Home > Spline (mathematics)

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.

1 Definition

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

a parametric curve

is called a spline of degree n if

and restricted to each subinterval

In other words on each subinterval or knot span

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.

2 Example

The most simple spline has degree 1. It is also called linear spline and is just a polygon.

3 Notes

For a given knot vector the splines of degree n form a vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for. A basisAbstract algebra Algebra Linear algebra In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V''. B is a mi for this vector space are the basis B-splines of degree n

4 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.

5 See also

• spline interpolation
• Hermite splineIn the mathematical subfield of numerical analysis a Hermite spline is a spline curve where each polynomial of the spline is in Hermite form. See also cubic Hermite spline Hermite polynomials Numerical analysis.
• Cubic Hermite splineIn the mathematical subfield of numerical analysis a cubic Hermite spline named in honor of Charles Hermite (pronounced air MEET , is a third-degree spline with each polynomial of the spline in Hermite form. The Hermite form consists of two control points
  • Cardinal splineA cardinal spline is a cubic Hermite spline whose tangents are defined by the points and a tension parameter. Given n 1 points p . p to be interpolated with n cubic Hermite curve segments, for each curve we have a starting point p and an ending point p wi
  • 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