2 bezier.cc -- implement Bezier and Bezier_bow
4 source file of the GNU LilyPond music typesetter
6 (c) 1998--2000 Jan Nieuwenhuizen <janneke@gnu.org>
11 #include "bezier-bow.hh"
15 static Real F0_1 (Real x)
17 return M_PI /2 * atan(2 * x / M_PI);
21 slur_height (Real width, Real h_inf, Real r_0)
23 return F0_1 (width * r_0 / h_inf) * h_inf;
27 For small w, the height should be proportional to w, for w ->
28 infinity, the height should rise to a limit asymptotically.
30 Hence we take F(x) such that
31 F(0) = 0 , F'(0) = 1, and F(infty) = 1
35 h = h_infinity * F(x * r_0 / h_infinity)
40 * F(x) = pi/2 * atan (2x/pi)
42 * F(x) 1/alpha * x^alpha / (1 + x^alpha)
46 [with the 2nd recipe you can determine how quickly the conversion from
47 `small' slurs to `big' slurs occurs.]
49 Although this might seem cand_idates to SCM-ify, it is not all clear
50 which parameters (ie. h_inf, r_0, F(.)) should be candidates for
51 this. At present h_inf and r_0 come from paper settings, but we did
52 no experiments for determining the best combinations of F, h_inf and
58 slur_shape (Real width, Real h_inf, Real r_0)
61 Real height = slur_height (width, h_inf, r_0);
64 curve.control_[0] = Offset (0, 0);
65 curve.control_[1] = Offset (indent, height);
66 curve.control_[2] = Offset (width - indent, height);
67 curve.control_[3] = Offset (width, 0);