lily/bezier-bow.cc

   1 /*
   2   bezier.cc -- implement Bezier and Bezier_bow
   3
   4   source file of the GNU LilyPond music typesetter
   5
   6   (c) 1998--2009 Jan Nieuwenhuizen <janneke@gnu.org>
   7 */
   8
   9 #include "misc.hh"
  10 #include "bezier.hh"
  11
  12 static Real
  13 F0_1 (Real x)
  14 {
  15   return 2 / M_PI *atan (M_PI *x / 2);
  16 }
  17
  18 Real
  19 slur_height (Real width, Real h_inf, Real r_0)
  20 {
  21   return F0_1 (width * r_0 / h_inf) * h_inf;
  22 }
  23
  24 /*
  25   ^                x                    x
  26   |
  27   height   <indent>
  28   |
  29   v      x                                       x
  30
  31
  32
  33   For small w, the height should be proportional to w, for w ->
  34   infinity, the height should rise to a limit asymptotically.
  35
  36   Hence we take F (x) such that
  37   F (0) = 0 , F' (0) = 1, and F (infty) = 1
  38
  39   and use
  40
  41   h = h_infinity * F (x * r_0 / h_infinity)
  42
  43
  44   Examples:
  45
  46   * F (x) = 2/pi * atan (pi x/2)
  47
  48   * F (x) = 1/alpha * x^alpha / (1 + x^alpha)
  49
  50   * (etc.)
  51
  52   [with the 2nd recipe you can determine how quickly the conversion from
  53   `small' slurs to `big' slurs occurs.]
  54
  55   Although this might seem cand_idates to SCM-ify, it is not all clear
  56   which parameters (ie. h_inf, r_0, F (.)) should be candidates for
  57   this.  At present h_inf and r_0 come from layout settings, but we did
  58   no experiments for determining the best combinations of F, h_inf and
  59   r_0.
  60
  61
  62   The indent is proportional to the height of the slur for small
  63   slurs.  For large slurs, this gives a certain hookiness at the end,
  64   so we increase the indent.
  65
  66   indent = G (w)
  67
  68   w -> 0,  G (w) -> .33 w
  69
  70
  71   (due to derivative constraints, we cannot have indent > len/3)
  72
  73   w -> inf, G (w) -> 2*h_inf
  74
  75   i.e.
  76
  77
  78   G (0) = 0 , G'(0) 1/3, G (infty) = 2h_inf
  79
  80   solve from
  81
  82   G (w) = r  + p/(w+q)
  83
  84   yields
  85
  86   G (w) = 2 h_inf - max_fraction * q^2/ (w + q)
  87
  88   with q = 2 h_inf
  89 */
  90
  91 void
  92 get_slur_indent_height (Real *indent, Real *height,
  93                         Real width, Real h_inf, Real r_0)
  94 {
  95   Real max_fraction = 1.0 / 3.1;
  96   *height = slur_height (width, h_inf, r_0);
  97
  98   Real q = 2 * h_inf / max_fraction;
  99   *indent = 2 * h_inf - sqr (q) * max_fraction / (width + q);
 100 }
 101
 102 Bezier
 103 slur_shape (Real width, Real h_inf, Real r_0)
 104 {
 105   Real indent;
 106   Real height;
 107
 108   get_slur_indent_height (&indent, &height,
 109                           width, h_inf, r_0);
 110
 111   Bezier curve;
 112   curve.control_[0] = Offset (0, 0);
 113   curve.control_[1] = Offset (indent, height);
 114   curve.control_[2] = Offset (width - indent, height);
 115   curve.control_[3] = Offset (width, 0);
 116   return curve;
 117 }