Let us recall the definition of the B-spline basis functions as follows:
This set of basis functions has the following properties, many of which resemble those of Bézier basis functions.
This observation can be generalized easily. In fact, ignoring the change of knot span endpoints, to create a knot of multiplicity k, k-1 basis functions will be affected. One of them loses one knot span, a second of them loses two, a third of them loses three and so on.
The figures below show the basis functions of degree 5 where the left end and right end knots have multiplicity 6, while all knots in between are simple (Figure (a))). Figure (b) is the result of moving u_{5} to u_{6}. Those basis functions ended at u_{6} have fewer knot spans on which they are non-zero. Then, u_{4} and then u_{3} are moved to u_{6}, making u_{6} a knot of multiplicity 4 (Figure (c) and (d). Figure (e) shows the result after moving u_{2} to u_{6}, creating a knot of multiplicity 5.
(a) | (b) | |
(c) | (d) | |
(e) |
In the above figures, since the multiplicity of knot u_{6} are 1 (simple), 2, 3, 4 and 5, the numbers of non-zero basis function at u_{6} are 5, 4, 3, 2 and 1.