> in message
>> > Dmitry,
>> > you may have a look at the Fourier Series approach:
>> > http://www.fho-emden.de/~hoffmann/softpoly23062003.pdf
>> > The N point polyline (N-1 segments) is closed by a Bzier
>> > path and represented by two Fourier Series for x and y.
>> > Fourier Series can be interpolated for any M. An example
>> > is in the chapter "Between Contours" which describes
>> > interpolations between paths with originally N1,N2 points.
>> > The polylines can be self intersecting, no restrictions.
>> > Best regards --Gernot Hoffmann
>> Thanks. From what I could figure out, with the maximum
>> number of harmonics the interpolating curve simply connects
>> all of the dots, and with fewer harmonics it fits a smooth
>> line to the point set, sort of like the least squares method.
>> I can see how I could use this to reduce the number of
>> points in a polyline, but I am not sure how I could refine a
>> polyline in this manner.
> Actually I dont know what you are meaning by "refine".
> You can use MORE new points than originally given - thats
> the pleasant feature of the Fourier interpolation, because
> this is based on a function system. As you say - its a least
> squares method, and the function system is orthogonal with
> respect to the range 0..2*pi or the parameter s=0..1.
> And these new points can be taken from the path with deleted
> higher harmonics - the smoothed path.
> Unfortunately the method doesnt create equal segment lengths
> for equally spaced parameters s+ds .
> Its simple to make all but the last segment equally long, but
> the last fills the gap and is therefore shorter.
> The method is rather crude, increment s+ds until the segment
> is long enough. A more subtle method would iterate ds,
> starting by the last ds. This fails easily if the curve was
> "regular" before and is "irregular" later.
> An example and additionally the formulas are in the updated
> Equal segment lengths (including the last) require IMO a
> global iteration.
> Best regards --Gernot Hoffmann
how best to choose the cutoff frequency, with a generally arbitrary
number of points in the original polynom?
Northwestern University, Evanston, IL. USA