Interpolation between Bezier curves
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From: deltaquattro on 16 Feb 2010 11:47 Hi, I have two different functions f(x) and g(x) described as plane Bezier curves, say of fourth degree: [f] =[(Pi_x,Pi_y)] [g] =[(Qi_x,Qi_y)] with i=1..5. I would like to compute some samples of a third curve h(x)=f(x)/g(x). Right now, I'm computing h by calculating f and g at many points, say 21 equispaced values of the parameter t: t=[0,....19/20,1] -> (x_i,f(x_i)); (y_i,g(y_i)) Clearly, x_i and y_i will not in generally coincide, so I interpolate the values of g at the abscissae of f and I compute 20 samples of h as (x_i, f(x_i)/g_interp(x_i)). However, I have two problems: 1) since I have the Bezier nodes, I'd like to somewhat interpolate them and then compute h, instead than evaluating f and g on a refined grid and then computing h. It seems to me that I'm wasting cpu time and information. Is there a way to do that? 2) even if I stick to my expensive sampling -> interpolation procedure, it won't always work. The reason is that, because of possible errors in the input, sometimes P1_x <> Q1_x or P5_x <> Q5_x i.e., f and g are not defined exactly on the same interval. So I am not able to interpolate the g samples to the abscissae of the f samples. How can I solve this? Thanks, Best Regards, deltaquattro
From: Gordon Sande on 16 Feb 2010 12:56 On 2010-02-16 12:47:26 -0400, deltaquattro <deltaquattro(a)gmail.com> said: > Hi, Both numerical analysis and graphics have specialized newsgroups that are a better match to what you seem to be asking. > I have two different functions f(x) and g(x) described as plane Bezier > curves, say of fourth degree: > > [f] =[(Pi_x,Pi_y)] > [g] =[(Qi_x,Qi_y)] So f and g are paths in the plane. They will be multivalued and otherwise ill behaved as functions y = f(x) or y = g(x). The Pi_x are not monotone so this is not a usual way to tabulate a function which one would expect to be single valued in any case. Or at least that is what one expects from the comment that they are plane Bezier curves. > with i=1..5. I would like to compute some samples of a third curve > h(x)=f(x)/g(x). Right now, I'm computing h by calculating f and g at > many points, say 21 equispaced values of the parameter t: What is t (other than arc length along the Bezier curve) and what is x as a function of t? Is it single valued? Same for y. But why isn't it (x_i,f(x_i)) and (x_i,g(x_i)), From your set up it seems y=f(x) or y=g(x) so why the change of notation which just confuses the reader (and makes the reader wonder if there are other confusions). One is curious about why the ratio is of interest. That is a question about how your problem arises. Perhaps there is an alternate solution to the actual problem. There are lots of instances of good answers to poor questions which end up being poor annswers in substance. Have you head about the patient who died when it was an completely successful operation for the sugeon! > t=[0,....19/20,1] -> (x_i,f(x_i)); (y_i,g(y_i)) > > Clearly, x_i and y_i will not in generally coincide, so I interpolate > the values of g at the abscissae of f and I compute 20 samples of h as > (x_i, f(x_i)/g_interp(x_i)). However, I have two problems: > > 1) since I have the Bezier nodes, I'd like to somewhat interpolate > them and then compute h, instead than evaluating f and g on a refined > grid and then computing h. It seems to me that I'm wasting cpu time > and information. Is there a way to do that? What is your hourly rate? (Zero if you are a graudate student according to the behaviour of the management! :-( ) What is the hourly cost of your computer? How many hours of yoour time will save how much computer time? > 2) even if I stick to my expensive sampling -> interpolation > procedure, it won't always work. The reason is that, because of > possible errors in the input, sometimes > > P1_x <> Q1_x or P5_x <> Q5_x > > i.e., f and g are not defined exactly on the same interval. So I am > not able to interpolate the g samples to the abscissae of the f > samples. How can I solve this? Thanks, If you just had numerical tables giving f and g then one would fit some form (low order spline would do well) and then evaluate the rational function on whatever set of points you wanted. But you have said enough side tracking other things to wonder what it is that you actually want to do. > Best Regards, > > deltaquattro
From: GianLuigi Piacentini on 16 Feb 2010 14:46 deltaquattro wrote: > Hi, > > I have two different functions f(x) and g(x) described as plane Bezier > curves, say of fourth degree: > > [f] =[(Pi_x,Pi_y)] > [g] =[(Qi_x,Qi_y)] > > with i=1..5. I would like to compute some samples of a third curve > h(x)=f(x)/g(x). Right now, I'm computing h by calculating f and g at > many points, say 21 equispaced values of the parameter t: > > t=[0,....19/20,1] -> (x_i,f(x_i)); (y_i,g(y_i)) > > Clearly, x_i and y_i will not in generally coincide, so I interpolate > the values of g at the abscissae of f and I compute 20 samples of h as > (x_i, f(x_i)/g_interp(x_i)). However, I have two problems: > > 1) since I have the Bezier nodes, I'd like to somewhat interpolate > them and then compute h, instead than evaluating f and g on a refined > grid and then computing h. It seems to me that I'm wasting cpu time > and information. Is there a way to do that? > 2) even if I stick to my expensive sampling -> interpolation > procedure, it won't always work. The reason is that, because of > possible errors in the input, sometimes > > P1_x <> Q1_x or P5_x <> Q5_x > > i.e., f and g are not defined exactly on the same interval. So I am > not able to interpolate the g samples to the abscissae of the f > samples. How can I solve this? Thanks, > > Best Regards, > > deltaquattro You could extrapolate the curves a little, so that their definition interval will be equal (I think that the book of Farin or that of Mortenson explains the math). Many years ago, when I watched colleagues working with similar problems, I remember that they used splines defined as y=f(x). I used piecewise linear interpolation (again in the form y=f(x)) since its precision was enough given the uncertainities in the data. Given that you want h(x), my opinion is that there is no point in having x as a (parametric) function, instead of the independent variable. Hoping this helps... G.L. Piacentini
From: Arjan on 17 Feb 2010 15:17 Don't extrapolate! This will lead to disasters! Get the interval that is covered by both functions and limit h to that domain! The Bezier coefficients may have been estimated using a criterion that will give you f and g with a certain precision, but this does not guarantee precision of h! Division can lead to asymtots. How do you expect to catch them? In case you estimated those Bezier coefficients yourself, possibly because f and g are too time-consuming to calculate from their definition, isn't it possible to use the original functions behind f and g just once again, in your construction of a new array of values for h? ;-) Arjan
From: deltaquattro on 19 Feb 2010 08:42
On 16 Feb, 18:56, Gordon Sande <g.sa...(a)worldnet.att.net> wrote: > On 2010-02-16 12:47:26 -0400, deltaquattro <deltaquat...(a)gmail.com> said: > > > Hi, > > Both numerical analysis and graphics have specialized newsgroups that > are a better match to what you seem to be asking. No. I crossposted to both comp.lang.fortran and sci.math.num-analysis because I felt that Fortran programmers may be familiar with such problems, and I was right since all the answers came from comp.lang.fortran. > > > I have two different functions f(x) and g(x) described as plane Bezier > > curves, say of fourth degree: > > > [f] =[(Pi_x,Pi_y)] > > [g] =[(Qi_x,Qi_y)] > > So f and g are paths in the plane. They will be multivalued and otherwise > ill behaved as functions y = f(x) or y = g(x). The Pi_x are not monotone > so this is not a usual way to tabulate a function which one would expect > to be single valued in any case. Or at least that is what one expects from > the comment that they are plane Bezier curves. They're not multivalued and the Pi_x, Qi_x are monotone. f and g are functions: they are only described as plane Bezier curves, which means that their graph (a curve in the plane) has been represented by interpolation with a plane Bezier curve. I thought that would be clear from the context, and since the other repliers understood me, that means that I was at least not too obscure. > > > with i=1..5. I would like to compute some samples of a third curve > > h(x)=f(x)/g(x). Right now, I'm computing h by calculating f and g at > > many points, say 21 equispaced values of the parameter t: > > What is t (other than arc length along the Bezier curve) and what is > x as a function of t? Is it single valued? Same for y. It's not the arc length, it's just the parameter of the Bernstein representation: f: x(t)=sum(i=0,...,4)Pi_x*B(t)^n_i y(t)=sum(i=0,...,4)Pi_y*B(t)^n_i g: x(t)=sum(i=0,...,4)Qi_x*B(t)^n_i y(t)=sum(i=0,...,4)Qi_y*B(t)^n_i where B(t)^n_i are the Bernstein polynomials. In both cases x(t) is a single valued function of x. > it (x_i,f(x_i)) and (x_i,g(x_i)), From your set up it seems y=f(x) or > y=g(x) so why the change of notation which just confuses the reader > (and makes the reader wonder if there are other confusions). > > One is curious about why the ratio is of interest. That is a question > about how your problem arises. Perhaps there is an alternate solution ? f(x) and g(x)) are two physical quantities which are given, and I have to calculate the graphs of another quantity which is their ratio. > to the actual problem. There are lots of instances of good answers to > poor questions which end up being poor annswers in substance. Have you > head about the patient who died when it was an completely successful > operation for the sugeon! > > > t=[0,....19/20,1] -> (x_i,f(x_i)); (y_i,g(y_i)) > > > Clearly, x_i and y_i will not in generally coincide, so I interpolate > > the values of g at the abscissae of f and I compute 20 samples of h as > > (x_i, f(x_i)/g_interp(x_i)). However, I have two problems: > > > 1) since I have the Bezier nodes, I'd like to somewhat interpolate > > them and then compute h, instead than evaluating f and g on a refined > > grid and then computing h. It seems to me that I'm wasting cpu time > > and information. Is there a way to do that? > > What is your hourly rate? (Zero if you are a graudate student according > to the behaviour of the management! :-( ) What is the hourly cost of your > computer? How many hours of yoour time will save how much computer time? I've never been asked such a question when coming to a ng for help. Can we keep the discussion to the point, please? > > > 2) even if I stick to my expensive sampling -> interpolation > > procedure, it won't always work. The reason is that, because of > > possible errors in the input, sometimes > > > P1_x <> Q1_x or P5_x <> Q5_x > > > i.e., f and g are not defined exactly on the same interval. So I am > > not able to interpolate the g samples to the abscissae of the f > > samples. How can I solve this? Thanks, > > If you just had numerical tables giving f and g then one would fit > some form (low order spline would do well) and then evaluate the rational > function on whatever set of points you wanted. But you have said enough > side tracking other things to wonder what it is that you actually want to do. > Thanks for the suggestion. Regards deltaquattro |