lily/bezier-bow.cc

   1 /*
   2   This file is part of LilyPond, the GNU music typesetter.
   3
   4   Copyright (C) 1998--2011 Jan Nieuwenhuizen <janneke@gnu.org>
   5
   6   LilyPond is free software: you can redistribute it and/or modify
   7   it under the terms of the GNU General Public License as published by
   8   the Free Software Foundation, either version 3 of the License, or
   9   (at your option) any later version.
  10
  11   LilyPond is distributed in the hope that it will be useful,
  12   but WITHOUT ANY WARRANTY; without even the implied warranty of
  13   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  14   GNU General Public License for more details.
  15
  16   You should have received a copy of the GNU General Public License
  17   along with LilyPond.  If not, see <http://www.gnu.org/licenses/>.
  18 */
  19
  20 #include "misc.hh"
  21 #include "bezier.hh"
  22
  23 static Real
  24 F0_1 (Real x)
  25 {
  26   return 2 / M_PI *atan (M_PI *x / 2);
  27 }
  28
  29 Real
  30 slur_height (Real width, Real h_inf, Real r_0)
  31 {
  32   return F0_1 (width * r_0 / h_inf) * h_inf;
  33 }
  34
  35 /*
  36   ^                x                    x
  37   |
  38   height   <indent>
  39   |
  40   v      x                                       x
  41
  42
  43
  44   For small w, the height should be proportional to w, for w ->
  45   infinity, the height should rise to a limit asymptotically.
  46
  47   Hence we take F (x) such that
  48   F (0) = 0 , F' (0) = 1, and F (infty) = 1
  49
  50   and use
  51
  52   h = h_infinity * F (x * r_0 / h_infinity)
  53
  54
  55   Examples:
  56
  57   * F (x) = 2/pi * atan (pi x/2)
  58
  59   * F (x) = 1/alpha * x^alpha / (1 + x^alpha)
  60
  61   * (etc.)
  62
  63   [with the 2nd recipe you can determine how quickly the conversion from
  64   `small' slurs to `big' slurs occurs.]
  65
  66   Although this might seem cand_idates to SCM-ify, it is not all clear
  67   which parameters (ie. h_inf, r_0, F (.)) should be candidates for
  68   this.  At present h_inf and r_0 come from layout settings, but we did
  69   no experiments for determining the best combinations of F, h_inf and
  70   r_0.
  71
  72
  73   The indent is proportional to the height of the slur for small
  74   slurs.  For large slurs, this gives a certain hookiness at the end,
  75   so we increase the indent.
  76
  77   indent = G (w)
  78
  79   w -> 0,  G (w) -> .33 w
  80
  81
  82   (due to derivative constraints, we cannot have indent > len/3)
  83
  84   w -> inf, G (w) -> 2*h_inf
  85
  86   i.e.
  87
  88
  89   G (0) = 0 , G'(0) 1/3, G (infty) = 2h_inf
  90
  91   solve from
  92
  93   G (w) = r  + p/(w+q)
  94
  95   yields
  96
  97   G (w) = 2 h_inf - max_fraction * q^2/ (w + q)
  98
  99   with q = 2 h_inf
 100 */
 101
 102 void
 103 get_slur_indent_height (Real *indent, Real *height,
 104                         Real width, Real h_inf, Real r_0)
 105 {
 106   Real max_fraction = 1.0 / 3.1;
 107   *height = slur_height (width, h_inf, r_0);
 108
 109   Real q = 2 * h_inf / max_fraction;
 110   *indent = 2 * h_inf - sqr (q) * max_fraction / (width + q);
 111 }
 112
 113 Bezier
 114 slur_shape (Real width, Real h_inf, Real r_0)
 115 {
 116   Real indent;
 117   Real height;
 118
 119   get_slur_indent_height (&indent, &height,
 120                           width, h_inf, r_0);
 121
 122   Bezier curve;
 123   curve.control_[0] = Offset (0, 0);
 124   curve.control_[1] = Offset (indent, height);
 125   curve.control_[2] = Offset (width - indent, height);
 126   curve.control_[3] = Offset (width, 0);
 127   return curve;
 128 }