9.2.5. Spline Curves References B-spline Basis

A nonuniform, nonrational B-spline of order k is a piecewise polynomial function of degree \(k - 1\) in a variable t.

A set of non-descending breaking points, called knot, \(t_0 \le t_1 \le \ldots \le t_m\) defines a knot vector \(T = (t_0, \ldots, t_{m})\).

If each knot is separated by the same distance h (where \(h = t_{i+1} - t_i\)) from its predecessor, the knot vector and the corresponding B-splines are called “uniform”.

Given a knot vector T, the associated B-spline basis functions, \(B_i^k(t)\) are defined as:

\[\begin{split}B_i^1(t) = \left\lbrace \begin{array}{l} 1 \;\textrm{if}\; t_i \le t < t_{i+1} \\ 0 \;\textrm{otherwise} \end{array} \right.\end{split}\]
\[\begin{split}\begin{split} B_i^k(t) &= \frac{t - t_i}{t_{i+k-1} - t_i} B_i^{k-1}(t) + \frac{t_{i+k} - t}{t_{i+k} - t_{i+1}} B_{i+1}^{k-1}(t) \\ &= w_i^{k-1}(t) B_i^{k-1}(t) + [1 - w_{i+1}^{k-1}(t)] B_{i+1}^{k-1}(t) \end{split}\end{split}\]


\[\begin{split}w_i^k(t) = \left\lbrace \begin{array}{l} \frac{t - t_i}{t_{i+k} - t_i} \;\textrm{if}\; t_i < t_{i+k} \\ 0 \;\textrm{otherwise} \end{array} \right.\end{split}\]

These equations have the following properties, for \(k > 1\) and \(i = 0, 1, \ldots, n\) :

  • Positivity: \(B_i^k(t) > 0\), for \(t_i < t < t_{i+k}\)
  • Local Support: \(B_i^k(t) = 0\), for \(t_0 \le t \le t_i\) and \(t_{i+k} \le t \le t_{n+k}\)
  • Partition of unity: \(\sum_{i=0}^n B_i^k(t)= 1\), for \(t \in [t_0, t_m]\)
  • Continuity: \(B_i^k(t)\) as \(C^{k-2}\) continuity at each simple knot B-spline Curve

A B-spline curve of order k is defined as a linear combination of control points \(p_i\) and B-spline basis functions \(B_i^k(t)\) given by

\[S^k(t) = \sum_{i=0}^{n} p_i\; B_i^k(t) ,\quad n \ge k - 1,\; t \in [t_{k-1}, t_{n+1}]\]

In this context the control points are called De Boor points. The basis functions \(B_i^k(t)\) is defined on a knot vector

\[T = (t_0, t_1, \ldots, t_{k-1}, t_k, t_{k+1}, \ldots, t_{n-1}, t_n, t_{n+1}, \ldots, t_{n+k})\]

where there are \(n+k+1\) elements, i.e. the number of control points \(n+1\) plus the order of the curve k. Each knot span \(t_i \le t \le t_{i+1}\) is mapped onto a polynomial curve between two successive joints \(S(t_i)\) and \(S(t_{i+1})\).

Unlike Bézier curves, B-spline curves do not in general pass through the two end control points. Increasing the multiplicity of a knot reduces the continuity of the curve at that knot. Specifically, the curve is \((k-p-1)\) times continuously differentiable at a knot with multiplicity \(p (\le k)\), and thus has \(C^{k-p-1}\) continuity. Therefore, the control polygon will coincide with the curve at a knot of multiplicity \(k-1\), and a knot with multiplicity k indicates \(C^{-1}\) continuity, or a discontinuous curve. Repeating the knots at the end k times will force the endpoints to coincide with the control polygon. Thus the first and the last control points of a curve with a knot vector described by

\[\begin{eqnarray} T = ( \underbrace{t_0, t_1, \ldots, t_{k-1},}_{\mbox{$k$ equal knots}} \quad \underbrace{t_k, t_{k+1}, \ldots, t_{n-1}, t_n,}_{\mbox{$n$-$k$+1 internal knots}} \quad \underbrace{t_{n+1}, \ldots, t_{n+k}}_{\mbox{$k$ equal knots}}) \end{eqnarray}\]

coincide with the endpoints of the curve. Such knot vectors and curves are known as clamped. In other words, clamped/unclamped refers to whether both ends of the knot vector have multiplicity equal to k or not.

Local support property: A single span of a B-spline curve is controlled only by k control points, and any control point affects k spans. Specifically, changing \(p_i\) affects the curve in the parameter range \(t_i < t < t_{i+k}\) and the curve at a point where \(t_r < t < t_{r+1}\) is determined completely by the control points \(p_{r-(k-1)}, \ldots, p_r\).

B-spline to Bézier property: From the discussion of end points geometric property, it can be seen that a Bézier curve of order k (degree \(k-1\)) is a B-spline curve with no internal knots and the end knots repeated k times. The knot vector is thus

\[\begin{eqnarray} T = ( \underbrace{t_0, t_1, \ldots, t_{k-1}}_{\mbox{$k$ equal knots}} ,\quad \underbrace{t_{n+1}, \ldots, t_{n+k}}_{\mbox{$k$ equal knots}} ) \end{eqnarray}\]

where \(n+k+1 = 2k\) or \(n = k-1\). Algorithms for B-spline curves Evaluation and subdivision algorithm

A B-spline curve can be evaluated at a specific parameter value t using the de Boor algorithm, which is a generalization of the de Casteljau algorithm. The repeated substitution of the recursive definition of the B-spline basis function into the previous definition and re-indexing leads to the following de Boor algorithm:

\[S(t) = \sum_{i=0}^{n+j} p_i^j B_i^{k-j}(t) ,\quad j = 0, 1, \ldots, k-1\]


\[p_i^j = \Big[1 - w_i^j\Big] p_{i-1}^{j-1} + w_i^j p_i^{j-1}, \; j > 0\]


\[w_i^j = \frac{t - t_i}{t_{i+k-j} - t_i} \quad \textrm{and} \; p_j^0 = p_j\]

For \(j = k-1\), the B-spline basis function reduces to \(B_l^1\) for \(t \in [t_l, t_{l+1}]\), and \(p_l^{k-1}\) coincides with the curve \(S(t) = p_l^{k-1}\).

The de Boor algorithm is a generalization of the de Casteljau algorithm. The de Boor algorithm also permits the subdivision of the B-spline curve into two segments of the same order. De Boor Algorithm

Let the index l define the knot interval that contains the position, \(t \in [t_l , t_{l+1}]\). We can see in the recursion formula that only B-splines with \(i = l-K, \dots, l\) are non-zero for this knot interval, where \(K = k - 1\) is the degree. Thus, the sum is reduced to:

\[S^k(t) = \sum _{i=l-K}^{l} p_{i} B_i^k(t)\]

The algorithm does not compute the B-spline functions \(B_i^k(t)\) directly. Instead it evaluates \(S(t)\) through an equivalent recursion formula.

Let \(d _i^r\) be new control points with \(d_i^1 = p_i\) for \(i = l-K, \dots, l\).

For \(r = 2, \dots, k\) the following recursion is applied:

\[ \begin{align}\begin{aligned}d_i^r = (1 - w_i^r) d_{i-1}^{r-1} + w_i^r d_i^{r-1} \quad i = l-K+r, \dots, l\\w_i^r = \frac{t - t_i}{t_{i+1+l-r} - t_{i}}\end{aligned}\end{align} \]

Once the iterations are complete, we have \(S^k(t) = d_l^k\).

De Boor’s algorithm is more efficient than an explicit calculation of B-splines \(B_i^k(t)\) with the Cox-de Boor recursion formula, because it does not compute terms which are guaranteed to be multiplied by zero. Knot insertion

A knot can be inserted into a B-spline curve without changing the geometry of the curve. The new curve is identical to

\[\begin{split}\begin{array}{lcl} \sum_{i=0}^n p_i B_i^k(t) & \textrm{becomes} & \sum_{i=0}^{n+1} \bar{p}_i \bar B_i^k(t) \\ \mbox{over}\; T = (t_0, t_1, \ldots, t_l, t_{l+1}, \ldots) & & \mbox{over}\; T = (t_0, t_1, \ldots, t_l, \bar t, t_{l+1}, \ldots) & & \end{array}\end{split}\]

when a new knot \(\bar t\) is inserted between knots \(t_l\) and \(t_{l+1}\). The new de Boor points are given by

\[\bar{p}_i = (1 - w_i) p_{i-1} + w_i p_i\]


\[\begin{split}w_i = \left\{ \begin{array}{ll} 1 & i \le l-k+1 \\ 0 & i \ge l+1 \\ \frac{\bar{t} - t_i}{t_{l+k-1} - t_i} & l-k+2 \le i \leq l \end{array} \right.\end{split}\]

The above algorithm is also known as Boehm’s algorithm. A more general (but also more complex) insertion algorithm permitting insertion of several (possibly multiple) knots into a B-spline knot vector, known as the Oslo algorithm, was developed by Cohen et al.

A B-spline curve is \(C^{\infty}\) continuous in the interior of a span. Within exact arithmetic, inserting a knot does not change the curve, so it does not change the continuity. However, if any of the control points are moved after knot insertion, the continuity at the knot will become \(C^{k-p-1}\), where p is the multiplicity of the knot.

The B-spline curve can be subdivided into Bézier segments by knot insertion at each internal knot until the multiplicity of each internal knot is equal to k.