样条（spline）是什么？

1. 样条是函数，由多项式分段定义。
2. 样条通常是指分段定义的多项式参数曲线。

样条曲线的定义

样条曲线的分类

常用的样条有许多种，由它们的特征命名。以下列出其中几种：

• 由表达方式命名：

  • 如果样条是基曲线的线性组合， 则称为B样条(B-spline)
  • 如果每个子区间的多项式由伯恩施坦多项式(Bernstein polynomial)表达，则称为贝塞尔样条(Bézier splines)

• 由节点的特征命名:

  • 若使用单个节点，每个子区间长度相等且$C^{n-1}$连续，则称为均匀样条(uniform splines)
  • 若对子区间长度没有要求则称为非均匀样条(nonuniform splines)

• 由特殊条件限制命名：

  • 若要求在a与b二次导数为零则称为自然样条(natural splines)
  • 若要求样条曲线穿过实际数据点则称为插值样条(interpolating splines)

下面详细介绍几种样条：

B-spline

Definition

B 是基（basis）样条的缩略，B-spline是基曲线的线性组合。
B-样条是贝塞尔样条的一种一般化。
当节点数和多项式次数相等时，B-spline退化为Bézier spline。
A B-spline curve is a curve that consists of Bezier curves as segments.

Property

1. 样条包含在它的控制点的凸包中

B-spline vs. Bézier curve

1. B-splines are piecewise polynomials. The area of validity for each piece is limited by so called “knot points”. Usually some constraits are put at knot points, for example that we should have a continous curve, maybe also first and second derivatives should be the same there.
   
2. While a Bézier curve is defined by a single particular polynomial (a Bernstein polynomial), a spline is defined piecewise by polynomials (meaning you can even have a spline comprising Bézier curves。

How can we prove that a Bezier curve is a specific case of a B-spline curve by the definition of B-splines?

An informal sketch of a proof goes as follows.

A Bezier curve is a parametric curve C(t) = (x(t), y(t)) where x and y are each real-valued polynomials of some degree d. The polynomials x and y are each represented as linear combinations of Bernstein polynomials of degree d, which form a basis for the linear space of all polynomials of degree d.

Now a spline curve is a parametric curve whose x and y components are each piecewise polynomials of degree d. In general, there are n pieces, where n > 1, but we can also have the case where n = 1. In that special case, we have a Bezier curve; the x and y components of the spline curve are each just polynomials, which can be represented as linear combinations of Bernstein polynomials.

Furthermore, it can be proved, using the definition of B-spline (basis) functions of degree d, that if you let the underlying knot sequence be 0,…0, 1…1 — where the knots 0 and 1 each have multiplicity d + 1, then the B-spline basis functions are exactly the Bernstein polynomials.

A note on terminology: frequently spline curves are referred to as B-spline curves, as is done in the wording for this question. But in my answer, I reserve the term B-spline for a basis function, and a spline curve is then a linear combination of B-splines.

Bézier spline

这里注意区分Bézier spline和Bézier curve：Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier surfaces.

样条插值

样条插值就是分段低次多项式、在分段处具有一定光滑性的函数插值，它克服了高次多项式插值可能出现的振荡现象，具有较好的数值稳定性和收敛性，由这种插值过程产生的函数就是多项式样条函数。