Spline (mathematics)

In the mathematical subfield of numerical analysis a spline is a special curve defined piecewise by polynomials. The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes.

In interpolating 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 a k points t_i called knots in an interval [a,b] with

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

a parametric curve

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

is called a spline of degree n if

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

and restricted to each subinterval

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

In other words on each subinterval or knot span

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

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. (t₀,...,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.

Notes

For a given knot vector the splines of degree n form a vector space. A basis for this vector space are the basis B-splines of degree n

History

Before computers were used numerical calculations were done by hand. And 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. First as a replacment 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
• Bézier spline

de:Spline pt:Spline