I have the following question. If I have a NURB curve where one of the interior knots has multiplicity higher than the degree of curve (I did not chose to have such curves, I am writing a code for importing CAD drawings and there is a possibility of such ill defined cases) what does this unnecessary multiplicity do to the control points. In the specific example I have, I see that it results in multiplicity of the control points at the those knot values. For example, if I have a curve of degree $p$, and a internal knot $u$ of multiplicity $p+1$ this results in control point coinciding with the curve value at $u$ to have multiplicity two. Now, would this always be the case? Or is it just a coincidence. Because if it is always the case, I can just remove the knot until I have multiplicity $p$ and remove the corresponding control point corresponding amount of times and get a well parametrized curve. In case this is not always true, how would you suggest me to reduce the knots to get a curve where the knot vector does not have internal knots of multiplicity higher than the curve degree? I though maybe one could do curve degree elevation, but then again, I would somehow have to add knots and the end of knot vector (for them to have multiplicity of $degree+1$) and this seems somehow redundant.
Thank you very much for your help and time, Donatas
I had encountered the exact problem a few times when importing DXF files. The problematic B-spline curve in DXF file is of degree 3 and has interior knots of multiplicity 4. Luckily for such B-spline curves I had encountered they always have coincident control point at the knot value. Therefore, I can simply remove the extra knot and ignore the duplicated control point during importing. When the control points are not coincident, the B-spline curve data does not really represent a single curve and you will have to import the B-spline curve as multiple curves.
Suppose the B-spline curve you want to import is of degree 3 with knot sequence $[0, 0, 0, 0, 0.5, 0.5, 0.5, 0.5, 1, 1, 1, 1]$. This curve would have 8 control points $P_0,P_1$ ~ $P_7$. When $P_3$ and $P_4$ are not identical, you can import it as two curves, with first curve defined by $P_0$~$P_3$ and 2nd one defined by $P_4$~$P_7$.