The Spline (mathematics) reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Spline (mathematics)

A spline is a smooth curve defined mathematically by two or more control points. The control points that lie on the curve are called knots.

Splines can be divided into two main categories:

• interpolating splines which pass through all of the control points
• approximating splines which pass near all of the control points

Usually, approximating splines are smoother curves since interpolating splines may have bumps near the knots. In the picture, the curve that passes through A, B, C, and D is an interpolating spline (specifically, a linear spline) and the curve that passes through A and D, but not B and C, is an approximating spline (specifically, a Bézier spline).

Within numerical analysis, interpolating splines are used for piecewise polynomial interpolation. An important characteristic of splines is that they are given by polynomials, but only piecewise: different polynomials may be used in different parts of a curve. A significant advantage of this approach is that it can follow a large, complex curve and still avoid Runge's phenomenon by using low degree polynomials. Cubic polynomials are the most common type used.

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 but also for other kinds of interpolation, such as smoothing of digital audio.

The term spline comes fom the spline gadget used by shipbuilders to draw smooth shapes.

Table of contents

1 Formal definition of polynomial spline 2 Interpolating splines 3 Approximating splines

Formal definition of polynomial spline

A function S is called a spline of degree k if:

1. The domain of S is an interval [a, b]
2. are all continuous functions on [a, b]
3. There are knots (t_i,y_i) such that a = t₀ < t₁ < ... < t_n = b and such that S is a polynomial of degree at most k on each subinterval .

In general, the continuity of a function f at s can be defined by the condition:

Interpolating splines

The interpolating splines include:

• Linear spline
• Quadratic spline
• Cubic spline
  • Natural cubic spline
  • Clamped cubic spline
  • Periodic cubic spline

The de Boor algorithm is an efficient method for evaluating an interpolating spline curve.

Approximating splines

The approximating splines include:

• Bézier spline
• B-spline
• Nonuniform rational B-spline (NURBS)
• Cubic Hermite spline
  • Cardinal spline
  • Catmull-Rom spline
  • Kochanek-Bartels spline
• Subbotin quadratic spline

The de Casteljau's algorithm is a recursive method to create Bézier splines.

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