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From: paul beach <sniffyraven@fa...>  20100731 22:28:08

Bezier curves for graphics, theory, and, sound wave applications are rather different. Perhaps this will be usefull in deciding which is which. ********** I don't see any reason to use a Bezier curve for smoothing. Global effects are generally unwanted for interpolation. But for wave synthesis, Bernstien polynomials may be just thing. Expert comment has to be in context. There seem to be a few flaws in the aurgument, even if it is only used for smoothing "1. Undesirable properties of Bézier curves are their numerical instability for large numbers of control points, and the fact that moving a single control point changes the global shape of the curve. 2. The former is sometimes avoided by smoothly patching together loworder Bézier curves." 1. The constants are determined by binomial coefficients, for large values, Sterling's aproximation might be used. 2. What the text says is true for a graphics package. It is called a bicubic patch. Each region may have a different color or reflection coefficient; and there may be thousands of squares stitched together like a quilt. Indeed! there may be numerical instability. There can be a three dimensional object in fourspace if you want to do research on the frontiers of mathematics, otherwise the version for wave files is the simplest case. A SURFACE in 3d: This is for graphics, as mentioned above. This might be of some use in modeling waves in a ripple tank, though the theory of interference patterns done by other means. A CURVE in 2d: This has two parameters, two control points in Windows paint. There can be many control points, sometimes called a spline. DO not use two parameters for sound files since there may be two or more values, for one value of time. If you want two parameters, use separate sound tracks. One parameter in 2d: There can be many control points. Over 130 may be awkward for Nyquist. Note that musical instrument waveforms are sometimes called beat frequencies, multiple tracks might be appropriate. Is it HARD? A plugin formula can seems to be the usual thing. For many control points you have to inspect the formula and do the monkey see, monkey do thing. http://mathworld.wolfram.com/BernsteinPolynomial.html For Audacity the pairs, (t, bt) would be pushed into list, and used in a pwl function. bt is just one line of calculations but; would be rather long, if there are many control pointsand you are using a plugin method. A flexible program where the user specifies the number of control points is not without some effort, though it is hardly intractable. Sub b1() bt = 0# t = 0 p1 = 1#: p2 = 3#: p3 = 0#: p4 = 1# For i = 1 To 11 bt = (1  t) ^ 3 * p1 + 3 * (1  t) ^ 2 * t * p2 + 3 * (1  t) * t ^ 2 * p3 + t ^ 3 * p4 Cells(i, 1) = t Cells(i, 2) = bt t = t + 0.1 Next i End Sub On Wed, 28 Jul 2010 14:38:31 0400, "Roger Dannenberg" <rbd@...> said: > Sorry I didn't respond to this while it was fresh. I filed this request > for smooth envelopes away to possibly implement or at least think more > about. From mathworld.wolfram.com, " Undesirable properties of Bézier > curves are their numerical instability for large numbers of control > points, and the fact that moving a single control point changes the > global shape of the curve. The former is sometimes avoided by smoothly > patching together loworder Bézier curves." But patching together > loworder curves also creates discontinuities at least in higher > derivatives, and I'm not too clear on the bandlimit properties of > Bézier curves. Furthermore, it's relatively expensive to implement > highorder polynomials as functions of time without a primitive to do > it. I think a better idea is to use piecewise linear functions as > provided in Nyquist. This has been a standard approach in computer music > and audio signal processing for decades. If the discontinuities worry > you, then you can run the pwl function through a lowpass filter using > any of the Nyquist primitives, which are fast, efficient, and give you > simple control over how much smoothing is done. One particular case > where pwl functions are a problem is for quick onsets. If you want to > smoothly turn on a signal without a click or pop, the usual technique is > a "raised cosine", or (1  cos(t))/2, which generates a sort of > "S"curve from 0 to 1. In my experience, this is noticeably better than > a simple linear ramp. You can find an implementation for RAISEDCOSINE > and examples of its use in nyquist/demos/fft_tutorial.htm. > > Roger >  paul beach sniffyraven@... 