Knot multiplicity

Hi everybody,

as I've just started studying the theory of parametric curves and their representation by grasshopper algorithms, I would need to find an answer to two questions about knot vectors.

In many books (e.g. Piegl, Tiller - The NURBS Book) I've read that in a non-periodic B-spline curve and, consequently, in a NURBS curve, the first and the last knots should have degree+1 multiplicity, while in Rhino and Grasshopper this multiplicity equals the degree.

Second, I've found that, if N is the number of control points and p the degree of the curve,

number of knots = p + N +1,

not p + N -1 as I read in Rhino help or in Grasshopper docs.

I think I'm missing something important, please help me to understand better!

Thanks in advance,

Gabriele (Italy)

Replies to This Discussion

Permalink Reply by David Rutten on April 20, 2013 at 1:59am

Hi Gabriele,

I'm no Nurbs expert, you'll have to talk to Dale Lear or Chuck Welsh (dalel@mcneel.com & chuck@mcneel.com respectively) in the Seattle main office if you want a definitive answer. Wikipedia agrees with our use of knot-vectors; "The number of knots is always equal to the number of control points plus curve degree minus one."

I do not know where this difference comes from. If it's merely an implementation detail (optimization?) or whether there are different mathematical definitions for Nurbs curves.

David Rutten

david@mcneel.com

Poprad, Slovakia

Permalink Reply by Jon Mirtschin on April 20, 2013 at 2:05am

Different software store slightly different knot vectors as you've noticed. Revit for example does store the additional knots, Rhino doesn't. The plugins translating from one to the other should generate or ignore the additional when translating, the geometry should be identical.

Permalink Reply by Gabriele Adamo on April 20, 2013 at 3:58am

Hi David and Jon,

thanks very much for your answers. My aim is to re-create the most significant parametric curves and surfaces algorithms in grasshopper, following the standard (i.e. academic) definitions.

I'm just a GH newbie, and this is my way to introduce myself to this exciting and powerful piece of software.

Thanks again

Gabriele Adamo

gabrieleadamo@gmail.com

Palermo, Italy

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