Let $\boldsymbol{s}:=\{s_{i}\mid i\in\mathbb{Z}\}$ be a sequence of nondecreasing real numbers ($s_{i}\leq s_{{i+1}}$). For each $i$, we recursively define a set of real-valued functions $N_{{i,j}}$ (for $j=0,1,\ldots$) as follows:

where

Definitions. Using the notations above:

1. 1.

   the sequence $\boldsymbol{s}$ is known as a knot sequence, and the individual term in the sequence is a knot;

2. 2.

   the functions $N_{{i,j}}$ are called the $i$-th B-spline basis functions of order $j$, and the recurrence relation is called the de Boor recurrence relation, after its discoverer Carl de Boor;

3. 3.

   given any non-negative integer $j$, the vector space $V_{j}(\boldsymbol{s})$ over $\mathbb{R}$, generated by the set of all B-spline basis functions of order $j$ is called the B-spline of order $j$. In other words, the B-spline $V_{j}(\boldsymbol{s}):=\operatorname{span}\{N_{{i,j}}(t)\mid i=0,1,\ldots\}$ over $\mathbb{R}$.

4. 4.

   Any element of $V_{j}(\boldsymbol{s})$ is a B-spline function of order $j$.

The $0$th order B-spline basis functions are nothing more than characteristic functions on half-open intervals (or the zero function if $s_{i}=s_{{i+1}}$). When $j=1,2,3$, the B-spline basis functions are said to be linear, quadratic, or cubic.

The calculations of the higher order B-spline basis functions can be easily understood by use of triangular difference tables. For example, to calculate $N_{{53}}$, one would use

Remarks.

• Each B-spline basis function $N_{{i,j}}$ is completely defined by the finite set $\{s_{i},\ldots,s_{{i+j+1}}\}$ of knots. Furthermore, it is

  1. (a)

     non-zero in the open interval $(s_{i},s_{{i+j+1}})$,

  2. (b)

     $N_{{i,j}}$ restricted to each subinterval $[s_{k},s_{{k+1}})$ is a polynomial function of degree $j$ for $k=i,\ldots,i+j$, and

  3. (c)

     identically $0$ outside the interval $[s_{i},s_{{i+j+1}})$.

• In $V_{j}(\boldsymbol{s})$, the non-zero B-spline basis functions of order $j$ are linearly independent. In other words, the set of all non-zero $N_{{i,j}}$ forms a basis for $V_{j}(\boldsymbol{s})$ and hence the name B-spline basis functions.

• Given a knot sequence $\boldsymbol{s}$, a knot $s_{i}$ is said to be knotted if $s_{i}=s_{{i+1}}$. It is a double knot if $s_{{i-1}}<s_{i}=s_{{i+1}}<s_{{i+2}}$. Triple knots and more generally $n$-multi knots are defined analogously. A knot sequence is knotted if it contains a knotted knot.

• If a knot sequence is not knotted, then it can be shown that the B-spline basis function $N_{{i,j}}$ is $C^{{j-1}}$ but not of $C^{j}$. For example, $N_{{i,1}}$ is continuous but not differentiable.

• If a knot sequence $\boldsymbol{s}$ is finite and not knotted, then $V_{j}(\boldsymbol{s})$ is finite dimensional. If $\boldsymbol{s}$ has $n$ knots, then $V_{j}(\boldsymbol{s})$ has dimension $n-j$ for $j<n$, and $0$ otherwise.

In most applications of B-splines, finite knot sequences are considered. A finite knot sequence $\{s_{0},\ldots,s_{n}\}$ is also known as a knot vector.

Let $\boldsymbol{s}=\{s_{0},\ldots,s_{n}\}$ be a knot vector. Let $P_{0},\ldots,P_{m}$ be $m+1$ points in $\mathbb{R}^{k}$ (usually $k=2$ or $3$). Then we may form a linear combination

where $P_{i}N_{{i,j}}(t)$ is the scalar multiplication of the scalar $N_{{i,j}}(t)$ by the vector $P_{i}$. This is possible only when there are at least as many $N_{{i,j}}$ (there are $n+1-j$ of them) as there are $P_{i}$ (there are $m$ of them). In other words, $m\leq n+1-j$. Set $p=n+1-m$. Then the function defined by

is called a B-spline curve with control points $P_{1},\ldots,P_{m}$. Note that $P(t)$ is completely determined by the control points and the knot vector $\boldsymbol{s}$, and each coordinate of $P(t)$ is a B-spline function in $V_{p}(\boldsymbol{s})$.

More to come…