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Knot Multiplicity
Figure 2-11: All Third-Order B-Splines for a Certain Knot Sequence with Various Knot Multiplicities
For example, for a B-spline of order 3, a simple knot would mean two smoothness conditions, i.e., continuity of function and first derivative, while a double knot would only leave one smoothness condition, i.e., just continuity, and a triple knot would leave no smoothness condition, i.e., even the function would be discontinuous.
Figure 2-11 shows a picture of all the third-order B-splines for a certain mystery knot sequence t. The breaks are indicated by vertical lines. For each break, try to determine its multiplicity in the knot sequence (it is 1,2,1,1,3), as well as its multiplicity as a knot in each of the B-splines. For example, the second break has multiplicity 2 but appears only with multiplicity 1 in the third B-spline and not at all, i.e., with multiplicity 0, in the last two B-splines. Note that only one of the B-splines shown has all its knots simple. It is the only one having three different nontrivial polynomial pieces. Note also that you can tell the knot-sequence multiplicity of a knot by the number of B-splines whose nonzero part begins or ends there. The picture is generated by the following MATLAB statements, which use the command spcol from this toolbox to generate the function values of all these B-splines at a fine net x.
t=[0,1,1,3,4,6,6,6]; x=linspace(-1,7,81); c=spcol(t,3,x);[l,m]=size(c); c=c+ones(l,1)*[0:m-1]; axis([-1 7 0 m]); hold on for tt=t, plot([tt tt],[0 m],'-'), end plot(x,c,'linew',2), hold off, axis off
Further illustrated examples are provided by the demo spalldem.
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