Theory Of A B-Spline Basis Function

Abstract

This paper extends the work of a previous paper of the author. A theoretical argument is provided to justify the heuristic algorithm used in the former paper. On the basis of the theory one derives, the previous algorithm can be further simplified. In the simplified basis function algorithm, the regular basis function (where $N_i^1(t)$ is 1 for $t_i \\le t \\lt t_{i + 1}$ and zero elsewhere) can be used for all cases except the case of the last point of a clamped B-spline where the basis function is modified to $N_{i,1} (t)$ where is 1 for $t_i \\le t \\le t_{i + 1}$ and zero elsewhere. Under this simplified algorithm, the knots ( i.e. , $t_{0}$ , $t_{1}, \\ldots, t_{n+k}$ ) are a k -extended partition in the interior of the knot vector with a possibility that two ends of the knot vector could be a $(k + 1)$ -extended partition (case of clamped B-spline). It is shown that given a set of $(n + 1)$ control points, $V_{0}$ , $V_{1}, \\ldots, V_{n}$ , the order of k , and the knots $(t_{0}, t_{1}, \\ldots, t_{n+k})$ , the B-spline $P(t) = \\sum_{i = 0}^{n} N_{i}^{k}(t)V_{i}$ is a continuous function for $t \\in [t_{k - 1}, t_{n + 1}]$ and maintains the partition of unity. This algorithm circumvents the problem of generating a spike at the last point of a clamped B-spline. The constraint of having k -extended partition interior knots overcomes the problem of disconnecting the B-spline at the k repeated knot.