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4th order bezier bows

From: Rune Zedeler
Subject: 4th order bezier bows
Date: Fri, 22 Feb 2002 16:56:12 +0100

The slurs of lilypond are ugly - they are not "tense" enough.
I have done some simple by-hand-testing and found out that 4th order
bezier bows will be enough to do nice curves.
To draw a curve from a to b, one needs two extra points, c1 and c2, to
specify a 3rd order (normal) bezier bow, and 3 points, f1, f2 and f3, to
specify a 4th order bezier bow.
To get a 4th order bezier bow with the exact same curvature as a given
3rd order bow, on uses  f1 = .75[a,c1] ; f2 = .5[c1,c2] ; f3 = .25[c2,b]
My idea is to keep this definition for f1 and f3 and then to control the
tension of the curve by moving f2 along the line from .5[a,b] (very
tense) to .5[c1,c2] (no tension). Imho .5[f1,f3] usually produces
something very nice compared to 3rd order bezier bows.
I know no postscript at all, and I couldn't find where in the sources
you define the postscript commands for drawing 3rd order bezier bows.

Please comment.

Attached a small slammed java-program in which you can play around with
bezier bows. Save the file as, compile it and run it with
$java Bezier <n> where <n> is the order.


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