> 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.

>> Dmitry

> Dmitry,

> 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

> doc.

> 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?

--

Dmitry Epstein

Northwestern University, Evanston, IL. USA

mitia(at)northwestern(dot)edu