Spline (mathematics)

In the mathematics subfield of numerical analysis a spline is a special differential geometry of curves defined piecewise by polynomials. The term spline comes from the flexible spline (device) devices used by shipbuilders and draftsmen to draw smooth shapes. In interpolation problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results even when using low degree polynomials, thus avoiding Runge's phenomenon. In curve fitting splines are used to approximate complex shapes. The simplicity of representation and the ease with which a complex spline's shape may be computed make splines popular representations for curves in computer science, predominantly in computer graphics. == Definition == Given ''k'' points ''t''_''i'' called knots in an interval [''a'',''b''] with :

a=t_0 < t_1 < \ldots < t_{k-2} < t_{k-1} = b

a parametric curve :

S:[a,b] \to \mathbb{R}

is called a spline of degree ''n'' if :

S \in \mathrm{C}^{n-1}(a,b)

and restricted to each subinterval :

S_{[t_i,t_{i+1})} \in \Pi_n \mbox{ , } i = 0,\ldots k-2

In other words on each subinterval or knot span :

[t_i,t_{i+1}) \mbox{ , } i = 0,\ldots k-2

''S'' is identical to a polynomial of degree ''n''. ''S''(''t''_''i'') is called knot value and (''t''_''i'', ''S''(''t''_''i'')) is called internal control point. (''t''₀,...,''t''_''k-1'') is called the knot vector. If the knots are equidistantly distributed in the Interval [''a'',''b''] we say the spline is uniform otherwise we say it is non-uniform. ==Example== The most simple spline has degree 1. It is also called linear spline and is just a polygon. A common spline is the natural cubic spline, of degree 3. Natural is defined by setting the second derivatives of the spline polynomials equal to each other at the endpoints of the interval of interpolation.

S''(a) = S''(b) = 0

on the interval

[a,b]

This forces the spline to be a straight line outside of the interval, while not disrupting its smoothness. ==Notes== For a given knot vector the splines of degree ''n'' form a vector space. A basis_(linear algebra) for this vector space are the basis B-splines of degree ''n'' ==History== Before computers were used, numerical calculations were done by hand. Although piecewise-defined functions like the signum function or step function were used, polynomials were generally preferred because they were easier to work with. With 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. == See also == *spline interpolation *Hermite spline *Cubic Hermite spline **Cardinal spline **Catmull-Rom spline **Kochanek-Bartels spline *B-spline *Nonuniform rational B-spline (NURBS) *the de Boor algorithm is an efficient method for evaluating B-splines *B�zier spline == External links == * [http://ibiblio.org/e-notes/Splines/Intro.htm An Interactive Introduction to Splines] *[http://www.vias.org/simulations/simusoft_spline.html Learning by Simulations] Interactive simulation of various cubic splines Splines

Spline (mathematics)

==Bezier curve not a spline ?== I did a complete rewrite of the article. A Bezier curve is not a spline. User:MathMartin 18:02, 19 Sep 2004 (UTC) What is the difference? 20 Mar 05 To explain what a spline is I think it is best to contrast splines with polynomials. Splines are then defined as piecewiese polynomials (of course you can consider a polynomial a spline with only one piece). After this difference is clear you can discuss different forms of the polynomials used to construct the spline (like Bernstein form, Hermite form, Monomial form etc.). So although in some sense a Bezier curve is a spline I think it is clearer not use it as a central example in the main spline article. User:MathMartin 16:04, 21 Mar 2005 (UTC)

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Spline_(mathematics)
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