NURBS curve based on Control points,Weights & Knots

Hi Everyone

We are working on a project and the goal is to find a curve's function. While working on Nurbs, I figured out that we can define it with Control points, weight and Knots. OK! So the question pops up! Is there any mathematical equation for Nurbs?? I Put the points, weights and knots in the equation and get the curve? Afterall Nurbs modeling is based on mathematics so perhaps we can have its equation for research.

I found this paper related to my question

https://www.cs.ubc.ca/~heidrich/Papers/CHAMONIX.96.pdf

Fitting Uncertain Data with NURBS

Fitting of uncertain data, that is, tting of data points

that are subject to some error, has important applications for example

in statistics and for the evaluation of results from physical experiments.

Fitting in these problem domains is usually achieved with polynomial

approximation, which involves the minimization of an error at discrete

data points. Norms typically used for this minimization include the l1, l2

and l1 norms, which are chosen depending on the problem domain and

the expected type of error on the data points.

In this paper we describe how the l1 and l1 norms can be applied

to integral and rational B-spline tting as a linear programming problem.

This allows for the use of B-splines and NURBS for the tting of uncertain

data.

Fitting Uncertain Data with NURBS

Replies to This Discussion

Permalink Reply by David Rutten on September 24, 2015 at 3:56am

Sort of, nurbs curves are polynomial piece-wise curves. So (as I understand) the shape of the span between two knots is governed by three polynomial equations {f(t)=x, f(t)=y, f(t)=z}, where t is the curve parameter.

The exact formulation of the functions depends on the control-points positions, the weights and the degree.

Permalink Reply by Parametric House on September 24, 2015 at 4:11am

Thanks David

So there is a parametric equation for each span!? Unfortunately, I'm looking for a single equation for a specific arch so I guess going for Nurbs is not a good choice. Perhaps Genetic programming is the best solution for finding the equation.

Permalink Reply by David Rutten on September 24, 2015 at 5:38am

Oh yes, different pieces of the curve are governed by different equations.

What sort shape/equation are you looking to get, and how will you know when you get it?

Permalink Reply by Parametric House on September 24, 2015 at 6:03am

For example, we have a kind of arch called elliptical 6-5 in ancient Persian architecture.

The arch has been shown bellow

I'm trying to use Galapagos to write my equation by using the Genetic Programming Method. The parameters will be symbols and variables which I will further feed into my evaluate function.

https://en.wikipedia.org/wiki/Genetic_programming

I then divide my arch to points and check if the x and y components fit in the ∑|f(x)-y|=0 equation. I mean the goal for Galapagos will be to minimum the ∑! I can also go for a parametric equation like f(u,v) = x and g(u,v)= y. I'm in the early stages, deciding which way to go!

https://en.wikipedia.org/wiki/Genetic_programming

Permalink Reply by David Rutten on September 27, 2015 at 12:13pm

I'm not sure it will work, the problem with making an equation tree like that is that it is very discontinuous. Sure, just changing numbers into other (nearby) numbers will yield small changes in outcome, but replacing division with addition or with a Sin will change the outcome of the equation drastically. Galapagos can only work when small changes to variables result in small changes in results, it can handle a certain amount of suddenness, but this may well be pushing it.

Permalink Reply by Parametric House on September 29, 2015 at 2:00pm

Your right David. Because of the rapid changes in the equations, Galapagos can't reach the answer. The only solution I'm thinking of is making a database of equations with constants and then feeding them to Galapagos.

Permalink Reply by Parametric House on September 24, 2015 at 6:08am

For example look at this:

Single equation for the graph of a marijuana leaf