Rebuild a straight line

Rhino Rhino for Windows
1

Hi everybody,
I just wonder… If I rebuild a straight line (2 points, degree 1) with let’s say 6 points en degree 3 the resulting controlpoints are not equally distributed. Why is that so?

Is there a rule behind this? Is there a way to rebuild a line like this with equally distributed points?

thanks, Tobias

2

Hi Tobias.
Just guessing here, not sure about that at all …
Maybe the line gets divided into 3 spans of equal length.
Distance from first to third point looks the same as from third to fourth and as from fourth to sixth.

3

Hi Emilio,
You are actually right that the curve gets equally divided except for the additional points at he end (2nd point from either side).
So the question remaining is what is the reason for the additional points at he end?

4

The points are there because the curve is now a degree-3 nurbs curve.
… And that is the CV arrangement for nurbs curves … :smile:
Sorry, I’m not able to explain that.

BTW I think that we have CVs at the joint points between the spans only under certain circumstances …
which again I’m not able to tell … :blush:
( I think we need to run ConvertToBeziers to check the length of the curve spans )

I believe Rajaa also talks about nurbs here:

5

Yeah, I have to admit that my knowledge about the math behind nurbs is not exactly profound…

But I found out, if I want to divide a curve equally with controlpoints I should rebuild it with one point more than the degree.
I still don’t know why though. Time to study again.

thanks, Tobias

6

Right - then the curve has no internal knots. Your rebuild to degree 3 and 6 points is two more points than the minimum for degree 3 curves and so it gets two internal knots. You get the same curve if you make a degree 3 InterpCrv with four equally spaced input points in line with each other.
In general, up to degree 5 anyway, it’s better (i.e. smoother curve) if you rebuild to a single span (Degree + 1 points)

-Pascal

7

Yes, that’s correct, AFAIK.
I thinks this is because the simplest nurbs curve has a number of CVs that is equal to its degree plus one.
( This one should be a Bezier curve, that is a nurbs curve composed by a single span )
Nurbs curve are defined in a way so that any additional CV we add to the curve (without changing the degree), the curve gets one more span.
Generally speaking, these spans must not have the same length … but I think Rebuild should give us same-length spans.

EDIT:
Sorry, I misunderstood your words and was talking about a nurbs curve (with more CVs), while you were referring to a bezier curve with equally spaced CVs (but then again, the CVs don’t have to be placed that way, we always have to check)
/EDIT

… But I’d better let you spend you time with reliable infos, like those in Rajaa’s text :wink:
Cheers

8

They should be, as far as I know- use ConvertToBeziers to check.

-Pascal

9

Thanks all for the info!
And Rajaa’s doc on nurbs math is quite excellent!

gr, Tobias