 Spline (mathematics)

In mathematics, a spline is a sufficiently smooth piecewisepolynomial function. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using lowdegree polynomials, while avoiding Runge's phenomenon for higher degrees.
In computer graphics splines are popular curves because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design.
The term spline comes from the flexible spline devices used by shipbuilders and drafters to draw smooth shapes^{[citation needed]}.
The most commonly used splines are cubic spline, i.e., of order 3—in particular, cubic Bspline and cubic Bézier spline. They are common, in particular, in spline interpolation simulating the function of flat splines.
Contents
Definition
A spline is a piecewisepolynomial real function
on an interval [a,b] composed of k ordered disjoint subintervals [t_{i − 1},t_{i}] with
 .
The restriction of S to an interval i is a polynomial
 ,
so that
The highest order of the polynomials P_{i}(t) is said to be the order of the spline S. If all subintervals are of the same length, the spline is said to be uniform and nonuniform otherwise.
The idea is to choose the polynomials in a way that guarantees sufficient smoothness of S. Specifically, for a spline of order n, S is required to be continuously differentiable to order n1 at the interior points t_{i}: for all and all ,.
Derivation of a Cubic Spline interpolating between points
This is one of the most important uses of splines. The algorithm for this is given in the article Spline interpolation
Examples
A simple example of a quadratic spline (a spline of degree 2) is
for which S'(0) = 2.
A simple example of a cubic spline is
as
and
An example of using a cubic spline to create a bell shaped curve is the IrwinHall polynomials:
History
Before computers were used, numerical calculations were done by hand. Functions such as the step function were used but polynomials were generally preferred. With the advent of computers, splines first replaced polynomials in interpolation, and then served in construction of smooth and flexible shapes in computer graphics.^{[1]}
It is commonly accepted that the first mathematical reference to splines is the 1946 paper by Schoenberg,^{[2]} 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),^{[3]} Robin Forrest describes "lofting", a technique used in the British aircraft industry during World War II to construct templates for airplanes by passing thin wooden strips (called "splines") through points laid out on the floor of a large design loft, a technique borrowed from shiphull 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 replotted 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 J. C. Ferguson^{[4]} at Boeing and (somewhat later) by M.A. Sabin at British Aircraft Corporation.
The word "spline" was originally an East Anglian dialect word.^{[5]}
The use of splines for modeling automobile bodies seems to have several independent beginnings. Credit is claimed on behalf of de Casteljau at Citroën, Pierre Bézier at Renault, and Birkhoff,^{[6]} Garabedian, and de Boor at General Motors (see Birkhoff and de Boor, 1965),^{[7]} 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 General Motors resulted in a number of papers being published in the early 1960s, including some of the fundamental work on Bsplines.^{[8]}
Work was also being done at Pratt & Whitney Aircraft, where two of the authors of (Ahlberg et al., 1967)^{[9]} — the first booklength treatment of splines — were employed, and the David Taylor Model Basin, by Feodor Theilheimer. The work at General Motors is detailed nicely in (Birkhoff, 1990) and (Young, 1997).^{[10]} Davis (1997) summarizes some of this material.
See also
References
 ^ Epperson, History of Splines, NA Digest, vol. 98, no. 26, 1998.
 ^ Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math., vol. 4, pp. 45–99 and 112–141, 1946.
 ^ Bartels, Beatty, and Barsky, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, 1987.
 ^ Ferguson, James C, Multivariable curve interpolation, J. ACM, vol. 11, no. 2, pp. 221228, Apr. 1964.
 ^ "spline". Oxford English Dictionary. Oxford University Press. 2nd ed. 1989.
 ^ Birkhoff, Fluid dynamics, reactor computations, and surface representation, in: Steve Nash (ed.), A History of Scientific Computation, 1990.
 ^ Birkhoff and de Boor, Piecewise polynomial interpolation and approximation, in: H. L. Garabedian (ed.), Proc. General Motors Symposium of 1964, pp. 164–190. Elsevier, New York and Amsterdam, 1965.
 ^ Davis, Bsplines and Geometric design, SIAM News, vol. 29, no. 5, 1997.
 ^ Ahlberg, Nilson, and Walsh, The Theory of Splines and Their Applications, 1967.
 ^ Young, Garrett Birkhoff and applied mathematics, Notices of the AMS, vol. 44, no. 11, pp. 1446–1449, 1997.
Further reading
 Stoer; Bulirsch. Introduction to Numerical Analysis. Springer Science+Business Media. pp. 93–106. ISBN 0387904204.
 Chapra, Canale. Numerical Methods for Engineers (5th ed.).
External links
Theory
 Cubic Splines Module Prof. John H. Mathews California State University, Fullerton
 Spline Curves, Prof. Donald H. House Clemson University
 An Interactive Introduction to Splines, ibiblio.org
 Introduction to Splines, codeplea.com
Excel functions
 Open source Excel cubic spline User Defined Function
 SRS1 Cubic Spline for Excel  Free Excel cubic spline function (with utility to embed spline function code into any workbook)
Online utilities
 Online Cubic Spline Interpolation Utility
 Learning by Simulations Interactive simulation of various cubic splines
 Symmetrical Spline Curves, an animation by Theodore Gray, The Wolfram Demonstrations Project, 2007.
Computer code
 Notes, PPT, Mathcad, Maple, Mathematica, Matlab, Holistic Numerical Methods Institute
 various routines, NTCC
 Sisl: Opensource Clibrary for NURBS, SINTEF
 Closed Bezier Spline, C#, WPF, Oleg V. Polikarpotchkin
 Bezier Spline from 2D Points, C#, WPF, Oleg V. Polikarpotchkin
Categories: Splines
 Interpolation
Wikimedia Foundation. 2010.
Look at other dictionaries:
Spline — can refer to:* Flat spline, a device to draw curves * Rotating spline, a mating mechanism on a driveshaft. * Spline (mathematics), a mathematical function used for interpolation or smoothing. * Spline cord, a type of thin rubber cord used to… … Wikipedia
Spline interpolation — See also: Spline (mathematics) In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. Spline interpolation is preferred… … Wikipedia
SplineInterpolation — Beispiel eines Splines mit 8 Knoten Bei der Spline Interpolation versucht man, gegebene Stützstellen, auch Knoten genannt, mit Hilfe stückweise stetiger Polynome, genauer Splines, zu interpolieren. Während das Ergebnis einer Polynominterpolation… … Deutsch Wikipedia
spline — [splʌɪn] noun 1》 a rectangular key fitting into grooves in the hub and shaft of a wheel, especially one formed integrally with the shaft which allows movement of the wheel on the shaft. ↘a corresponding groove in a hub along which the key may … English new terms dictionary
List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia
Bspline — In the mathematical subfield of numerical analysis, a B spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. A fundamental theorem states that every spline function of a given… … Wikipedia
Kubisch Hermitescher Spline — In dem mathematischen Teilgebiet der Numerik wird unter einem kubisch hermiteschen Spline (auch cSpline genannt) ein Spline verstanden der zwischen n Kontrollpunkten interpoliert. Die Kontrollpunkte sind durch n − 1 Segmente verbunden, die aus… … Deutsch Wikipedia
Polyharmonic spline — In mathematics, polyharmonic splines are used for function approximation and data interpolation.They are very useful for interpolation of scattered datain many dimensions.Polyharmonic splines are a special case of radial basis functions andare… … Wikipedia
List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… … Wikipedia
List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… … Wikipedia