n
3577:i
3572:)
3570:x
3568:(
3565:i
3563:S
3558:n
3544:,
3539:]
3534:)
3529:n
3525:y
3521:,
3516:n
3512:x
3508:(
3505:,
3502:.
3499:.
3496:.
3493:,
3490:)
3485:1
3481:y
3477:,
3472:1
3468:x
3464:(
3461:,
3458:)
3453:0
3449:y
3445:,
3440:0
3436:x
3432:(
3427:[
3422:=
3419:C
3397:.
3392:3
3388:)
3382:j
3378:x
3371:x
3368:(
3363:j
3359:d
3355:+
3350:2
3346:)
3340:j
3336:x
3329:x
3326:(
3321:j
3317:c
3313:+
3310:)
3305:j
3301:x
3294:x
3291:(
3286:j
3282:b
3278:+
3273:j
3269:a
3265:=
3262:)
3259:x
3256:(
3251:j
3247:S
3213:=
3210:)
3207:b
3204:(
3197:S
3193:=
3189:)
3186:a
3183:(
3176:S
3162:C
3129:t
3127:2
3107:3
3101:t
3095:2
3089:,
3084:2
3080:t
3076:2
3070:t
3067:+
3064:1
3058:=
3055:)
3052:t
3049:(
3044:2
3040:P
3036:=
3029:)
3026:t
3023:(
3020:S
3013:2
3007:t
3001:1
2995:,
2990:2
2986:t
2982:+
2979:t
2976:6
2970:1
2967:=
2964:)
2961:t
2958:(
2953:1
2949:P
2945:=
2938:)
2935:t
2932:(
2929:S
2922:1
2916:t
2910:0
2904:,
2899:2
2895:t
2891:2
2885:2
2879:=
2876:)
2873:t
2870:(
2865:0
2861:P
2857:=
2850:)
2847:t
2844:(
2841:S
2809:3
2803:t
2797:2
2791:,
2786:2
2782:t
2778:+
2775:t
2769:2
2766:=
2763:)
2760:t
2757:(
2752:2
2748:P
2744:=
2737:)
2734:t
2731:(
2728:S
2721:2
2715:t
2709:1
2703:,
2700:t
2697:2
2694:=
2691:)
2688:t
2685:(
2680:1
2676:P
2672:=
2665:)
2662:t
2659:(
2656:S
2649:1
2643:t
2637:0
2631:,
2626:2
2622:t
2615:t
2612:4
2609:+
2606:1
2600:=
2597:)
2594:t
2591:(
2586:0
2582:P
2578:=
2571:)
2568:t
2565:(
2562:S
2546:)
2544:t
2542:(
2540:S
2533:t
2526:t
2509:Y
2505:X
2487:]
2484:b
2481:,
2478:a
2475:[
2469:t
2465:,
2460:)
2455:)
2452:t
2449:(
2446:Y
2443:,
2440:)
2437:t
2434:(
2431:X
2426:(
2421:=
2418:)
2415:t
2412:(
2409:G
2390:k
2386:i
2380:i
2378:j
2373:i
2371:t
2355:)
2350:k
2346:t
2342:,
2337:1
2331:k
2327:t
2323:,
2317:,
2312:1
2306:k
2302:t
2298:,
2293:2
2287:k
2283:t
2279:,
2273:,
2268:3
2264:t
2260:,
2255:2
2251:t
2247:,
2241:,
2236:2
2232:t
2228:,
2223:1
2219:t
2215:,
2209:,
2204:1
2200:t
2196:,
2191:0
2187:t
2183:(
2167:n
2157:i
2155:r
2151:n
2146:i
2144:j
2129:,
2126:]
2121:1
2118:+
2115:i
2111:t
2107:=
2102:i
2098:t
2094:[
2087:1
2084:+
2081:i
2077:j
2068:i
2064:j
2057:n
2053:C
2046:)
2043:t
2040:(
2037:S
2024:i
2020:t
2014:i
2012:t
2008:)
2006:t
2004:(
2000:i
1996:P
1991:)
1989:t
1987:(
1983:i
1979:P
1974:)
1972:t
1970:(
1967:i
1965:P
1957:i
1953:t
1947:i
1945:t
1929:.
1926:)
1922:t
1918:(
1912:r
1906:n
1902:S
1886:r
1880:t
1874:n
1867:t
1861:r
1856:n
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1827:i
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1807:i
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1798:C
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1775:1
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1747:.
1744:)
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1735:t
1731:(
1726:)
1721:i
1717:r
1713:(
1708:i
1704:P
1700:=
1693:)
1688:i
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1680:(
1675:)
1670:i
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1662:(
1657:1
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1631:,
1628:)
1623:i
1619:t
1615:(
1610:)
1607:1
1604:(
1599:i
1595:P
1591:=
1584:)
1579:i
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1571:(
1566:)
1563:1
1560:(
1555:1
1549:i
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1537:,
1534:)
1529:i
1525:t
1521:(
1516:)
1513:0
1510:(
1505:i
1501:P
1497:=
1490:)
1485:i
1481:t
1477:(
1472:)
1469:0
1466:(
1461:1
1455:i
1451:P
1432:i
1430:r
1426:n
1416:i
1414:r
1409:i
1407:P
1400:i
1396:P
1390:i
1388:t
1383:i
1381:t
1363:i
1359:r
1354:C
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1309:C
1302:S
1287:n
1278:n
1269:n
1264:i
1262:P
1243:)
1240:k
1238:t
1234:0
1231:t
1227:t
1217:i
1215:t
1209:k
1188:.
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1172:t
1164:1
1158:k
1154:t
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1144:)
1141:t
1138:(
1133:1
1127:k
1123:P
1119:=
1112:)
1109:t
1106:(
1103:S
1086:,
1081:2
1077:t
1070:t
1062:1
1058:t
1051:,
1048:)
1045:t
1042:(
1037:1
1033:P
1029:=
1022:)
1019:t
1016:(
1013:S
1006:,
1001:1
997:t
990:t
982:0
978:t
971:,
968:)
965:t
962:(
957:0
953:P
949:=
942:)
939:t
936:(
933:S
916:i
914:P
910:S
906:i
892:.
888:R
881:]
876:1
873:+
870:i
866:t
862:,
857:i
853:t
849:[
846::
841:i
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825:i
823:P
819:k
799:b
796:=
791:k
787:t
778:1
772:k
768:t
753:1
749:t
740:0
736:t
732:=
729:a
721:]
716:k
712:t
708:[
702:)
697:k
693:t
689:,
684:1
678:k
674:t
670:[
664:)
659:1
653:k
649:t
645:,
640:2
634:k
630:t
626:[
614:)
609:2
605:t
601:,
596:1
592:t
588:[
582:)
577:1
573:t
569:,
564:0
560:t
556:[
553:=
550:]
547:b
544:,
541:a
538:[
528:1
522:k
519:,
513:,
510:0
507:=
504:i
500:,
497:]
492:1
489:+
486:i
482:t
478:,
473:i
469:t
465:[
442:k
438:S
424:.
420:R
413:]
410:b
407:,
404:a
401:[
398::
395:S
371:,
367:R
356:S
331:)
325:(
320:)
316:(
312:.
302:.
44:C
34:.
20:)
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