# NURBS knot multiplicity

In Splitting of NURBS curves there the answer relied on "maximum knot multiplicity". In order not to mix-up different topics I would like to kindly ask to answer it in another question: what is this "knot multiplicity" all about?

When you have a curve you can adjust the knots so that they lie on top of each other. This is essentially a bit like having several control points on top of each other, except there's only one point. This is sometimes known as multiplicity or duplicity.

When you have as many knots on top of each other as you have degrees in the curve smoothness, you end up with a cusp, also known as as sharp corner. Once you have a sharp corner you can just go and delete the points on the opposite side as they no longer affect points no the other side of the corner.

Image 1: Curve and control cage (Top) and basis functions (bottom). In this case the curve acts as 2 adjoining bezier curves, The image is using the knot vector [0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2].

If we look at a knot vector then the multiplicity looks like a seqence of n knots that have the same number. Note if you do not add ecactly as many knots on top as there is smoothness then you get a partially sharp corner.

Image 2: Animation with interpolation of knot form [0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2] to [0, 0, 0, 0, 0.34, 1, 1.66, 2, 2, 2, 2] and back. If you do proper point insertion then the curve does not change. Just talking about what multiplicity is.

Alternatively to changing parametrization you can just add points on top of each other. This is equivalent although slight misuse of resources, but useful in surface modeling. Having many control points on top could also be useful for the uniform nature of the knot distribution.

• When you say "knots lie on top of each other", you mean that their values don't change in the knot vector? So in your example [0, 0, 0, 0], [1, 1 ,1], [2, 2, 2] all "lie on top of each other"? Aug 27, 2015 at 12:33
• @EcirHana yes that is what i mean a more normal parametrisation would be [0, 0, 0, 0, 0.5, 1, 1.5, 2, 2, 2, 2] Ill add a animation Aug 27, 2015 at 12:45
• Just to check: so to split a NURBS curve the main task is to insert knots so that the overall shape remains unchanged, correct? Aug 27, 2015 at 13:28
• Yes, once you have enough knots you can delete whats on the other side. Aug 27, 2015 at 13:31
• There is a different strategy also you can insert one knot and delete the points that nolonger affect the knot you get a slightly different parametrisation though. I prefer this method for most of my own modeling. But nobody else seems to have heard of this. Aug 27, 2015 at 13:34