I want to make a closed interpolated curve based on four points.

But i don’t know how the Rhino generated the curve, because with 4 points a infinite number of curves can be generated.

I’d like to ask, what is the principle of interpolated curve in Rhino (Grasshopper), is there any formula？

Please explain in detail, thank you very much！

There is only one degree 3 curve which will go through 4 points.

Thank you for your reply.

Can you suggest a mathematical formula ？

I should expand a bit. Assuming the curve will be parametric, then there are an infinite number of parameter values which can be assigned to each point. Rhino provides three choices for parameterization: uniform, chord and square root of chord. If the curve is rational with weighting factors then there also would be an infinite number of weights would could be assigned to each point. Rhino appears to not use weights other than 1 for interpolated curves. So yes, there are an infinite number of curves. Also, the number of spans could be increased over the minimum number needed to fit the NURBS though the points (number of spans needed = number of points - degree except for special cases). Or if the number of points equals the degree or less then there will also be an infinite number of curves.

So there are an infinite number of NURBS which will go through 4 points. But once the degree of the NURB, the parameterization and the weights are selected there is a unique NURB which will go through the points and have the minimum number of spans (assuming number of points is greater than the degree). The control point locations can be solved as a set of simultaneous equations.

What is the reason for your question? Is part of an assignment or an exam?

I got it.

When the parameters were determined ,the curve would be certain. Right?

I made an experiment with Rhino software and i put the results in one of my paper.

The reviewer asked me about this question, so i have to make it clear.

Could you give me some suggestion about replying this question exactly？

Thank you very much!

Sorry, I don’t provide exact answers for students.

Thank you all the same

Can you recommend some books?

Thank you!