why do the nurbs describe the conical and b-spline well?

[lucaam86]

Hello everyone,
as from the title of the post I would like to better understand why the nurbs effectively describe the conicals (ellips, circumferences, parables, hyperboles) while the b-splines do not.
I know it's a mathematical question about rational polynomials, but is there someone who could give me an example to better understand this concept?
Thank you.
♪

 

[Counterblow Hammer]

b-spline are splines made up of a series of bezier curves combined with each other in which continuity constraints are imposed at junction points (which I can vary from class c0 to class c2). However, they retain the limits of bezier curves, i.e. the impossibility to approximate the conicals as the polynomials of bernestein that describe the individual bezier curves have the equispaced knots on the carrier of the nodes themselves.

the nurbs (not uniform ration bezier spline) are born taking into account that they must pass through the interpolar points, therefore the condition of uniformity of the nodes comes to decay. to make this happen the function of mixing the curve (i.e. the one that is multiplied by the polynomials of benstein) is weighed (to avoid the phenomenon of runge when the points are too close) and is given by the ratio of two polynomials (to allow the condition of non-equispace between the knots).


if you do a search on the internet you will find slides much more explanatory than the rapid nod from me now proferred. :finger:

 

[lucaam86]

b-spline are splines made up of a series of bezier curves combined with each other in which continuity constraints are imposed at junction points (which I can vary from class c0 to class c2).

However, they retain the limits of bezier curves, i.e. the impossibility to approximate the conicals as the polynomials of bernestein that describe the individual bezier curves have the equispaced knots on the carrier of the nodes themselves.

first of all thanks to the answer not easy to get. I must say that you explained a lot more to me with this short speech than the slides I had read on the internet (maybe too complex for me, I am a three-year degree).
When you say "the impossibility to approximate the conicals" do you also refer to the circumference? or from the ellipse onwards?

 

[Counterblow Hammer]

However both the discourse to approximate a circle with b-splines (or bezier curves) is in the fact that if you did not notice you run the risk of not creating a good "fitting" of the circumference itself (i.e. the approximation error becomes no more negligible!); Such a problem could be a black beast especially in case you want to extrapolate from a b-spline (or bezier curve) that approximates a circumference arc.

If I wasn't clear, let me know that I try to explain myself better. . .

 

[lucaam86]

However both the discourse to approximate a circle with b-splines (or bezier curves) is in the fact that if you did not notice you run the risk of not creating a good "fitting" of the circumference itself (i.e. the approximation error becomes no more negligible!); Such a problem could be a black beast especially in case you want to extrapolate from a b-spline (or bezier curve) that approximates a circumference arc.

If I wasn't clear, let me know that I try to explain myself better. . .

Thank you again.
I have to tell you that I have not understood very well but not because you have not explained well but because I have some limits in this sense.. .

 

[Counterblow Hammer]

Thank you again.
I have to tell you that I have not understood very well but not because you have not explained well but because I have some limits in this sense.. .

to approximate a circumference arc with a bezier curve it is necessary to use a polynomial of bernestein at least grade 4, if not higher, to make sure that the "fitting" (i.e. approximation) is acceptable.
and so far there is nothing difficult (if not verifying why it is necessary that the degree of bernestein polynomial is at least 4).
as you will know from numerical analysis the high degree polynomials suffer from the runge phenomena, that is, they become unstable by oscillating; on the bezier curve that approximates the newly drawn arc may be necessary to carry out extrapolation operations (possibly you can think of this as an extension of the curve): If the polynomial describing the curve suffers from the runge phenomena you will find yourself having an extrapolation that has too high approximation errors compared to the effects values that you would get by extending the starting arc.

Moreover, if the degree of polynomial is sufficiently high starting, oscillation phenomena may occur already in interpolation, thus making the bezier curve created useless (you must think that bezier curves are used to create surfaces: if the curve starts sucks the same fate will touch the surface from it originated).

 

[lucaam86]

to approximate a circumference arc with a bezier curve it is necessary to use a polynomial of bernestein at least grade 4, if not higher, to make sure that the "fitting" (i.e. approximation) is acceptable.
and so far there is nothing difficult (if not verifying why it is necessary that the degree of bernestein polynomial is at least 4).
as you will know from numerical analysis the high degree polynomials suffer from the runge phenomena, that is, they become unstable by oscillating; on the bezier curve that approximates the newly drawn arc may be necessary to carry out extrapolation operations (possibly you can think of this as an extension of the curve): If the polynomial describing the curve suffers from the runge phenomena you will find yourself having an extrapolation that has too high approximation errors compared to the effects values that you would get by extending the starting arc.

Moreover, if the degree of polynomial is sufficiently high starting, oscillation phenomena may occur already in interpolation, thus making the bezier curve created useless (you must think that bezier curves are used to create surfaces: if the curve starts sucks the same fate will touch the surface from it originated).

now it is all quite clear at least theoretically.
if I with rhino design a curve and imposed degree = 9 start to see that the whole curve, for each point of control that design, becomes strongly unstable. that is, every time I insert a new point of control of the curve all the portion of the "retront" curve becomes unstable (it moves continuously). Obviously this does not happen with grade = 2, for example.
is this, practically, the runge phenomenon we talk about?

 

[Counterblow Hammer]

This is the phenomenon of runge!. .