superfici nurbs - alcune domande

[CADEnrico]

Hi.

I am an electronic engineer and for work reasons I need to make a 3d design with a program that has nurbs surfaces as primitive drawing.

I have a few questions because I want to understand some things.

1) What kind of surfaces are the nurbs? rational functions of x,y,z? are polynomials?

2) the intersection in the space of 2 nurbs surfaces what type of line generates?

Hello and thank you,
best enrico

 

[Fulvio Romano]

Although you do not understand what you need this information, however:

1) non-uniform rational b-spline, i.e. basically a spline with some game you do not uniformity on the weights of the knots, this to make it more handy (if you need more details ask as well)

2) Of course... a b-spline, result of the equation that equals two polynomials in the same variables.

However, to model a nurbs you do not need to handle these concepts.

 

[CADEnrico]

Hi.

thanks for the info.


> However, to model a nurbs does not need to handle these concepts.

in principle do not serve. But knowing them is not sin

My questions arise from the fact that I have noticed that sometimes with rhino you can not keep the continuity g2 when fitting 2 nurbs.

Hi.
best enrico

 

[Fulvio Romano]

My questions arise from the fact that I have noticed that sometimes with rhino you can not keep the continuity g2 when fitting 2 nurbs.

or use the wrong commands, or the nurbs are dirty. continuity between surfaces is always possible.

for "sporty" I mean with too many isoparametrics, or with multiple points (think for example at the top of a cone).

 

[CADEnrico]

Hi.

> or use the wrong commands, or the nurbs are dirty. the continuity between
> surfaces are always possible.

You mean the continuity of the curvature?

do you mean that it is always possible to collect 3 or 4 nurbs with a nurbs that makes the overall surface of class g2?

Hi.
best enrico

 

[Fulvio Romano]

mathematically yes, if the surfaces are of sufficient degree and if there are no contradictory boundary conditions.

 

[CADEnrico]

Hi.

> mathematically yes, if the surfaces are of sufficient degree and if there is no
> are contradictory boundary conditions.

and how come every time I put a surface to close 3 or 4 surfaces zebra tells me it's not g2?

Hi.
best enrico

 

[flaviobrio88]

and how come every time I put a surface to close 3 or 4 surfaces zebra tells me it's not g2?

without image (still better with igs file) it is really difficult to answer this question

 

[Fulvio Romano]

Hi.

> mathematically yes, if the surfaces are of sufficient degree and if there is no
> are contradictory boundary conditions.

and how come every time I put a surface to close 3 or 4 surfaces zebra tells me it's not g2?

Hi.
best enrico

without image (still better with igs file) it is really difficult to answer this question

not only, but zebra does not need to verify continuity. Please do not always make this mistake, I have already corrected it several times on this forum.

One thing is mathematical continuity, there are appropriate tools to verify it (discussed, but on rhino I do not remember the command, but on alias there is check continuity). If you use zebra or any environmental check (I use chrome because I find myself better) you are limited by shade resolution and you can also escape g0 continuity (holes).

What is different is the concept of "class a", that is a "beautiful" surface. the beauty of a surface, also related to the complete model, is not a mathematically expressible concept. then environmental check is used to see if a surface "turns well". and a surface can turn well even if it has holes.

Clear?

 

[PaoloColombani]

I share everything that says fulvio, I add maybe just something.

1) What kind of surfaces are the nurbs? rational functions of x,y,z?

rational functions of (x,y,z,w). are to all effects four-dimensional equations.

1) are polynomials reports?

precisely because of having to reinterpret a four-dimensional equation in a homogeneous space 3d the nurbs is expressed as a ratio of polynomials.

In practice the nurbs is a four-dimensional b-spline.
a b-spline is a bézier curve connection.
a bézier is a polynomial function of bernstein expressed in paramentary form with the restriction (domain) of the parameter between [0,1].

the polynomial functions of bernstein therefore constitute the basic functions (basis functions or blending functions) of the equation of the b-spline and therefore also of the nurbs.

2) the intersection in the space of 2 nurbs surfaces what type of line generates?

not a line but more generally a curve, generate an always curve of nurbs type.

 

[Fulvio Romano]

rational functions of (x,y,z,w). are to all effects four-dimensional equations.

It's the kind of detail I'd missed, and that I'd look into it on request:finger:

more than equations are polynomials. more than "quadridimensional" are "to four parameters", such as four parameters of the axis-angle type or quaternions.
the notation to four parameters and redundant in the three dimensions, and the redundancy is exploited to "addomesticate" the behaviors a little too effervescent of the splines.

 

[flaviobrio88]

not only, but zebra does not need to verify continuity. Please do not always make this mistake, I have already corrected it several times on this forum.

One thing is mathematical continuity, there are appropriate tools to verify it (discussed, but on rhino I do not remember the command, but on alias there is check continuity). If you use zebra or any environmental check (I use chrome because I find myself better) you are limited by shade resolution and you can also escape g0 continuity (holes).

What is different is the concept of "class a", that is a "beautiful" surface. the beauty of a surface, also related to the complete model, is not a mathematically expressible concept. then environmental check is used to see if a surface "turns well". and a surface can turn well even if it has holes.

Clear?

look not to tell me that when I try to explain the difference between zebra and curvature they look at me as if I were a Martian

 

[PaoloColombani]

more than equations are polynomials. more than "quadridimensional" are "to four parameters"

fulvio excuse if dissent, on your first statement I think you will agree that a polynomial in itself could not be evaluated if not written in the form of equation, for the second statement I should say that it is equations written in parametric form it would be good not to confuse the dimesions with the parameters that are one for each direction, therefore a parameter for the curves (normally noted with u) and two surface parameters (usually noted with u,v).
the dimensions remain (x,y,z) for b-spline and (x,y,z,w) for the nurbs.

However, both our friend who threw the rock in the pond seems to have disappeared from the discussion.

 

[Fulvio Romano]

I think you will agree that a polynomial in itself could not be evaluated if not written in the form of equation,

Of course, but an equation could also be of the type sin(x)+cos(x)=0. If you call it polynomial at least you know what shape it has, do you think?

for the second statement I should say that dealing with equations written in parametric form it would be good not to confuse the dimesions with the parameters that are one for each direction, therefore a parameter for the curves (normally noted with u) and two surface parameters (usually noted with u,v).
the dimensions remain (x,y,z) for b-spline and (x,y,z,w) for the nurbs.

I don't understand that. parameters are the "dimensions" of the object (curve or surface) and are one or two (descriptive ones, then in polynomial you can use as many as you want, provided redundant). The dimensions are those of the "world in which the object is described", and I wouldn't shake the fourth dimension... do you think? or did I not understand what you mean?

However, both our friend who threw the rock in the pond seems to have disappeared from the discussion.

In fact.. .

 

[CADEnrico]

Hi.

However, both our friend who threw the stone in the pond seems to me to be
> missing from the discussion.

I have not disappeared, I was nursing my son with the bottle only now I have 5 minutes to write

Meanwhile I'm reading the book "an introduction to nurbs" by f. rogers.
and something I started to understand.

Thank you for your clear and precise contributions.

Hi.
best enrico

 

[PaoloColombani]

I don't understand that. parameters are the "dimensions" of the object ...

I think if you try to write the equation of the straight in parametric form you will be forced to make a distinction between the parameter (u) and the size (x,y,z).

The dimensions are those of the "world in which the object is described", and I wouldn't shake the fourth dimension... do you think? or did I not understand what you mean?

who establishes that the "world in which the object is described" should be uniquely a three-dimensional Euclidean space? nothing prohibits us from describing an object in a four-dimensional space. That said, we know that to represent a 3d object in a 2d system it is necessary to project its points on the plane, not very differently to represent a 4d object in a three-dimensional space it is necessary to project its points on a 3d hyperplane. the objective geometry allows this transformation in a fairly easy way using the homogeneous coordinates. I try to explain:

any point in space 4d can be written in Cartesian coordinates:
pi = (x)i, yi, zi, wi)

to make any point in space 3d you can express it in homogeneous coordinates:
p(h)i = (x)i, yi, zi(1)

...and since by multiplying the homogeneous coordinates of a point by the same ≠ 0 value does not change its position, you can write:
wp(h)i =wxi, wyi, wzi, wi= p(h)ithe equivalent euclide of these homogeneous coordinates is obtained by dividing the first three coordinates for the fourth and this is equivalent to projecting point 4d on the hyperplane w = 1.

all this can be done on the equation of the b-spline that written in this way:
c(u) = o o(da i=0 ad n)ni,p(u)pican be rewritten in homogeneous coordinates so:
c(u) = o o(da i=0 ad n)ni,p(u)[wpi♪

and as mentioned before, in a three-dimensional Euclidean space becomes:
c(u) = o o(da i=0 ad n)ni,p(u)wpi News o o(da i=0 ad n)ni,p(u)wi)

that is exactly the equation of the nurbs. the term w (w ≥ 1) stands for weight (weight). placing w = 1 the nurbs becomes a b-spline.


references:

kuang shene site at the mtu computer science department:http://www.cs.mtu.edu/~shene/courses/cs3621/notes/
(al mtu vi ha insegnato anche carl de boor!)

john fisher, john lowther and ching-kuang shene,
if you know b-splines well, you also know nurbs!
acm 35th annual sigcse technical symposium, norfolk, virginia, march 3-7, 2004, pp. 343-347.

 

[PaoloColombani]

I have not disappeared, I was nursing my son with the bottle only now I have 5 minutes to write

Meanwhile I'm reading the book "an introduction to nurbs" by f. rogers.
and something I started to understand.

compliments for both. I have seen the site of the book you mention, you can also download beautiful examples in c, very interesting.

 

[Fulvio Romano]

who establishes that the "world in which the object is described" should be uniquely a three-dimensional Euclidean space? nothing prohibits us from describing an object in a four-dimensional space.

I am an electronic engineer and for work reasons I need to make a 3d design with a program that has nurbs surfaces as primitive drawing.

Of course, we can hypothesize the euclideous space (considering also the trims), we can think of non-euclide geometries, such as elliptical spaces or hyperbolic spaces, we can also uncomfortable poincarè, or perhaps call in cause hausdorff, but what pro?

any point in space 4d can be written in Cartesian coordinates:
pi = (x)i, yi, zi, wi)

to make any point in space 3d you can express it in homogeneous coordinates:
p(h)i = (x)i, yi, zi, 1)

I don't want to, and I don't think you're confusing the homogeneous coordinates with "an extra dimension."
homogeneous coordinates, as well as the axis-angle system and unitary quaternion, are notations that do not change the substance of things, they simplify only writing making possible some mathematical operations.

I think that all this barrel and response is reduced to mere philosophical speculation of the type "who has it longer" (the brain of course! :tongue, and I don't know how well it can do to the discussion.

I don't think I have to prove my skills, as I don't think you have to prove them to me, since the forum is zeppo of our posts that show that a little with the numbers we know how to do.

So we can also stop here for me. What do you say?

 

[PaoloColombani]

I think that all this barrel and response is reduced to mere philosophical speculation...

Until mathematics is an opinion, I don't think that this can be solved in a simple philosophical question. I'm sorry, but you didn't just say, "and I wouldn't shake the fourth dimension... do you think? or did I not understand what you mean? »I meant you didn't like an answer?

I don't want to, and I don't think you're confusing the homogeneous coordinates with "an extra dimension."

I don't think I messed up something that isn't flour of my bag. I didn't invent the rational b-spline. I quote:
«let us reinterpret this 4d point as a point in 3d homogeneous space. its euclidean equivalent is obtained by dividing the first three coordinate values by the fourth (i.e., projecting a 4d point to the hyperplane w = 1).»this and other you find it in the references I left and maybe you can continue your contest privately with kuang shene or even with carl de boor.

Hi.